Matthew A Hawkins
email: matthewallen.hawkins@esade.edu

Market identification to generation: a practice theory market orientation

Keywords: practice theory, marketing, branding, market orientation, narrative

whether they are published or not. The documents may come   L'archive ouverte pluridisciplinaire

Market identification to generation: a practice theory market orientation Introduction

The new millennium has ushered in an era of connectedness. Understandings of human behavior are shifting from being completely based on biological explanations to those that embrace a socially constituted, networked view of society [START_REF] Bruner | Acts of meaning[END_REF]. Meaning is not an objective fact rather meaning and society are constituted through engaging in practice, observing, interpreting and doing [START_REF] Giddens | The constitution of society: Outline of the theory of structuration[END_REF]. Therefore society is continuously socially constituted [START_REF] Searle | Construction of social reality[END_REF].

Accordingly, the meaning of discourse, a human movement, an object and even a brand rests on how it is used to fulfill a purpose. An individual's practice narrative provides the conscious and unconscious rationales that motivate and guide the way they enactment the practice [START_REF] Hawkins | The omnipresent personal narrative: Story formulation and the interplay among narratives[END_REF]. However, repeated and regularly enacted practices produce structures, norms and memory traces on how to enact specific practices across individuals [START_REF] Giddens | The constitution of society: Outline of the theory of structuration[END_REF][START_REF] Schatzki | On organizations as they happen[END_REF][START_REF] Sewell | A theory of structure: Duality, agency, and transformation[END_REF].

Marketers are now being urged to recognize that the meaning of their brand is based on how it is actually used and not how they desire the offering to be used. The rise of co-creation branding (i.e., [START_REF] Payne | Managing the co-creation of value[END_REF], for instance, recognizes that consumer interaction and brand use create meaning and value. Consumer Culture Theory (CCT) similarly embraces the notion that dynamic relationships exist among resources, consumers and brand meaning [START_REF] Arnould | Consumer culture theory (CCT): Twenty years of research[END_REF], 2007). Despite advancements in brand meaning theory, the role of practice in market constitution has yet to be incorporated into a multi-perspective market orientation. Because the consumer has a significant impact on brand meaning researchers have been focused on improving a firm's market reading and learning capabilities [START_REF] Cayla | Ethnographic stories for market learning[END_REF][START_REF] Neill | Developing the organization's sensemaking capability: Precursor to an adaptive strategic marketing response[END_REF][START_REF] Sinkula | Market information processing and organizational learning[END_REF]. However, market orientation approaches typically focus on learning about existing consumer practices and not generating new consumer practices. Conversely, design-driven innovation research is concerned with radical innovation that changes the way consumers think and act and less concerned with inserting products into existing practices [START_REF] Verganti | Design driven innovation: Changing the rules of competition by radically innovating what things mean[END_REF].

Building on Storbacka andNenonen's (2011a, 2011b) view of markets as dynamic, evolving resource configuration systems, the purpose of this article is to formalize a resourcemarket orientation based on practice theory principles. This perspective argues firms provide resources for other consumers to enroll into their socially situated practices; and, recognizes that a firm's market scripting activities can affect users' practices and routines. Under the resourcemarket perspective there are four main market orientation models: resource identifier, resource activator, resource configurer and resource generator. The main features of this perspective are: only when the brand is resourced is the actual market constituted; a brand is resourced across numerous practices of each user or network member; and, firms have the ability to create new practices and routines.

The following section explains the practice-based theory of resourcing as it provides the foundation for the resource-market perspective. Then, the four practice theory derived market orientations are conceptually developed.

Theoretical foundation: practice-based theory of resourcing

To develop the resource-market perspective, practice-based theory of resourcing is consulted to explain how a brand or resource enables individuals to enact practices. Practice refers to the act of enrolling resources to accomplish a goal in a specific time and space [START_REF] Jarzabkowski | Toward a theory of coordinating: Creating coordinating mechanisms in practice[END_REF][START_REF] Levina | Understanding shifting power relations within and across organizations: A critical genre analysis[END_REF]. Practice theory argues that the meaning of discourse, actions, and objects are (re)constituted through doing. Therefore, social reality is not stable but continuously (re)constituted. Accordingly, to understand the meaning of any resource it cannot be isolated and abstracted from its usage context [START_REF] Brown | Organizational learning and communities-of-practice: Toward a unified view of working, learning, and innovation[END_REF][START_REF] Lave | The practice of learning[END_REF][START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF]. [START_REF] Polanyi | Personal knowledge: Towards a post-critical philosophy[END_REF] echoes the importance of context by arguing that interpreting objects, actions and discourse "are intelligible only within the framework of a useful performance which they successfully serve" (p. 185). Therefore, practice theory provides a strong theoretical foundation for contemporary market approaches by recognizing that the resource system the offering is integrated into determines its meaning [START_REF] Hatch | Toward a theory of brand co-creation with implications for brand governance[END_REF][START_REF] Payne | Co-creating brands: diagnosing and designing the relationship experience[END_REF] and by embracing the idea that resource systems can exhibit relative stability allowing for market identification.

Resources are considered to be any object, action, or discourse that an agent expects to assist them in enacting a practice [START_REF] Feldman | Resources in emerging structures and processes of change[END_REF][START_REF] Feldman | Resources, resourcing, and ampliative cycles in organizations[END_REF][START_REF] Sewell | A theory of structure: Duality, agency, and transformation[END_REF]. As a resource, individuals rely on their narratives or schemas when deciding to resource a brand into a practice. Narratives are an individuals sensemaking cognitive framework that gives meaning to objects, brands and practices [START_REF] Hawkins | The omnipresent personal narrative: Story formulation and the interplay among narratives[END_REF].

It is helpful to distinguish between abstracted and contextualized brand meanings as there is a significant body of literature that relies on brand perceptions abstracted from context.

Accordingly, brand-as-artifact refers to brand perceptions that are decontextualized and are not socially situated [START_REF] Orlikowski | Using technology and constituting structures: A practice lens for studying technology in organizations[END_REF]. It is through repeated, structuralized interactions that produce similar results that viewing brands and markets as stable artifacts makes sense.

Conversely, brand-in-practice refers to brand meanings stemming from the brand being resourced in a particular context [START_REF] Orlikowski | Using technology and constituting structures: A practice lens for studying technology in organizations[END_REF]. Methodologically, surveys that assess brand meaning using contextually devoid indicators (e.g., brand personality [START_REF] Aaker | Dimensions of brand personality[END_REF]) investigate brand-as-artifact meanings while ethnographic and cultural studies are better positioned to capture brand-in-practice meanings.

Resourcing corollary

As brands are only considered a resource when in-practice, the resourcing corollary to practice theory explains how brands are enrolled into practices. Under the resourcing corollary, a resource is created when it is enrolled into a practice, before integration it was only a potential resource [START_REF] Feldman | Theorizing practice and practicing theory[END_REF][START_REF] Feldman | Resources, resourcing, and ampliative cycles in organizations[END_REF]. In relation to branding, a brand is not defined by its innate qualities but by how the brand is (not) used to accomplish a goal in-situ. Accordingly, marketers attempt to have their offering viewed as a potential resource for practice enactment. The value placed on practice then brings the dynamic nature of brand meaning to the fore. An individual's reality is structured through repeated actions resulting in a structured resource system that can lead to relatively stable in-practice meanings [START_REF] Giddens | The constitution of society: Outline of the theory of structuration[END_REF]. However, using brands-as-artifact methods to assess brand meanings fail to recognize the contextually-based nature of brand-in-practice meanings and can result in overlooking market changes. Accordingly, marketing perspectives that internalize practice's role in (re)constituting resources are better suited to not only develop a more refined understanding of a particular brand's meaning but, as to be argued, they can also facilitate generating new practices and markets that resource the brand.

Resource-market perspective

A forward-looking managerial perspective of the market should account for traditional marketing orientations that focus on identifying markets as well as contemporary orientations that strive to create new cultural meanings, practices and markets. As Storbacka andNenonen (2011a, 2011b) argue, markets that exhibit relative stability can be said to have high marketness. At the other end of the continuum, markets exhibiting low marketness are resource systems still being contested.

Accordingly, approaches to the market should explicitly recognize the degree of marketness of the system they are contributing to.

The four market orientations under a resource-market perspective are: 1) resource identifier (RI), 2) resource activator (RA), 3) resource configurer (RC), and; 4) resource generator (RG) (see Table 1). Multiple views of marketing activities are necessary to capture the dynamism of the market, since as soon as other firms begin to target new practices promoted under a market generating approach their marketing orientations are either identifying, activating or configuring.

- ------------------------------Insert Table 1

about here -------------------------------

Resource identifier

The first resource-market approach is the resource identifier and is a commonly advocated branding strategy. As [START_REF] Mccracken | Culture and consumption: A theoretical account of the structure and movement of the cultural meaning of consumer goods[END_REF] and [START_REF] Escalas | Self-construal, reference groups, and brand meaning[END_REF] argue, this is where culturally relevant attributes are embedded into goods and brands through advertising and opinion leaders. In particular, RI strategies attempt to identify and insert their brand into the socially constituted world in a manner that allows it to be seen as a resource to enroll into identifiable, previously enacted practices. An RI strategy requires a mature market with high marketness that has at least one practice repeatedly enacted by a relatively large number of actors to target.

As practices are repeated through time and space, the resource system exhibits stability thereby increasing the attractiveness and applicability of the RI orientation. A brand can obtain strong market differentiation by owning or associating with relevant resource(s). Implementing brand-as-artifact research methods, such as surveys can be useful under RI. This research can then be used to identify and assess brand attributes to develop market differentiation strategies.

The strategic challenges, besides marketing mix development and implementation, are resisting organizational inertia and believing the market is stable [START_REF] Hannan | The population ecology of organizations[END_REF][START_REF] Giddens | The constitution of society: Outline of the theory of structuration[END_REF].

Resource activator

Market strategies that pursue differentiation within the practice's dominate resource configuration but provide branded resources outside actors' current decision making sets are approaching the market under a resource activator (RA) orientation. This strategy is particularly attractive for new market entrants or firms with limited resources who want to avoid directly competing with brand(s) that own market attributes by juxtaposing their brand against dominant discourses or stories presented by market leaders. But, as the activated market grows in size it becomes easier and more attractive for other firms to identify. Firms with multiple product lines could experiment with the RA market approach to grow market share or decrease costs through economies of scale.

The challenge presenting managers under an RA orientation is correctly reading culture to avoid promoting a branded resource that is too far from the dominate practice narrative.

Adjusting the way actors configure resources requires clout (Storbacka & Nenonen, 2011a[START_REF] Storbacka | Scripting markets: From value propositions to market propositions[END_REF] and the more novel the promoted configuration the more challenging it will be to adjust users' narratives and practices. Research methods that not only gauge where culture is but where culture is going are vital in this orientation. Projective research methods that ask respondents to visualize their behavior in particular situations can be helpful. Work on consumption collectives highlight that understanding brand-in-practice meanings offers various benefits to firms [START_REF] Mcalexander | Building brand community[END_REF] including identifying future trends [START_REF] Von Hippel | Democratizing innovation[END_REF]. Therefore, qualitative and cultural research on consumption collectives can help guide market scripting activities.

Resource configurer

Rather than working within a practice's dominant resource configuration narrative, a resource configurer (RC) orientation promotes alternative practice resource configuration narrative when approaching the market. Marketers then attempt to strategically embed these attributes into their brand in hopes of being viewed as a potential resource. This allows dynamic firms to overtake competing firms suffering from organizational inertia and less likely to adjust marketing strategies to meet the newly configured market.

Constructing and promoting novel resource configurations involve considerable risk and uncertainty. Users can be reluctant to view their practice in the way promoted by the firm's marketing mix. As [START_REF] Feldman | Resources in emerging structures and processes of change[END_REF] shows the resulting change in the constitution of one practice will change how resources in other practices are viewed. A resource can be, and is often, resourced across numerous overlapping practices; the ripple effect from a resource change in one practice could change more important practices thereby preventing (or promoting) brand adoption. Thus, brands that are resourced across multiple practices can be challenging to replace.

The resource configurer market orientations can be viewed as precursors to the RI and RA models. This is because only one firm can promote a new practice; other marketers are then working within the newly promoted resource configuration narrative. Conducting contextually rich research enables managers to strategically develop marketing mixes based on new resourcing stories that are unique, attractive, and easy for actors to integrate into their practice narrative. Qualitative research, such as interviews and focus groups are relatively inexpensive ways to learn brand-in-practice meanings. However, observation-and participant-based research methods allow for social and tacit knowledge to be gathered.

Resource generator

Resource generator (RG) strategies focus on developing and promoting new practices and providing the necessary resource(s) for enacting the practice. RG orientations embrace the firm's ability to co-construct new practices through the promotion of new narratives. Similarly, agency enables individuals to overcome the suggestive power of structures to create new ways of performing [START_REF] Reckwitz | Toward a theory of social practices: A development in culturalist theorizing[END_REF]. The constitution of new practices and markets relying upon branded objects offer strategists the opportunity to tightly intertwine their offered resource(s) into the practice narrative. This is similar to studying lead-users since they may be generating new practices or re-configuring existing practices using resources that could be produced by a firm [START_REF] Von Hippel | Lead users: A source of novel product concepts[END_REF]. As the practice is (re)invented [START_REF] Shove | Consumers, producers and practices: Understanding the invention and reinvention of Nordic walking[END_REF] through time additional practices can become interwoven into this new practice potentially allowing the brand to become a core resource across multiple practices or within a performance. This point further supports the notion that brands enrolled across multiple practices are difficult to replace and can elevate brand loyalty and repurchasing rates.

Creating a practice that is replicable at a collective level and across time and contexts is a challenging endeavor. The practice needs to be accepted by others and this involves considerable organizational learning that demands firm investments of thought, time, and money into supporting the new practice during a period of low marketness. [START_REF] Verganti | Design driven innovation: Changing the rules of competition by radically innovating what things mean[END_REF] discusses the extraordinary efforts Italian design firms go through to promote their design-driven innovations.

Designers spend years conducting cultural research (archival and ethnographic) to understand where culture is going. Then, they expend considerable resources (time, money) to promote their new view, such as targeting opinion leaders and trend setters [START_REF] Verganti | Design driven innovation: Changing the rules of competition by radically innovating what things mean[END_REF]. Furthermore, market players who delay entrance can benefit from learning; both learning from the failures of the market generator and from consumers learning how to use the product that might increase future acceptance. Overall, holistic research [START_REF] Patton | Qualitative research & evaluation methods[END_REF] strategies should be executed in order to develop a contextually rich understanding of where culture is going. This includes participantbased research, as gaining legitimacy within a consumption collective can be leveraged to encourage practice adoption and brand use [START_REF] Canniford | How to manage consumer tribes[END_REF].

Discussion

The resource-market perspective has three main features. First, while marketers desire to identify or generate the meanings and markets they would like to work within, it is only when the brand is resourced is the actual market constituted. Taking a brand-as-artifact approach or developing a brand image solely through surveys do not provide significant insight into in-use meaning which stems from how a brand is actually integrated into practices.

Second, stories and narratives from all relevant stakeholders should be gathered to assess in-use meaning across all supply chain members (Storbacka & Nenonen, 2011a). Creating a clear picture of brand-in-practice meanings needs to include the views of all supply chain members, which the resource-market perspective lends itself to. Therefore, collecting and analyzing the stories from buyers, regulators, stockholders, employees and other major brand network members will provide a more complete view of a brand's-in-practice meanings than just focusing on end-users, for instance. marketing mixes that avoid direct confrontation with market leaders. Accordingly, brand-asartifact meanings tend to be more valuable to brands operating with RI and RA orientations while brands pursuing RC and RG strategies value brand-in-practice meanings. It is recommended that firms select positioning strategies based upon market research capabilities.

Firms that are still developing interpretive research competencies should consider RI and RA strategies. Conversely, firms with active ethnographic research departments could consider pursuing RC or RG strategies. Conducting field research has an additional benefit of providing firms with an opportunity to legitimately, but peripherally participate in relevant consumption collectives [START_REF] Brown | Knowledge and organization: A social-practice perspective[END_REF][START_REF] Lave | Situated learning: Legitimate peripheral participation[END_REF]. Once community legitimacy is earned, the firm can exert influence in guiding shared interests and practices towards areas the firm can exploit to generate revenue or to shield the community from branded resources offered by competing firms.

The ability of a brand to serve as a resource across multiple practices is an important concept to internalize. Understanding that brands are resourced to support the enactment of multiple, overlapping practices can help managers implement changes that minimize impacts across practices thereby improving adoption rates [START_REF] Feldman | Resources in emerging structures and processes of change[END_REF]. This finding deserves further attention as Storbacka andNenonen (2011a, 2011b) argue competitive advantages stem from non-redundant network relationships while the resource-market perspective suggests that redundant relationships among resources not only creates the market but also offers the opportunity to firmly and deeply embed offerings into a brand users' practices.

Conclusion

The resource-market perspective builds upon Storbacka andNenonen's (2011a, 2011b) view of markets as dynamic and evolving resource configuration systems to offer managers and researchers a framework to view brand meaning and to develop marketing mixes that explicitly embraces the prominent role of the usage context in market formation [START_REF] Vargo | It's all B2B…and beyond: Toward a systems perspective of the market[END_REF]. It is argued that despite the socially constituted nature of markets, the structuralization provided by repeated, collective action leads to relative market stability. This stability allows marketers and entrepreneurs to identify markets. However, firms are able to configure new ways to view the market or to generate new markets through the promotion of new practices and routines.

Acknowledgements

The author would like to thank Polina Landgraf, Russ Nelson, Tanusree Jain and Núria Nadal for their thoughtful feedback during the development and writing processes. Additionally, a version of this paper was presented at the 3 rd Annual GIKA Conference and received valuable feedback from the participants. Portions of this paper were written while visiting the University of Arkansas-Fayetteville

Author

Third, generating a new practice and market are difficult and other companies can start promoting marketing mixes under RI, RA or RC orientations to position themselves within the emerging market. Accordingly, firms need a certain level of market power or clout to continue to influence market configurations, especially within low marketness contexts. The promotion of resourcing stories: expressed frameworks on how practice should enroll a particular brand can help with marketing scripting. A resourcing story has a different objective than developing a unified core brand value in stakeholders (de Chernatony and Harris, 2000; de Chernatony and Riley, 1998) through storytelling. Core brand values are static attributes incorporated into stakeholders' knowledge set, such as viewing the brand as trust-worthy or innovative. I argue managers should strive to develop a shared narrative that includes what the brand does and how the brand should be resourced into practices. This would addresses how the brand fits into a larger assemblage of resources in a continuously transitioning social system and is not a value or singular attribute. Brands needs to be seen as a resources that does something-not is something-in overlapping systems of practices and performances.

Theoretical contributions

Theoretically, the conclusion that a brand can serves as a core resource offers a novel explanation for increased brand loyalty. As repeated actions structuralize the environment [START_REF] Giddens | The constitution of society: Outline of the theory of structuration[END_REF], brands that can integrate themselves into multiple practices should be better able to survive in competitive markets than brands that are not. And, brand value increases as the in-practice knowledge becomes utilizable across multiple practices and routines [START_REF] Boisot | Knowledge assets: Securing competitive advantage in the information economy[END_REF].

This suggests that brands utilized across multiple practices will have higher repurchasing rates and levels of brand loyalty compared to brands that are not enrolled across multiple practices.

Managerial recommendations

The four market orientations of the resource-market perspective are particularly valuable for strategists and marketing managers. Brands that are repeatedly enrolled in practices and enjoy relatively high marketness need to be aware that the market is not stable. Therefore, by analyzing competing brands firms can offer guidance on how to strategically (re)position a brand so that it continues to maintain a strong presence within a dense but always adjusting collection of individual practices. While, new market entrants can analyze practice narratives to develop


Nicolas Jacon
email: njacon@univ-fcomte.fr

Cédric Lecouvey
email: cedric.lecouvey@lmpa.univ-littoral.fr

Crystal isomorphisms for irreducible highest weight U v ( sl e )-modules of higher level

We study the crystal graphs of irreducible U v ( sl e )-modules of higher level ℓ. Generalizing results of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally label these graphs: the Uglov multipartitions. By works of Ariki, Grojnowski and Lascoux-Leclerc-Thibon, it is then known that these bijections permit also to link the distinct parametrizations of the simple modules in modular representation theory of Ariki-Koike algebras. Our main tool is to make explicit an embedding of the U v ( sl e )-crystals of level ℓ into U v (sl ∞ )-crystals associated to highest weight modules.

Introduction

Let U v ( sl e ) be the affine quantum group of type A

(1) e-1 and {Λ i | i ∈ Z/eZ} its set of fundamental weights. Consider ℓ ∈ N and s = (s 0 , ...., s ℓ-1 ) ∈ (Z/eZ) ℓ . Denote by V e (Λ s ) the irreducible U v ( sl e )-module of highest weight Λ s = ℓ-1 i=0 Λ s i . The general theory of Kashiwara provides a crystal basis and a global basis for V e (Λ s ). The crystal basis of V e (Λ s ) comes equipped with the crystal graph B e (Λ s ) which encodes much information on the module structure. There are different possible realizations of V e (Λ s ) depending on the choice of a representative s = (s 0 , ..., s ℓ-1 ) ∈ Z ℓ of the class s ∈ (Z/eZ) ℓ . Indeed, to s is associated a Fock space F s e which provides an explicit construction of V e (Λ s ). We will denote it V s e (Λ s ) and write B s e (Λ s ) for the corresponding crystal graph. The crystals B s e (Λ s ) with s ∈ s are thus all isomorphic to the abstract crystal B e (Λ s ).

The purpose of the paper is to make explicit the isomorphisms between the crystals B s e (Λ s ) when s runs over s. By the works of Ariki [START_REF] Ariki | Representations of Quantum algebras and combinatorics of Young tableaux[END_REF], Grojnowski [START_REF] Grojnowski | Representations of affine Hecke algebras (and affine quantum GL n ) at roots of unity[END_REF] and Lascoux-Leclerc-Thibon [START_REF] Lascoux | Hecke algebras at roots of unity and crystal bases of quantum affine algebras[END_REF], an important application of these isomorphisms is to provide bijections between the distinct parametrizations of the simple modules in modular representation theory of Ariki-Koike algebras. To be more precise, let η be a primitive e th -root of unity. Write H = H(η; η s 0 , ..., η s ℓ-1 ) for the Ariki-Koike algebra defined over an algebraically closed field F of characteristic 0. This algebra is generated by T 0 , • • • , T n-1 subject to the relations (T 0 -η s 0 )...(T 0 -η s ℓ-1 ) = 0, (T i -η)(T i +1) = 0, for 1 ≤ i ≤ n and the type B braid relations

(T 0 T 1 ) 2 = (T 1 T 0 ) 2 , T i T i+1 T i = T i+1 T i T i+1 (1 ≤ i < n), T i T j = T j T i (j ≥ i + 2).
The algebra H is not semisimple in general, and, by a deep Theorem of Ariki, its representation theory is intimately connected to the global bases of the irreducible U v ( sl e )-modules. In particular, the simple modules of H are labelled by the vertices of any crystal B s e (Λ s ) such that s ∈ s. In fact the Fock space F s e admits a crystal basis indexed by multipartitions of length ℓ. This implies that the vertices of the crystal B s e (Λ s ) can be identified with certain multipartitions of length ℓ which are called the Uglov multipartitions. Note that when ℓ = 2, remarkable results of Geck [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF] show that these multipartitions also naturally appear in the context of Kazhdan-Lusztig theory and cellular structure of Hecke algebras. Other connections between Uglov multipartitions and modular representation theory of Ariki-Koike algebras are also known to hold for ℓ > 2 [START_REF] Jacon | Crystal graphs of higher level q-deformed Fock spaces, Lusztig a-values and Ariki-Koike algebras[END_REF]. Although nice properties have been given in particular cases (see [START_REF] Ariki | On the tensor product of two basic representations of U v ( sl e )[END_REF] and [START_REF] Ariki | Dipper-James-Murphy's conjecture for Hecke algebras of type B[END_REF]), the combinatorics of the Uglov multipartitions and their associated crystal graphs are not really well understood. For example, we do not even have a non recursive characterization of the Uglov multipartitions for any s ∈ s.

This paper extends the results obtained in [START_REF] Jacon | Crystal graphs of irreducible U v ( sl e )-modules of level two and Uglov bipartitions[END_REF] for ℓ = 2 by giving a combinatorial description of the isomorphisms between the crystals B s e (Λ s ). Nevertheless, the ideas we use are quite different. Indeed, we show that the crystals B s e (Λ s ) can be embedded in crystals corresponding to irreducible highest weight U v (sl ∞ )-modules. This result allows us to prove that most of the crystal isomorphisms in type A [START_REF] Ariki | On the classification of semi-simple modules for cyclotomic Hecke algebras in type G(m, 1, n) and Kleshchev multipartitions[END_REF] e-1 can be derived from crystal isomorphisms in type A ∞ . Now the combinatorial description of the isomorphisms of A ∞ -crystals is very close to that of the isomorphisms of A r -crystals in finite rank r. This permits us to use some elegant results of Nakayashiki and Yamada [START_REF] Nakayashiki | Kostka-Foulkes polynomials and energy functions in solvable lattice models[END_REF] on combinatorial R-matrices in type A r . One of the advantages of this new method is to avoid cumbersome case by case verifications unavoidable in [START_REF] Jacon | Crystal graphs of higher level q-deformed Fock spaces, Lusztig a-values and Ariki-Koike algebras[END_REF]. We would like to mention also that there is another way to realize the abstract crystals B e (Λ s ) by using Fock spaces of level ℓ which are tensor products of Fock spaces of level 1. The crystal B a e (Λ s ) so obtained notably appears in the works by Ariki (see [START_REF] Ariki | Representations of Quantum algebras and combinatorics of Young tableaux[END_REF]). The vertices of B a e (Λ s ) are parametrized by multipartitions called the "Kleshchev multipartitions" [START_REF] Kleshchev | On the decompositions numbers and branching coefficients for symmetric and special linear groups[END_REF]. The crystals B s e (Λ s ) and B a e (Λ s ) do not coincide in general. In particular, our method does not permit one to embed B a e (Λ s ) in a crystal of type A ∞ (but see the remark after Theorem 4.2.2).

The present paper is organized as follows. The second section is devoted to the combinatorial description of certain isomorphisms of U v (sl ∞ )-crystals. In section 3, we recall basic results on U v ( sl e )-crystals. By using two natural parametrizations of the Dynkin diagram in type A

(1) e-1 , we link in particular the two usual presentations of the crystals B s e (Λ s ) which appear in the literature. We then show in Section 4 that the crystals B s e (Λ s ) can be embedded in crystals corresponding to irreducible highest weight U v (sl ∞ )-modules. This embedding allows us to give in Section 5 a description of the isomorphisms between the crystals B s e (Λ s ) and to obtain another characterization for the sets of Uglov multipartitions. This characterization does not necessitate an induction on the sum of the parts of the Uglov multipartitions contrary to the original one [START_REF] Uglov | Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF].

Crystals in type A ∞

In this section, we study crystal isomorphisms in type A ∞ .

Background on

U v (sl ∞ )
Let sl ∞ be the Lie algebra associated to the doubly infinite Dynkin diagram in type A ∞ (see [START_REF] Kac | Bombay Lectures on highest weight representations of infinite dimensional Lie algebras[END_REF] and [START_REF] Kac | Infinite Dimensional Lie Algebras[END_REF]).

A ∞ : • • • - -m • - 1-m • -• • • - -1 • - 0 • - 1 • -• • • - m-1 • - m • -• • •. (1)
We denote by U v (sl ∞ ) the corresponding quantum group and we write ω i , i ∈ Z for the fundamental weights of the corresponding root system. We associate to the sequence s = (s 0 , ..., s ℓ-1 ) ∈ Z ℓ the dominant weight ω s = ℓ-1 i=0 ω s i . Then the irreducible highest weight U v (sl ∞ )-modules are parametrized by the sequences s of arbitrary length ℓ. We denote by V ∞ (ω s ) the irreducible U v (sl ∞ )-module of highest weight ω s . The module V ∞ (ω s ) admits a crystal basis. We refer the reader to [START_REF] Hong | Introduction to quantum groups and crystals bases[END_REF] for a complete review on crystal bases. We write B(ω s ) for the crystal graph corresponding to V ∞ (ω s ). When s = (s 0 ) we write for short V ∞ (ω s 0 ) and B ∞ (ω s 0 ) instead of V (ω s ) and B(ω s ).

In addition to the irreducible highest weight modules V ∞ (ω s ), it will be convenient to consider also irreducible modules V ∞ (k) indexed by nonnegative integers which are not of highest weight. By a column of height k, we mean a column shaped Young diagram

C = x 1 • • x k (2)
of height k filled by integers

x i ∈ Z such that x 1 > • • • > x k . When a ∈ C
i , f i , k i , i ∈ Z of U v (sl ∞ ) are given by f i (v C ) = 0 if δ i (C) = 0 or δ i+1 (C) = 1 v C ′ with C ′ = C \ {i} ∪ {i + 1} otherwise , e i (v C ) = 0 if δ i (C) = 1 or δ i+1 (C) = 0 v C ′ with C ′ = C \ {i + 1} ∪ {i} otherwise , k i (v C ) = v δ i (C)-δ i+1 (C) v C .
where for any i ∈ Z,

δ i (C) = 1 if i ∈ C and δ i (C) = 0 otherwise.
U v (sl a,b ) is isomorphic to the quantum group U v (sl d ) with d = b -a + 1. The subspace V a,b (k) of V ∞ (k) generated by the basis vectors v C such that C contains only letters x i with a ≤ x i ≤ b has the structure of a U v (sl a,b )-module. Moreover V a,b (k) is isomorphic to the k th fundamental module of U v (sl a,b ).
We follow the convention of [START_REF] Kashiwara | Decomposition of q-deformed Fock spaces[END_REF] and consider U v (sl ∞ ) as a Hopf algebra with coproduct given by

∆(k i ) = k i ⊗ k i , ∆(e i ) = e i ⊗ k i + 1 ⊗ e i and ∆(f i ) = f i ⊗ 1 + k -1 i ⊗ f i .
Given M 1 and M 2 two U v (sl ∞ )-modules with crystal graphs B 1 and B 2 , the crystal graph structure on B 1 ⊗ B 2 is then given by

f i (u ⊗ v) = f i (u) ⊗ v if ϕ i (v) ≤ ε i (u) u ⊗ f i (v) if ϕ i (v) > ε i (u) , (3)
e i (u ⊗ v) = u ⊗ e i (v) if ϕ i (v) ≥ ε i (u) e i (u) ⊗ v ifϕ i (v) < ε i (u) . (4)
Note that this convention is the reverse of that used in many references on crystals bases (see for example [START_REF] Hong | Introduction to quantum groups and crystals bases[END_REF]) but it is the natural one for working with multipartitions.

The crystals graphs of irreducible U v (sl ∞ )-modules

For any s ∈ Z, the U v (sl ∞ )-module V ∞ (ω s ) can be obtained as an irreducible component of the Fock space F ∞ (ω s ) defined by considering semi-infinite wedge products of V ∞ (1). The crystal B ∞ (ω s ) is identified with the graph whose vertices are the infinite columns

C = x 1 • • x k s -k + 1 s -k • • •
that is, the infinite columns shaped Young diagrams filled by decreasing integers x i from top to bottom and such that

x k = s -k + 1 for k sufficiently large. Given C 1 and C 2 in B ∞ (ω s ), we have an arrow C 1 i → C 2 if and only if i ∈ C 1 , i + 1 / ∈ C 2 and C 2 = C 1 -{i} + {i + 1}. The highest weight vertex of B ∞ (ω s ) is the column C (s) such that x k = s -k + 1 for all k ≥ 1. We associate to the U v (sl ∞ )-module V ∞ (k) a graph B ∞ (k).
Its vertices are the columns of height k (see [START_REF] Ariki | Representations of Quantum algebras and combinatorics of Young tableaux[END_REF]) and we draw an oriented arrow

C 1 i → C 2 if and only if i ∈ C 1 , i + 1 / ∈ C 2 and C 2 = C 1 -{i} + {i + 1}. The graph B ∞ (k) can be regarded as the crystal graph of V ∞ (k).
In fact, one can define a notion of crystal basis for the module V ∞ (k) despite the fact it is not of highest weight. This can be done essentially by considering the direct limit of the directed system formed by the crystal bases of the U v (sl a,b )-modules V a,b (k) [START_REF] Lecouvey | Crystal bases and combinatorics of infinite rank quantum groups[END_REF]. The corresponding crystal graph then coincide with B ∞ (k). In the sequel we will only need the crystal B ∞ (k) and not the whole crystal basis of B ∞ (k). Consider an infinite column C ∈ B ∞ (ω s ) with letters x i , i ≥ 1. We write π a (C) for the finite column obtained by deleting the infinite sequence of letters x i such that x i < a in C. Note that the height of π a (C) depends then of the integer a chosen. Let s = (s 0 , ..., s ℓ-1 ) ∈ Z ℓ . The abstract crystal B ∞ (ω s ) is then isomorphic to the connected component

B s ∞ (ω s ) of B ∞ (s) = B ∞ (ω s 0 ) ⊗ • • • ⊗ B ∞ (ω s ℓ-1 ) with highest weight vertex b (s) = C (s 0 ) ⊗ • • • ⊗ C (s ℓ-1 ) . Consider b = C 0 ⊗ • • • ⊗ C ℓ-1 ∈ B s ∞ (ω s ) with C k ∈ B ∞ (ω s k ), k = 0, ..., ℓ -1.
For any integer a and any k = 0, ..., ℓ -1, let h k be the height of the finite column π a (C k ). We set

π a (b) = π a (C 0 ) ⊗ • • • ⊗ π a (C ℓ-1 ) ∈ B ∞ (h 0 ) ⊗ • • • ⊗ B ∞ (h ℓ-1 ).
Consider b 1 and b 2 two vertices of B s ∞ (ω s ). Let K be any path between b 1 and b 2 in B s ∞ (ω s ), that is any sequence of crystal operators such that b 1 = K(b 2 ). Lemma 2.2.1 With the previous notation, for any integer a ≥ 1 sufficiently large, we have

π a (b 1 ) ∈ B ∞ (h 0 ) ⊗ • • • ⊗ B ∞ (h ℓ-1 ) and π a (b 2 ) ∈ B ∞ (h 0 ) ⊗ • • • ⊗ B ∞ (h ℓ-1 ). In this case π a (b 1 ) = K(π a (b 2 )) in B ∞ (h 0 ) ⊗ • • • ⊗ B ∞ (h ℓ-1 ).
Proof. We can choose a sufficiently large so that the infinite columns appearing in every vertex b of the path joining b 1 to b 2 contain all the letters x < a. Then, for each k = 0, ..., ℓ -1, the height of the column obtained by deleting the letters x < a in the k-th column of b does not depend on b. Set h k = s k + 1a. The deleted letters x < a do not interfere during the computation of the crystal operators defining the path between b 1 and b 2 . This implies that the path joining π a (b 1 ) to π a (b 2 ) obtained by applying the same sequence K of crystal operators also exists in

B ∞ (h 0 ) ⊗ • • • ⊗ B ∞ (h ℓ-1 ). Thus π a (b 1 ) = K(π a (b 2 )).
The vertices of B ∞ (ω s ) are also naturally labelled by partitions. Recall that a partition λ = (λ 1 , ..., λ p ) of length p is a weakly decreasing sequence λ 1 ≥ • • • ≥ λ p of nonnegative integers called the parts of λ. In the sequel we identify the partitions having the same nonzero parts and denote by P the set of all partitions. It is convenient to represent λ by its Young diagram. In the sequel we use the French convention for the Young diagram (see example below). To any pair (a, b) of integers, we associate the node γ = (a, b) which is the box obtained after a north moves and b east moves starting from the south-west box of λ. Observe that the node γ does not necessarily belong to λ. We say that γ is removable when γ = (a, b) ∈ λ and λ\{γ} is a partition. Similarly γ is said addable when γ = (a, b) / ∈ λ and λ ∪ {γ} is a partition. We associate to the infinite column C ∈B ∞ (ω s ) with letters x k the partition λ such that λ k = x ks + k -1. Since λ k = 0 for k sufficiently large, this permits us to label the vertices of B ∞ (ω s ) by

P. Let γ = (a, b) be a node for λ ∈ B ∞ (ω s ). The content of γ is c(γ) = b -a + s. (5)
In B ∞ (ω s ), we have an arrow λ i → µ if and only if µ/λ = γ where γ is addable in λ with c(γ) = i. For any partitions λ = (λ 1 , ..., λ p ) we set |λ| = λ 1 + • • • + λ p . Then |λ| is equal to the length of the directed path joining ∅ to λ in the crystal B ∞ (ω s ).

Example 2.2.2

To C =

7 6 4 1 1 2 • • • ∈ B ∞ (ω 3 ) corresponds λ = R A R A R A . We have f 4 (C) = 7 6 5 1 1 2 • • • or equivalently f 4 (λ) = R A R A for the node γ = (3, 4) is addable in λ with content c(γ) = 4 -3 + 3 = 4.
Here we have written k instead of -k for any positive integer k.

Since B s ∞ (ω s ) is labelled by partitions, the crystal B ∞ (s) is labelled by multipartitions λ = (λ (0) , ..., λ (ℓ-1) ). More precisely, to each vertex b = C 0 ⊗•••⊗C ℓ-1 ∈ B ∞ (s) such that C k ∈ B(ω s k ) for k = 0, ..., ℓ -1, corresponds the multipartition λ = (λ (0) , ..., λ (ℓ-1)
) where λ (k) is obtained from C k as described above. By a part of the multipartition λ, we mean a part of one of the partitions λ (k) . The action of the operators e i and f i on b are deduced from ( 3) and ( 4). One verifies easily that they can be also obtained by applying the following algorithm. Let w i be the word obtained by considering the letters i and i + 1 successively in the columns C 0 , C 1 , ..., C ℓ-1 of b when these columns are read from top to bottom and left to right. The word w i is called the i-signature of b. Encode in w i each integer i by a symbol + and each integer i + 1 by a symbol -. Choose any factor of consecutive -+ and delete it. Repeat this procedure until no factor -+ can be deleted. The final sequence w i is uniquely determined and has the form w i = + pq . It is called the reduced i signature of b. Then f i (b) is obtained from b by replacing the integer i corresponding to the rightmost symbol + in w i by i + 1. Similarly, e i (b) is obtained from λ by replacing the integer i + 1 corresponding to the leftmost symbolin w i by i. One can describe similarly the action of e i and f i on b considered as the multipartition λ = (λ (0) , ..., λ (ℓ-1) ). This time, one considers the word W i obtained by reading the addable and removable nodes of λ with content i successively in the partitions λ (0) , ..., λ (ℓ-1) . This reading is well defined since a partition cannot both content addable and removable nodes with content i. Encode in W i each removable node by R and each addable node by A. Let W i be the word obtained by deleting the factors RA successively in the encoding. One can write

W i = A p R q . ( 6
)
Then f i (b) is obtained from λ by adding the node γ corresponding to the rightmost symbol A in W i .

The crystal isomorphism between B

∞ (ω k ) ⊗ B ∞ (ω l ) and B ∞ (ω l ) ⊗ B ∞ (ω k )
Consider k and l two integers. We denote by ψ k,l the crystal graph isomorphism

ψ k,l : B ∞ (ω k ) ⊗ B ∞ (ω l ) ≃ → B ∞ (ω l ) ⊗ B ∞ (ω k ). (7)
To give the explicit combinatorial description of ψ k,l , we start by considering the crystal graph isomorphism

θ k,l between B ∞ (k)⊗B ∞ (l) and B ∞ (l)⊗B ∞ (k). Consider C 1 ⊗C 2 in B ∞ (k)⊗B ∞ (l).
We are going to associate to

C 1 ⊗ C 2 a vertex C ′ 2 ⊗ C ′ 1 ∈ B ∞ (l) ⊗ B ∞ (k). Suppose first k ≥ l. Consider x 1 = min{t ∈ C 2 }. We associate to x 1 the integer y 1 ∈ C 1 such that y 1 = max{z ∈ C 1 | z ≤ x 1 } if min{z ∈ C 1 } ≤ x max{z ∈ C 1 } otherwise . ( 8
)
We repeat the same procedure to the columns C 1 \ {y 1 } and C 2 \ {x 1 }. By induction this yields a sequence {y 1 , ..., y l } ⊂ C 1 . Then we define C ′ 2 as the column obtained by reordering the integers of {y 1 , ..., y l } and C ′ 1 as the column obtained by reordering the integers of

C 1 \ {y 1 , ..., y l } + C 2 . Now, suppose k < l. Consider x 1 = min{t ∈ C 1 }. We associate to x 1 the integer y 1 ∈ C 2 such that y 1 = min{z ∈ C 2 | x 1 ≤ z} if max{z ∈ C 2 } ≥ x min{z ∈ C 2 } otherwise . ( 9
)
We repeat the same procedure to the columns C 1 \ {x 1 } and C 2 \ {y 1 } and obtain a sequence {y 1 , ..., y k } ⊂ C 2 . Then we define C ′ 1 as the column obtained by reordering the integers of {y 1 , ..., y k } and C ′ 2 as the column obtained by reordering the integers of

C 2 \ {y 1 , ..., y l } + C 1 . We denote by θ k,l the map defined from B ∞ (k) ⊗ B ∞ (l) to B ∞ (l) ⊗ B ∞ (k) by setting θ k,l (C 1 ⊗ C 2 ) = C ′ 2 ⊗ C ′ 1 . Example 2.3.1 Consider C 1 = 9
. Thus θ 6,5 (C 1 ⊗ C 2 ) = C ′ 2 ⊗ C ′ 1 .
One can also easily verifies that

θ 5,6 (C ′ 2 ⊗ C ′ 1 ) = C 1 ⊗ C 2 . Proposition 2.3.2 The map θ k,l is an isomorphism of U v (sl ∞ )-crystals that is, for any integer i and any vertex C 1 ⊗ C 2 ∈ B ∞ (k) ⊗ B ∞ (l), we have θ k,l • e i (C 1 ⊗ C 2 ) = e i (C ′ 2 ⊗ C ′ 1 ) and θ k,l • f i (C 1 ⊗ C 2 ) = f i (C ′ 2 ⊗ C ′ 1 ). ( 10
)
Proof. Choose a < b two integers such that the letters of C 1 and C 2 belong to [a, b]. It follows from Proposition 3.21 of [START_REF] Nakayashiki | Kostka-Foulkes polynomials and energy functions in solvable lattice models[END_REF], that the isomorphism between the finite U v (sl a-1,b+2 )crystals B a-1,b+2 (k) ⊗ B a-1,b+2 (l) and B a-1,b+2 (l) ⊗ B a-1,b+2 (k) is given by the map θ k,l . Since the actions of the crystal operators e i and f i with i ∈ [a -1, b + 2[ on the crystals B a-1,b+2 (l) and B a-1,b+2 (k)) can be obtained from the crystal structures of B ∞ (k) and B ∞ (l), this implies the commutation relations [START_REF] Hong | Introduction to quantum groups and crystals bases[END_REF]

for any i ∈ [a -1, b + 2[. Now if i / ∈ [a -1, b + 2[ we have e i (C 1 ⊗ C 2 ) = f i (C 1 ⊗ C 2 ) = 0 because the letters of C 1 and C 2 belong to [a, b]. Similarly e i (C ′ 2 ⊗ C ′ 1 ) = C ′ 2 ⊗ C ′ 1 = 0 because the letters of C 1 ∪ C 2 are the same as those of C ′ 1 ∪ C ′ 2 .
Hence (10) holds for any integer i.

Now consider C 1 ⊗ C 2 ∈ B ∞ (ω k ) ⊗ B ∞ (ω l ).
Let a be any integer such that C 1 and C 2 both contain all the integers x < a.

Set C 1 = π a (C 1 ), C 2 = π a (C 2 ) and θ k,l (C 1 ⊗ C 2 ) = C ′ 2 ⊗ C ′ 1 .
Since C 1 and C 2 both contain only letters x ≥ a, it is also the case for the columns C ′ 1 and C ′ 2 by the previous combinatorial description of the isomorphism θ k,l . Write C ′ 1 and C ′ 2 for the infinite columns obtained respectively from C ′ 1 and C ′ 2 by adding boxes containing all the letters x < a.

Then C ′ 1 ∈ B ∞ (ω k ) and C ′ 2 ∈ B ∞ (ω l ). Indeed C 1 ∈ B ∞ (ω k ), C 2 ∈ B ∞ (ω l ) and C 1 , C ′ 1 (resp. C 2 , C ′
2 ) have the same height.

Corollary 2.3.3 (of Proposition 2.3.2) For any C 1 ⊗ C 2 ∈ B ∞ (ω k ) ⊗ B ∞ (ω l ) we have with the above notation ψ k,l (C 1 ⊗ C 2 ) = C ′ 2 ⊗ C ′ 1 . Proof. Recall that C (k) ⊗ C (l) is the highest weight vertex of B ∞ (ω k ) ⊗ B ∞ (ω l ). Write C 1 ⊗ C 2 = F (C (k) ⊗ C (l) ) where F is a sequence of crystal operators f i , i ∈ Z. The crystal graph isomorphism ψ k,l must send the highest vertex of B ∞ (ω k ) ⊗ B ∞ (ω l ) on the highest weight vertex of B ∞ (ω l ) ⊗ B ∞ (ω k ). Thus, we have ψ k,l (C (k) ⊗ C (l) ) = C (l) ⊗ C (k) . In particular, the Corollary is true for C 1 ⊗ C 2 = C (k) ⊗ C (l) . Now consider C 1 ⊗ C 2 ∈ B ∞ (ω k ) ⊗ B ∞ (ω l
) and choose a an integer such that C 1 and C 2 both contain all the integers x < a. Set C (k) = π a (C (k) ) and

C (l) = π a (C (l) ). Similarly write C 1 = π a (C 1 ), C 2 = π a (C 2 ). Since C 1 ⊗ C 2 = F (C (k) ⊗ C (l) ) we obtain from Lemma 2.2.1 the equality C 1 ⊗ C 2 = F (C (k) ⊗ C (l)
). By definition of the crystal isomorphism θ k,l , we thus have

C ′ 2 ⊗ C ′ 1 = F (C (l) ⊗ C (k) ). The columns C (k) , C (l) , C ′ 1 and C ′ 2 contains only letters x ≥ a. Hence we can write F = f i 1 • • • f ir with i m ≥ a for any m ∈ {1, ..., r}. Now the infinite columns C (l) , C (k) , C ′ 1 and C ′
2 are respectively obtained from C (k) , C (l) , C ′ 1 and C ′ 2 by adding all the letters x < a. We can thus deduce from the equality

C ′ 2 ⊗ C ′ 1 = F (C (l) ⊗ C (k) ), the equality C ′ 2 ⊗ C ′ 1 = F (C (l) ⊗ C (k) ). This implies that ψ k,l (C 1 ⊗ C 2 ) = C ′ 2 ⊗ C ′ 1 . Example 2.3.4 Consider λ (1) = (5, 5, 5, 4, 4, 3) ∈ B ∞ (ω 4 ) and λ (2) = (4, 4, 4, 3, 2) ∈ B ∞ (ω 3 ). The columns C 1 , C 2 such that C 1 ⊗ C 2 = (λ (1) , λ (2) ) ∈ B ∞ (ω 4 ) ⊗ B ∞ (ω 3 ) are C 1 = 9 8 7 5 4 2 2 • • • and C 2 = 7 6 5 3 1 2 • • • . Hence C 1 = 9 8 7 5 4 2 and C 2 = 7 6 5 3 1 for a = 2.
We deduce from Example 2.3.1 that

C ′ 2 = 9 7 5 4 2 2 • • • and C ′ 1 = 8 7 6 5 3 1 2 • • • .
Thus we can write ψ 4,3 (λ (1) , λ (2) ) = (µ (2) , µ (1) ) with µ (1) = (4, 4, 4, 4, 3, 2) and µ (2) = (6, 5, 4, 4, 3).

Remark:

The columns C 1 and C 2 obtained in the previous algorithm depend on the integer a considered. Nevertheless, this is not the case for the resulting infinite columns C ′ 1 and C ′ 2 as long as a << 0.

Crystals in type A

(1) e-1

We now turn to the problem of studying the crystals in type A

(1) e-1 . In this section, we recall and show basic facts on their combinatorial descriptions.

Background on U v ( sl e )

In order to link the different labellings of the crystal graphs in type A

(1) e-1 appearing in the literature, we shall need the two following Dynkin diagrams

A (1),+ e-1 : 0 • 1 • -• • • - e-1 • and A (1),- e-1 : 0 • e-1 • -• • • - 1 • . ( 11
)
Let U + v ( sl e ) and U - v ( sl e ) be the affine quantum groups defined respectively from the root systems in type A

(1),+ e-1 and A

(1),e-1 (see for example [START_REF] Mathas | Iwahori-Hecke algebras and Schur algebras of the symmetric group[END_REF], Chapter 6). Observe that this notation, which is not related to the triangular decomposition ofU - v ( sl e ), only means we are considering two copies of U + v ( sl e ) with Dynkin diagrams A

(1),+ e-1 and A

(1),e-1 . Write {Λ + 0 , Λ + 1 , ..., Λ + e-1 } and {Λ - 0 , Λ - 1 , ..., Λ - e-1 } for the dominant weights of the root systems A

(1),+ e-1 and A

(1),e-1 . We have then Λ

+ k = Λ - e-k . Write {E + i , F + i , K + i | i = 0, ..., e -1} and {E - i , F - i , K - i | i = 0, ...,
   U + v ( sl e ) → U - v ( sl e ) v d + -→ v d- E + i -→ E - e-i , F + i -→ F - e-i and K + i -→ K - e-i (12)
is an isomorphism of algebras. We associate to s = (s 0 , ..., s ℓ-1 ) ∈ Z ℓ the dominants weights

Λ + s = ℓ-1 k=0 Λ + s k (mod e) and Λ - s = ℓ-1 k=0 Λ - s k (mod e) . Let V e (Λ + s ) (resp. V e (Λ - s
)) be the irreducible U + v ( sl e )-module (resp. U - v ( sl e )module) of highest weight Λ + s (resp. Λ - s ). These modules can be constructed by using the Fock space representation F s e of level ℓ. Let Π ℓ,n be the set of multipartitions λ = (λ (0) , ..., λ (ℓ-1) ) of length ℓ with rank n, i.e. such that λ (0) + • • • λ (ℓ-1) = n. The Fock space F s e is defined as the C(v)-vector space generated by the symbols | λ, s with λ ∈ Π ℓ,n

F s e = n≥0 λ∈Π ℓ,n C(v) | λ, s .
The Fock space F s e can be endowed with the structure of a U + v ( sl e )-module or equivalently with the structure of a U - v ( sl e )-module. These actions are defined by introducing total orders ≺ + s and ≺ - s on the i-nodes of the multipartitions. Consider λ = (λ (0) , ..., λ (ℓ-1) ) a multipartition and suppose s (called the multicharge) is fixed. Then the nodes of λ can be identified with the triplet γ = (a, b, c) where c ∈ {0, ..., ℓ -1} and a, b are respectively the row and column indices of the node γ in λ (c) . The content of γ is the integer c (γ) = ba + s c and the residue res(γ) of γ is the element of Z/eZ such that res(γ) ≡ c(γ)(mod e).

(

) 13
We will say that γ is an i-node of λ when res(γ) ≡ i(mod e). Let γ 1 = (a 1 , b 1 , c 1 ) and γ 2 = (a 2 , b 2 , c 2 ) be two i-nodes in λ. We define the total order ≺ + s and ≺ - s on the addable and removable i-nodes of λ by setting

γ 1 ≺ + s γ 2 ⇐⇒ c(γ 1 ) < c(γ 2 ) or c(γ 1 ) = c(γ 2 ) and c 1 < c 2 , ( 14
)
γ 1 ≺ - s γ 2 ⇐⇒ c(γ 1 ) < c(γ 2 ) or c(γ 1 ) = c(γ 2 ) and c 1 > c 2 . ( 15
)
Using these orders, it is possible to define an action of U + v ( sl e )-module and an action of U - v ( sl e )module on F s e . These modules will be denoted by F s,+ e and F s,e . For these actions the empty multipartition ∅ = (∅, ..., ∅) is a highest weight vector respectively of highest weight Λ + s and Λ - s .

We denote by V s e (Λ + s ) and V s e (Λ - s ) the irreducible components with highest weight vector ∅ in F s,+ e and F s,e . We refer the reader to [START_REF] Jimbo | Combinatorics of representations of U q ( sl(n)) at q = 0[END_REF] for a detailed exposition. Note that the modules V s e (Λ + s ) (resp. V s e (Λ - s )) such that s ∈ s are all isomorphic to the abstract module V e (Λ + s ) (resp. V e (Λ - s )). However, the corresponding actions of the Chevalley operators do not coincide in general. This will yield different parametrizations of the associated crystals.

Crystal structure on B s,+ e

Consider the set of addable and removable i-nodes of λ (see Section 2.2 for the definition of removable and addable nodes). Let w i be the word obtained by writing the addable and removable i-nodes of λ in decreasing order with respect to ≺ + s , next by encoding each addable i-node by the letter A and each removable i-node by the letter R. Write w i = A p R q for the word obtained from w i by deleting as many of the factors RA as possible. If p > 0, let γ be the rightmost addable i-node in w i . The node γ is called the good i-node. Then we have an arrow

λ i → µ in B s,+ e
if and only if µ is obtained from λ by adding the good i-node γ.

Crystal structure on B s,e

Consider the set of addable and removable i-nodes of λ. Let w i be the word obtained by writing these i-nodes in increasing order with respect to ≺ - s , next by encoding each addable i-node by the letter A and each removable i-node by the letter R. Write w i = A p R q for the word obtained from w i by deleting as many of the factors RA as possible. If p > 0, let γ be the rightmost addable i-node in w i . Then we have an arrow

λ i → µ in B s,- e
if and only if µ is obtained from λ by adding the good i-node γ.

Link between the crystals B s,+ e and B s,e

For any multicharge s = (s 0 , ..., s ℓ-1 ), we denote by s * the multicharge s * = (es 0 , ..., es ℓ-1 ). According to the isomorphism [START_REF] Jacon | Crystal graphs of higher level q-deformed Fock spaces, Lusztig a-values and Ariki-Koike algebras[END_REF], there should exist some bijections ν : B s,+ e → B s * ,e such that, given any two multipartitions λ, µ,

λ i → µ in B s,+ e ⇐⇒ ν(λ) e-i → ν(µ) in B s * ,- e . ( 16
)
Note that ν cannot be unique since each of the crystals B s,+ e and B s * ,e contains isomorphic connected components. To each multipartition λ = (λ (0) , ..., λ (ℓ-1) ) in B s,+ e , we associate its conjugate multipartition λ ′ = (λ ′(0) , ..., λ ′(ℓ-1) ) in B s * ,+ e where for any k = 0, ..., ℓ -1, λ ′(k) is the conjugate partition of λ (k) . Proposition 3.2.1 The map ξ : λ → λ ′ from F s,+ e to F s * ,e is a bijection and satisfies [START_REF] Kac | Infinite Dimensional Lie Algebras[END_REF].

Proof. Consider λ and λ ′ as vertices respectively of the crystals F s,+ e and F s * ,e

. Let γ = (a, b, c) be a node appearing at the right end of the a-th row of λ (c) in λ. One associates to γ the node γ ′ appearing on the top of the a-th column of λ ′(k) in λ ′ . Then by definition of the node γ ′ , we have γ ′ = (b, a, c). This gives for the content of the nodes γ and γ

′ c(γ) = b -a + s c and c(γ ′ ) = a -b + e -s c .
Now observe that γ is a removable (resp. addable) node if and only if γ ′ is a removable (resp. addable) node. We have thus

γ is an A (resp. R) i-node ⇐⇒ γ ′ is an A (resp. R) (e -i)-node. ( 17
)
Let w i be the word obtained by writing the addable or removable i-nodes of λ in decreasing with respect to ≺ + s as in Section 3.2.1. Similarly, let w ′ e-i be the word obtained by writing the addable or removable (ei)-nodes of λ ′ in increasing order with respect to ≺ + s as in Section 3.2.2. Write w i = γ 1 • • • γ r where for any m = 1, ..., r, γ m is an i-node that is addable or removable. Then by definition of the order ≺ + s and ≺ - s (see ( 14) and ( 15)), we have

w ′ e-i = γ ′ 1 • • • γ ′ r .
Write w i = A p R q for the word obtained from w i by the cancellation process of the pairs RA . We deduce from ( 17) that the word w ′ e-i = (A ′ ) p (R ′ ) q coincides with the word obtained from w ′ e-i by the cancellation process of the letters RA. In particular, γ is the good i-node for λ if and only if γ ′ is the good (ei)-node for λ ′ . Hence the bijection ξ satisfies (16). . Since ξ(∅) = ∅, we derive from the previous proposition the equivalence • The vertices of B s e (Λ + s ) are the e-restricted partitions, that is the partitions λ = (λ 1 , ..., λ p ) such that λ iλ i+1 ≤ e -1 for any i = 1, ..., p -1.

λ i → µ in B s e (Λ + s ) ⇐⇒ ξ(λ) e-i → ξ(µ) in B s * e (Λ - s * ). (18
• The vertices of B s e (Λ - s ) are the e-regular partitions, that is the partitions λ with at most e -1 parts equal.

Remarks:

(i) : The crystals B s e (Λ + s ) are essentially those used in [START_REF] Ariki | Representations of Quantum algebras and combinatorics of Young tableaux[END_REF] whereas the crystals B s e (Λ - s ) appear in [START_REF] Geck | Modular representations of Hecke algebras[END_REF] and [START_REF] Jacon | Crystal graphs of irreducible U v ( sl e )-modules of level two and Uglov bipartitions[END_REF]. (ii) : Consider s = (s 0 , ..., s ℓ-1 ) and s ′ = (s ′ 0 , ..., s ′ ℓ-1 ) two multicharges such that s k ≡ s ′ k for any k = 0, ..., ℓ -1. Then Λ + s = Λ + s ′ , thus the crystals B s e (Λ + s ) and B

s ′ e (Λ + s ) are isomorphic. The combinatorial description of this isomorphism is complicated in general (but see Section 5). Nevertheless, in the case when there exists an integer d such that s k = s ′ k + de for any k = 0, ..., ℓ -1, one derives easily from the description of the crystal structure on B s e (Λ + s ) that the relevant isomorphism coincide with the identity map. The situation is the same for the crystals B s e (Λ - s ) and B

s ′ e (Λ - s ).

When the multicharge s = (s 0 , ..., s ℓ-1 ) satisfies 0 1. λ is cylindric, that is, for every k = 0, ..., ℓ -2 we have λ

≤ s 0 ≤ • • • ≤ s ℓ-1 ≤ e -1,
(k) i ≥ λ (k+1)
i+s k+1 -s k for all i > 1 (the partitions are taken with an infinite numbers of empty parts) and λ

(ℓ-1) i ≥ λ (0) i+e+s 0 -s ℓ-1
for all i > 1 2. for all r > 0, among the residues appearing at the right ends of the length r rows of λ, at least one element of {0, 1, ..., e -1} does not appear.

Remarks:

(i) : By using [START_REF] Kashiwara | Decomposition of q-deformed Fock spaces[END_REF], it is easy to deduce a combinatorial characterization of the Uglov multipartitions appearing in B s e (Λ + s ) when 0 ≤ s ℓ-1 ≤ • • • ≤ s 0 ≤ e -1. These multipartitions are called the FLOTW multipartitions. (ii) : In the sequel we will essentially consider the crystals B s e (Λ + s ) because they can be embedded in a natural way in U v (sl ∞ )-crystals. Thanks to [START_REF] Kashiwara | Decomposition of q-deformed Fock spaces[END_REF], it will be easy to translate our statements to make them compatible with the crystals B s e (Λ - s ). (iii) : There exists also another realization B a e (Λ + s ) of the abstract crystal B e (Λ + s ) which is in particular used by Ariki [2]. This realization is obtained by defining the Fock space of level ℓ as a tensor product of Fock spaces of level 1. It is suited to the Specht modules theory for Ariki-Koike algebras introduced by Dipper, James and Mathas [START_REF] Dipper | Cyclotomic q-Schur Algebras[END_REF]. Note that in the level 2 case, Geck [START_REF] Geck | Hecke algebras of finite type are cellular[END_REF] has generalized this theory by defining Specht modules adapted to the definition of the Fock spaces used in this paper. It is expected that similar results hold in higher level. (iv) : The multipartitions which label the vertices of the crystals B a e (Λ + s ) are called the Kleshchev multipartitions. For any nonnegative integer n, the subgraph of the crystal B a e (Λ + s ) containing all the Kleshchev multipartitions of rank m ≤ n coincide with the corresponding subgraph of B s e (Λ + s ) when the multicharge s satisfies s i -s i+1 > n-1 for any i = 0, ..., ℓ-2. As a consequence Kleshchev multipartitions are particular cases of Uglov multipartitions.

Embedding of B

s e (Λ + s ) in B s ♦ ∞ (ω s )
The aim of this section is to show that we have an embedding of crystals from

B s e (Λ + s ) to B s ♦ ∞ (ω s ) with s ♦ = (s ℓ-1 , ..., s 0 ). 4.1 Embedding of B s e (Λ + s ) ≤n in B s f (Λ + s ) ≤n
In the sequel we denote by E i and F i , i = 0, ..., e -1 the crystal operators corresponding to the U + v ( sl e )-crystals. Consider λ and µ two multipartitions in F s,+ e and i ∈ {0, ..., e -1} such that F i (λ) = µ. Then µ is obtained by adding an i-node γ in λ. Set j = c(γ). In the sequel we slightly abuse the notation and write F j (λ) = µ although j does not belong to {0, ..., e -1} in general. Note that j is uniquely determined from the equality F i (λ) = µ. Observe that with this convention, an arrow in F s,+ e can be labelled by any integer. To recover the original labelling of a U + v ( sl e )-crystal, it suffices to read these labels modulo e. With this convention, when there exists an arrow from the multipartition λ to the multipartition µ in both F s,+ e and F

s,+ f

(with e = f ), this arrows can be pictured λ j → µ where j is the content of the node added to λ to obtain µ. In particular the label so obtained is independent of e and f and it makes sense to write F j (λ) = µ in B

= F j 1 • • • F jn (∅) in B s f (Λ + s ).
Proof. Since each crystal operator F j k adds a node on the multipartition

F j k+1 • • • F jn (∅), we have |λ| = n.
To prove the second assertion of our proposition, we proceed by induction on n. When n = 0, the proposition is immediate. Suppose our statement true for any multipartition λ with |λ| = n -1. Consider µ ∈ B s e (Λ + s ) with µ = n. With the above convention, there exist an integer j and a multipartition λ ∈ B s e (Λ + s ) such that F j (λ) = µ and µ is obtained by adding a node with content j to λ. Let i ∈ {0, ..., e -1} be such that i ≡ j(mod e). Write w e i for the word obtained by writing the addable or removable i-nodes of λ in decreasing order with respect to ≺ + s as in Section 3.2.1. Choose any integer f ≥ n + s . Observe that each j-node γ = (a, b, c) in λ verifies c(γ) = res(γ) when its residue is computed modulo f . Indeed, we have c(γ) = ba + s c and thus

-f < 1 -n + s c ≤ c(γ) ≤ n -1 + s c < f. (19)
Write w f j for the word obtained by writing the addable or removable j-nodes of λ in decreasing order with respect to ≺ + s . Then the nodes contributing to w f j are the addable and removable nodes of λ with content j. This implies that w f j is a subword of w e i , that is each node appearing in w f j also appears in w e i . If γ is a node in w e i which does not belong to w f j , we must thus have c(γ) = j. Since the nodes of w e i are ordered according to ≺ + s , we deduce that w f j is a factor of w e i (i.e., there is no node γ such that c(γ) = j between two nodes of w f j ). Write w e i and w f j respectively for the words obtained from w e i and w f j by the cancellation process of the factors RA. Since these words do not depend on the order of the consecutive factors RA which are deleted, w e j is a factor of w f i . Let γ 0 be the good i-node in w e i . Suppose that γ 0 is not a node contributing to w f j . Then, there exists a node δ 0 in w f j which can be paired with γ 0 to produce a factor RA. In this case, γ 0 and δ 0 are nodes of w e i because w f j is a factor of w e i . This gives a contradiction. Indeed, the word w e i does not depend on the cancellation process for the factors RA. So, if we can pair γ 0 and δ 0 together in order to obtain a factor RA in w f j , we can do the same pairing in w e i (for w f j is a factor of w e i ) and γ 0 cannot be the good i-node in w e i . Hence we have shown that γ 0 is a j-node of w f j . Since w f j is a factor of w e i , γ 0 is the rightmost addable j-node in w f j , thus is the good j-node for w f j . By the induction hypothesis, one has

λ = F j 1 • • • F jn (∅) in B s e (Λ + s ) and B s f (Λ + s ) because f ≥ n + s > n -1 + s . Moreover, we have F j (λ) = µ in B s e (Λ + s ) and B s f (Λ + s ) by the previous arguments. Thus µ = F j F j 1 • • • F jn (∅) in B s e (Λ + s ) and B s f (Λ + s ).
For any fixed nonnegative integer n, write

B s e (Λ + s ) ≤n = {λ ∈ B s e (Λ + s ) | |λ| ≤ n}.
We deduce from the above proposition, that the identity map ι e,f :

λ -→ λ from B s e (Λ + s ) ≤n to B s f (Λ + s ) ≤n is an embedding of crystals. 4.2 Embedding of B s e (Λ + s ) in B s ♦ ∞ (ω s )
For any multipartition λ = (λ (0) , ..., λ (ℓ-1) ) we write λ ♦ = (λ (ℓ-1) , ..., λ (0) ).

Proposition 4.2.1 Consider a multicharge s and f, n two nonnegative integers such that f ≥ n + s . Let λ be a multipartition in

B s f (Λ + s ) with |λ| = n. Suppose λ = F j 1 • • • F jn (∅) in B s f (Λ + s ). Then we have λ ♦ = f j 1 • • • f jn (∅) in the U v (sl ∞ )-crystal B s ♦ ∞ (ω s ).
Proof. We proceed by induction on n. When n = 0, the proposition is immediate. Suppose our statement true for any multipartition λ with |λ| = n. Consider µ ∈ B s f (Λ + s ) with µ = n. There exists an integer j and a multipartition λ ∈ B s e (Λ + s ) such that F j (λ) = µ and µ is obtained by adding a node with content j to λ. Since f ≥ n + s , we have res(γ) = c(γ) for each node γ in λ as in [START_REF] Lascoux | Hecke algebras at roots of unity and crystal bases of quantum affine algebras[END_REF]. Hence the word w f j obtained by writing the addable or removable j-nodes of λ in decreasing order with respect to ≺ + s contains exactly the addable and removable nodes of λ with content equal to j. Set λ = (λ (0) , ..., λ (ℓ-1) ). By definition of the order ≺ + s (see [START_REF] Jimbo | Combinatorics of representations of U q ( sl(n)) at q = 0[END_REF], this means that w f j is the word obtained by reading the addable and removable nodes with content equal to j successively in the partitions λ (ℓ-1) , λ (ℓ-2) , ..., λ (0) of λ. Moreover each partition λ (c) , c = 0, ..., ℓ -1 contains at most one addable or removable node with content j because the nodes with the same content in λ (c) must belong to the same diagonal. By the induction hypothesis, we know that λ

= F j 1 • • • F jn (∅) in B s f (Λ + s ) and λ ♦ = f j 1 • • • f jn (∅) in B s ♦ ∞ (ω s ).
The previous arguments show that the word w f j coincide with the word W j obtained by reading the addable and removable nodes of content j in λ ♦ as described in Section 2.2 just before [START_REF] Foda | Branching functions of A (1) n-1 and Jantzen-Seitz problem for Ariki-Koike algebras[END_REF]. Thus we

obtain µ = F j F j 1 • • • F jn (∅) in B s f (Λ + s ) and µ ♦ = f j f j 1 • • • f jn (∅) in B s ♦ ∞ (ω s ). Set B s ♦ ∞ (ω s ) ≤n = {λ ∈ B s ♦ ∞ (ω s ) | |λ| ≤ n}.
We deduce from the previous proposition that the map

ϕ (n) f,∞ : B s e (Λ + s ) ≤n → B s ♦ ∞ (ω s ) ≤n λ = (λ (0) , ..., λ (ℓ-1) ) -→ λ ♦ = λ (ℓ-1) ⊗ • • • ⊗ λ (0)
is an embedding of crystals for any f ≥ n + s .

B s e (Λ + s ) → B s ♦ ∞ (ω s ) λ = (λ (0) , ..., λ (ℓ-1) ) -→ λ ♦ = λ (ℓ-1) ⊗ • • • ⊗ λ (0)
is an embedding of crystals : for any λ ∈ B s e (Λ + s ) we have

λ = F j 1 • • • F jr (∅) in B s e (Λ + s ) =⇒ λ ♦ = f j 1 • • • f jr (∅) in B s ♦ ∞ (ω s ). Proof. Consider λ = (λ (0) , ..., λ (ℓ-1) ) ∈ B s e (Λ + s ) such that λ = F j 1 • • • F jr (∅) and set |λ| = n. By Proposition 4.1.1 we have λ = F j 1 • • • F jr (∅) in any crystal B s f (Λ + s ) with f ≥ n + s .
Fix such an integer f. We derive from Proposition 4.2.1 that

λ ♦ = f j 1 • • • f jr (∅) in B s ♦ ∞ (ω s ).
Clearly the map ϕ e,∞ is injective. Thus it is an embedding from the U v ( sl e )-crystal

B s e (Λ + s ) to the U v (sl ∞ )-crystal B s ♦ ∞ (ω s ).
Remark: According to the previous theorem, the crystal B s e (Λ + s ) can be embedded in B s ♦ ∞ (ω s ). The situation is more complicated for the crystal B a e (Λ + s ) labelled by Kleshchev multipartitions (see Remark (iv) after Proposition 3.2.4). Indeed, the subcrystal B a e (Λ + s ) ≤n can be embedded in a crystal B ∞ (ω s(n) ) where the multicharge s(n) = (s 0 (n), ..., s l-1 (n)) verifies s k (n)s k+1 (n) > n -1 for any k = 0, ..., l -2. Since s(n) depends on n, this procedure cannot provide an embedding of the whole crystal B a e (Λ + s ) in a crystal B ∞ (ω t ) where t is a fixed multicharge.

Isomorphism class of a multipartition

We can now use the above embedding to obtain a simple characterization of the set of Uglov multipartitions.

The extended affine symmetric group S ℓ

We write S ℓ for the extended affine symmetric group in type A ℓ-1 . The group S ℓ can be regarded as the group generated by the elements σ 1 , ..., σ ℓ-1 and y 0 , ...., y ℓ-1 together with the relations

σ i σ i+1 σ i = σ i+1 σ i σ i+1 , σ i σ j = σ j σ i for |i -j| > 1, σ 2 i = 1
with all indices in {1, ..., ℓ -1} and y i y j = y j y i , σ i y j = y j σ i for j = i, i + 1, σ i y i σ i = y i+1 .

We identify the subgroup of S ℓ generated by the transpositions σ i , i = 1, ..., ℓ -1 with the symmetric group S ℓ of rank ℓ. For any i ∈ {0, ..., ℓ -1}, we set

z i = y 1 • • • y i . Write also ξ = σ ℓ-1 • • • σ 1 and τ = y ℓ ξ. Since y i = z -1 i-1 z i , S
ℓ is generated by the transpositions σ i with i ∈ {1, ..., ℓ -1} and the elements z i with i ∈ {1, ..., ℓ}. Observe that for any i ∈ {1, ..., ℓ -1}, we have

z i = ξ ℓ-i τ i . ( 20
)
This implies that S ℓ is generated by the transpositions σ i with i ∈ {1, ..., ℓ -1} and τ. Consider e a fixed positive integer. We obtain a faithful action of S ℓ on Z ℓ by setting for any s = (s 0 , ..., s ℓ-1 ) ∈ Z ℓ σ i (s) = (s 0 , ..., s i , s i-1 , ..., s ℓ-1 ) and y i (s) = (s 0 , ..., s i-1 , s i + e, ..., s ℓ-1 ).

Then τ (s) = (s 1 , s 2 , ..., s ℓ-1 , s 0 + e). We denote by C(s) the orbit of the multicharge s under the action of S ℓ on Z ℓ . Clearly each class C(s) contains a unique multicharge s = ( s 0 , ..., s ℓ-1 ) such that 0

≤ s ℓ-1 ≤ • • • ≤ s 0 ≤ e -1. (21)
Hence the orbits C(s) are parametrized by the multicharges verifying [START_REF] Lecouvey | Crystal bases and combinatorics of infinite rank quantum groups[END_REF]. Given any multicharge s = (s 0 , ...., s ℓ-1 ) ∈ Z ℓ , it is easy to determinate w ∈ S ℓ such that s = w(s). To do this, we compute a sequence of multicharges as follows. Choose k ∈ N minimal to have s i + ke ≥ 0 for any i = 0, ..., ℓ -1. Then z k ℓ-1 (s) ∈ N ℓ . Consider σ ∈ S ℓ such that the coordinates of s (ℓ-1) = σz k ℓ-1 (s) weakly decrease. Write r ℓ-2 for the quotient of the division of s

(ℓ-2)
ℓ-2 by e and set s (ℓ-3) = z -r ℓ-2 ℓ-2 (s (ℓ-2) ). By induction one can compute a sequence s (ℓ-2) , ..., s (0) such that, for any i = 1, ..., ℓ -2, s (i-1) = z -r i i (s (i) ) where r i is the quotient of the division of s

(i) i -s (i) i+1
by e. We have then s = s (0) and ). According to Section 5.1, w decomposes as a product of the elements σ i i = 1, ..., ℓ -1 and τ. We write for short Ξ s and Σ s,i respectively for the crystal graph isomorphisms Γ s,τ (s) and Γ s,σ i (s) . The following proposition is a generalization of [START_REF] Jacon | Crystal graphs of irreducible U v ( sl e )-modules of level two and Uglov bipartitions[END_REF]Prop. 3.1]. Proposition 5.2.1 Consider λ = (λ (0) , ..., λ (ℓ-1) ) a multipartition and s a multicharge. Then Ξ s (λ) = (λ (1) , ..., λ (ℓ-1) , λ (0) ).

s = w(s) = z -r 0 0 • • • z -r ℓ-2 ℓ-2 σz k ℓ-1 (s). ( 22
Proof. Set s = (s 0 , ..., s ℓ-1 ). Then τ (s) = (s 1 , ...., s ℓ-1 , s 0 + e). Let λ = (λ (0) , ..., λ (ℓ-1) ) a multipartition and set λ # = (λ (1) , ..., λ (ℓ-1) , λ (0) ). Consider i ∈ {0, 1, ..., e -1} and γ

1 = (a 1 , b 1 , c 1 ), γ 2 = (a 2 , b 2 , c 2 ) two i-nodes of λ. Then γ # 1 = (a 1 , b 1 , c 1 -1(mod e)) and γ # 2 = (a 2 , b 2 , c 2 -1(mod e)) are two i-nodes of λ # . We then easily check that γ 2 ≺ + s γ 1 if and only if γ # 2 ≺ + τ (s) γ ′ 1 . This implies that Ξ s (λ) = λ # .
Consider λ = (λ (0) , ..., λ (ℓ-1) ) a multipartition and s a multicharge such that λ ∈ B s e (Λ + s ). Then we know that λ ♦ = (λ (ℓ-1) , ...λ

(0) ) ∈ B s ♦ ∞ (ω s ). Recall that, by definition of B s ♦ ∞ (ω s ), we can write λ ♦ = λ (ℓ-1) ⊗ • • • ⊗ λ (0) . For any integer i ∈ {0, ..., n -1}. Set ψ s i+1 ,s i (λ (i+1) ⊗ λ (i) ) = λ (i) ⊗ λ (i+1) (23)
where ψ s i+1 ,s i is the crystal graph isomorphism defined in [START_REF] Geck | Modular representations of Hecke algebras[END_REF].

Proposition 5.2.2 With the above notation, we have Σ s,i (λ) = (λ (0) , ..., λ (i+1) , λ (i) , ...λ (ℓ-1) ) that is Σ s,i (λ) is obtained by replacing in λ, λ (i) by λ (i+1) and λ (i+1) by λ (i) .

Proof. We have to prove that the diagram

B s e (Λ + s ) ϕe,∞ → B ∞ (ω s ℓ-1 ) ⊗ • • • ⊗ B ∞ (ω s i+1 ) ⊗ B ∞ (ω s i ) ⊗ • • • ⊗ B ∞ (ω s 0 ) Σ s,i ↓ ↓ ψ s i+1 ,s i B σ i (s) e (Λ + s ) ϕe,∞ → B ∞ (ω s ℓ-1 ) ⊗ • • • ⊗ B ∞ (ω s i ) ⊗ B ∞ (ω s i+1 ) ⊗ • • • ⊗ B ∞ (ω s 0 ) (24)
commutes. Consider a multipartition λ ∈ B s e (Λ + s ). Set λ = (λ (0) , ..., λ (ℓ-1) ). Let F i 1 , ..., F ir be a sequence of crystal operators such that λ

= F i 1 ••• F ir (∅). Then we can consider the multipartition µ = F i 1 , ..., F ir (∅) in the crystal B σ i (s) e (Λ + s ). Observe first that ψ s i+1 ,s i • ϕ e,∞ (∅) = ϕ e,∞ • Σ s,i (∅). (25)
Moreover, the maps ϕ e,∞ , Σ s,i and ψ s i+1 ,s i commute with the crystal operators. This permits us to write

ψ s i+1 ,s i • ϕ e,∞ (λ) = ψ s i+1 ,s i • ϕ e,∞ ( F i 1 • • • F ir (∅)) = f i 1 • • • f ir (ψ s i+1 ,s i • ϕ e,∞ (∅)).
One the other hand we have

ϕ e,∞ • Σ s,i (λ) = ϕ e,∞ • Σ s,i ( F i 1 • • • F ir (∅)) = f i 1 • • • f ir (ϕ e,∞ • Σ s,i (∅)).
Hence we derive the equality ψ s i+1 ,s i • ϕ e,∞ (λ) = ϕ e,∞ • Σ s,i (λ) from (25). This shows that the diagram (24) commutes and establish our proposition.

Example 5.2.3 Take ℓ = 3. Suppose s = (4, 0, 1) and λ = (λ (0) , λ (1) , λ (2) ) with λ (0) = (4, 3, 3, 2), λ

= (3, 3, 1) and λ (2) = (5, 3, 2). Let us compute Σ s,2 (λ). The infinite columns associated to λ (1) and λ (2) are respectively

C 1 = 3 2 1 3 4 • • • and C 2 = 6 3 1 2 3 4 • • • .
For a = 3 the corresponding finite columns are

C 1 = 3 2 1 3 and C 2 = 6 3 1 2 3 .
We have to determinate the image of C 2 ⊗ C 1 under the isomorphism θ 5,4 of Proposition 2.3.2. Note that the image of C 1 ⊗ C 2 under θ 4,5 is not relevant here because we must take into account the swap ⋄. We obtain {y 1 , y 2 , y 3 , y 4 } = {3, 1, 2, 3}. This gives θ 5,4 (C

2 ⊗ C 1 ) = C ′ 1 ⊗ C ′ 2 with C ′ 1 = 3 1 2 3 and C ′ 2 = 6 3 2 1 3 . Hence ψ 1,0 (C 2 ⊗ C 1 ) = C ′ 1 ⊗ C ′ 2 where C ′ 1 = 3 1 2 3 4 • • • and C ′ 2 = 6 3 2 1 3 4 • • • .
Finally we derive Σ s,2 (λ) = (λ (0) , λ (2) , λ (1) ) with λ (1) = (3, 2) and λ (2) = (5, 3, 3, 1).

Remark: Assume that λ = (λ (0) , λ (1) ) is a bipartition such that λ belongs to B s e (Λ + s ) where the multicharge s = (s 0 , s 1 ) verifies s 0 ≤ s 1 . Then the combinatorial procedure illustrated by the previous example which permits to compute the crystal isomorphisms Σ s,i , essentially reduces, up to a renormalization due to the change of labelling of the Dynkin diagram in type A

(1) e-1 (see [START_REF] Hong | Introduction to quantum groups and crystals bases[END_REF]), to the algorithm depicted in Theorem 4.6 of [START_REF] Jacon | Crystal graphs of irreducible U v ( sl e )-modules of level two and Uglov bipartitions[END_REF].

A non recursive characterization of the Uglov multipartitions

Consider a multicharge s and define the multicharge s as in [START_REF] Lecouvey | Crystal bases and combinatorics of infinite rank quantum groups[END_REF]. Then the crystals B We keep the notations of §5.1. For any i = 0, ..., ℓ -1, we have z i = ξ (ℓ-i) τ (i) . This permits to compute Γ s,z i (s) (λ) by using Propositions 5.2.1 and 5.2.2. We have then

I(λ) = Γ s,w(s) (λ)
with the notation [START_REF] Mathas | Iwahori-Hecke algebras and Schur algebras of the symmetric group[END_REF]. Conversely, given any FLOTW multipartition µ and any multicharge s, one can compute the multipartition λ ∈ B where w is obtained from s as in [START_REF] Mathas | Iwahori-Hecke algebras and Schur algebras of the symmetric group[END_REF].

Isomorphism class of a multipartition

Suppose that e is a fixed positive integer and s a multicharge of level ℓ. Consider λ a multipartition in B s e (Λ + s ). The isomorphism class of λ is the set

C(λ) = {Γ s,s ′ (λ) | s ′ ∈ C(s)}.
Thus C(λ) is the set of all multipartitions µ which appear at the same place as λ in a crystal

B

s ′ e (Λ + s ) where s ′ is a multicharge of the orbit of s under the action of S ℓ . Then C(λ) can be determined from λ by applying successive elementary transformations using Propositions 5.2.1 and 5.2.2. Observe that for any µ ∈ C(λ) we must have |λ| = µ . This implies in particular that C(λ) is finite. The cardinality of C(λ) is in general rather complicated to evaluate without computing the whole class C(λ). Nevertheless, we are going to see in Theorem 5.4.2, that it is possible to obtain an upper bound for card(C(λ)) and to determinate a finite subset S λ of S ℓ such that C(λ) = {Γ s,s ′ (λ) | s ′ ∈ S λ • s}.

Lemma 5.4.1 Let λ be a multipartition of rank n. Assume that s is a multicharge of level ℓ verifying : s js j+1 > n -1 (26)

for j = 0, ..., ℓ -2. Then for any k ∈ {0, ..., ℓ -1} we have Γ s,z k (s) (λ) = λ.

Proof. Let s = (s 0 , ..., s ℓ-1 ) be such that s js j+1 > n -1 for j = 0, ..., ℓ -2 and let µ be a multipartition in B are such that c(γ 1 ) > c(γ 2 ). Hence we have γ 2 ≺ + s γ 1 . Now, put (s ′ 0 , ..., s ′ l-1 ) := z k (s) = (s 0 +e, ...., s k +e, s k+1 , ..., s ℓ-1 ). As we have s ′ j -s ′ j+1 > n-1 for j = 0, ..., ℓ -2, the above discussion shows that the order ≺ + s and ≺ + z k (s) on the i-nodes of µ coincide.

  and b / ∈ C, we write C -{a} + {b} for the column obtained by replacing in C the letter a by the letter b. The module V ∞ (k) is defined as the vector space with basis B k = {v C | C is a column of height k}. The actions of the Chevalley generators e

Remark:

  Consider a, b two integers such that a < b. Denote by [a, b[ the set of integers i such that a ≤ i ≤ b -1. Write U v (sl a,b ) for the subalgebra of U v (sl ∞ ) generated by the Chevalley generators e i , f i , k i with i ∈ [a, b[. Then U v (sl a,b ) can be identified with the quantum group associated to the Dynkin diagram obtained by deleting the nodes i / ∈ [a, b[ in (1). In particular

.

  We obtain {y 1 , y 2 , y 3 , y 4 , y 5 } = {9, 2, 4, 5, 7}. This gives C

3. 2

 2 Crystal graph of the Fock space F s,integrable modules, thus by the general theory of Kashiwara, they admit crystal bases. Write B s,+ e and B s,efor the crystal graphs corresponding to the action of U + v ( sl e ) and U - v ( sl e ) on F s e . These crystals are labelled by multipartitions and their crystal graph structure can be explicitly described by using the total orders ≺ + s and ≺ - s . Consider two multipartitions λ, µ and an integer i ∈ {0, ..., e-1}. The crystal graph B highest weight vertex is the empty multipartition.

  s * ) are the irreducible components with highest weight vector ∅ in F s * ) can be realized as the connected components of highest weight vertex ∅ in B s,+ e and B s * ,e

) Definition 3 . 2 . 2 Example 3 . 2 . 3

 322323 The vertices of B s e (Λ + s ) and B s e (Λ - s ) are called the Uglov multipartitions. Suppose ℓ = 1 and s = (s).

  there exists a combinatorial description of the Uglov multipartitions labelling B s e (Λ - s ) due to Foda, Leclerc, Okado, Thibon and Welsh. Proposition 3.2.4 [6] [14] When 0 ≤ s 0 ≤ • • • ≤ s ℓ-1 ≤ e -1, the multipartition λ = (λ (0) , ..., λ (ℓ-1) ) belongs to B s e (Λ - s ) if and only if:

  s e (Λ + s ) and B s f (Λ + s ). For any multicharge s = (s 0 , ..., s ℓ-1 ) in Z ℓ , we set s = max{|s k | | k = 0, ..., ℓ -1} where |s k | = s k if s k is nonnegative and |s k | = -s k otherwise. The following proposition is a generalization in level ℓ of Proposition 4.1 in [13]. Proposition 4.1.1 Let n be a nonnegative integer. Consider λ a multipartition in B s e (Λ + s ) such that λ = F j 1 • • • F jn (∅). Then |λ| = n. Moreover, for any integer f ≥ n + s , we have λ

Theorem 4 . 2 . 2

 422 Given any positive integer e and any multicharge s the map ϕ e,∞ :

  s ) are isomorphic. For any multipartition λ ∈ B s e (Λ + s ), write I(λ) ∈ B s e (Λ + s ) for its image under this crystal isomorphism. It is possible to obtain I(λ) from λ by using results of §5.2.

Proposition 5 . 3 . 1

 531 s e (Λ + s ) such that I(λ) = µ. Indeed, we have then λ = Γ s,w -1 ( s) (µ). By remark (i) following Proposition 3.2.4, we thus derive a non recursive combinatorial description of the Uglov multipartitions labellingB + e (Λ + s ) n = {λ ∈ B s e (Λ + s ) | |λ| = n}. For any multicharge s B + e (Λ + s ) n = {Γ s,w -1 ( s) (µ) | µ ∈ B s e (Λ + s ) n }

  that µ ≤ n. Let i ∈ {0, 1, ..., e -1} and considerγ 1 = (a 1 , b 1 , c 1 ) and γ 2 = (a 2 , b 2 , c 2 ) two i-nodes in µ such that c 1 < c 2 . We have b 1a 1 + s c 1 -(b 2a 2 + s c 2 ) > b 1a 1 -(b 2a 2 ) + n -1 ≥ 0.Hence, the contains of γ 1 and γ 2 considered as nodes of µ ∈ B s,+ e

Theorem 5 . 4 . 2 - 2 ℓ- 2 • • • z r 0 0 .

 542220 Suppose that e is a fixed positive integer and s a multicharge of level ℓ such that0 ≤ s ℓ-1 ≤ • • • ≤ s 0 < eConsider λ a FLOTW multipartition in B s e (Λ + s ) of order n. For any j ∈ {0, ..., ℓ -2}, let p j be the minimal nonnegative integer such thats j + p j es j+1 > n -1.Then we have :C(λ) = {Γ s,z a 0 0 ...z a ℓ-2 ℓ-2 σ(s) (λ) |σ ∈ S ℓand 0 ≤ a j ≤ p j for any j ∈ {0, ..., ℓ -2}}. In particular, C(λ) is finite andcard(C(λ)) ≤ ℓ! ℓ-2 j=0 (p j + 1).Proof. Consider µ ∈ C(λ). According to[START_REF] Mathas | Iwahori-Hecke algebras and Schur algebras of the symmetric group[END_REF], one can writeµ = Γ s,w -1 (s) (λ) with w -1 = z -k ℓ-1 σz r ℓ-2 ℓ-2 • • • z r 0 0 .By Remark (ii) following Example 3.2.3, we have µ = Γ s,w -1 (s) (λ) = Γ s,u(s) (λ) where u = σz r ℓBy Lemma 5.4.1, we can thus deriveC(λ) = {Γ s,z a 0 0 ...z a ℓ-2ℓ-2 σ(s) (λ) |σ ∈ S ℓ and 0 ≤ a j ≤ p j for any j ∈ {0, ..., ℓ -2}} and our theorem follows.

  e -1} for the sets of Chevalley generators respectively in U + v ( sl e ) and U - v ( sl e ). Let v d + ∈ U + v ( sl e ) and v d -∈ U - v ( sl e ) be the quantum derivation operators. Then the map

ι :

)

  5.2 Action of the transformations s i and τ on a multipartitionConsider a multicharge s = (s 0 , ..., s ℓ-1 ) and w an element of the extended affine symmetric group. Set s ′ = w(s). Since the indices of the fundamental weights of U + v ( sl e ) belong to Z/eZ, we have Λ + s ′ = Λ + s . This implies that the crystals graphs B + s ), we are going to see how it is possible to determinate µ = Γ s,s ′ (λ) in a non-inductive way, that is, without computing a path joining λ to ∅ in B

	B	s e (Λ s e (Λ + s
				s e (Λ + s ) and B e (Λ + s ′ s ′ ) are isomorphic.
	Write Γ s,s ′ for the isomorphism between B	s e (Λ + s ) and B	s ′ e (Λ + s ′ ). Given a multipartition λ in

Acknowledgments: The authors thank the organizers of the workshop "Autour des conjectures de Broué" hold at the CIRM in Luminy (from 05/27/07 to 06/02/07) during which this paper was completed.


Rui Rodrigues

João M P Cardoso

An Infrastructure to Functionally Test Designs Generated by Compilers Targeting FPGAs *

This paper presents an infrastructure to test the functionality of the specific architectures output by a highlevel compiler targeting dynamically reconfigurable hardware. It results in a suitable scheme to verify the architectures generated by the compiler, each time new optimization techniques are included or changes in the compiler are performed. We believe this kind of infrastructure is important to verify, by functional simulation, further research techniques, as far as compilation to Field-Programmable Gate Array (FPGA) platforms is concerned.

Introduction

Compilation of software programs to reconfigurable hardware (e.g., FPGAs) is an important research subject (see, for instance, [START_REF] So | A Compiler Approach to Fast Design Space Exploration in FPGA-based Systems[END_REF]). It promises both to shrink the long design cycles needed to develop systems using reconfigurable hardware components and to take advantage of the large number of resources available in FPGA devices.

In order to test new techniques, researchers working on compilation techniques for FPGA-based runtime reconfigurable platforms usually use a HDL (Hardware Description Language) simulator or go directly to the implementation on the FPGA. Both flows are not appropriate to conduct an efficient test, which requires among other issues automation mechanisms. Although going to the implementation of the architecture on the FPGA permits a fast execution of the reconfigurable hardware structures, it requires long design cycles due to the need to perform logic synthesis and place and route. Furthermore, testing by implementation imposes other difficulties, such as the ones related to the following requirements: access to values on certain connections, assertions, inclusion of probes and stop mechanisms, automation needed to test the results for all the set of test cases used during the test, etc.

Certain algorithms, such as the ones computing on large data sizes (e.g., in image and video processing), may require simulation of a large number of clock cycles and therefore a fast simulation engine is needed. Previous work has shown that RTL (Register Transfer Language) simulation based on software languages can be faster than commercial HDL simulators [START_REF] Kuhn | Object Oriented Hardware Modeling and Simulation Based on Java[END_REF] [START_REF] Séméria | RTL C-Based Methodology for Designing and Verifying a Multi-Threaded Processor[END_REF].

Implementations with several configurations need an appropriate scheme to simulate the flow of configurations and the communication between them. Since HDLs have not been developed for programming dynamically reconfigurable hardware devices, they do not directly support those features. Trying to solve these issues, tools for simulation of partial runtime reconfigurable hardware have already been presented (e.g., [START_REF] Robertson | A Design Flow for Partially Reconfigurable Hardware[END_REF]).

Previous work on research and developing a compiler for FPGA-based reconfigurable platforms [START_REF] Cardoso | Compilation for FPGA-Based Reconfigurable Hardware[END_REF] has revealed difficulties to test designs output by the compiler, each time modifications were carried-on. Those modifications may include code restructuring or addition of new optimization techniques. Checking the overall test suite required long time efforts.

Furthermore, coupling the reconfigurable hardware component to a microprocessor model might be required and therefore hardware/software co-simulation features are also important. The use of the same language for modeling both components permits to mix both software and reconfigurable hardware components without specialized cosimulation environments [START_REF] Séméria | RTL C-Based Methodology for Designing and Verifying a Multi-Threaded Processor[END_REF].

Aiming at addressing these issues, we developed an infrastructure suitable to verify, by functional simulation, research efforts on compilation techniques at highabstraction levels. It extensively uses XML (eXtensible Markup Language) [START_REF]W3C: Extensible markup language (XML[END_REF], XSL (eXtensible Stylesheet Language), and Java technologies. The infrastructure is currently being used in Galadriel and Nenya [START_REF] Cardoso | Compilation for FPGA-Based Reconfigurable Hardware[END_REF], a framework to compile Java algorithms to FPGA-based platforms.

Infrastructure

The test infrastructure (see Figure 1) uses Hades [START_REF] Hendrich | A Java-based Framework for Simulation and Teaching[END_REF] as simulation engine. Hades is a publicly available, eventbased simulator developed in Java. Since it supports behavioral Java descriptions, we are able to take profit of all the Java features.

The Galadriel and Nenya compiler [START_REF] Cardoso | Compilation for FPGA-Based Reconfigurable Hardware[END_REF] has been modified to output the specification of the datapaths, control units, and the Reconfiguration Transition Graph (RTG), using XML dialects. Note that the RTG is used when the compiler maps the input algorithm onto multiple configurations (temporal partitions) [START_REF] Cardoso | Compilation for FPGA-Based Reconfigurable Hardware[END_REF]. Those XML files are then translated to the required language by XSLT (XSL Transformation) engines. This permits users to define their own XSL translation rules to output representations using the chosen language (e.g., Verilog, VHDL, SystemC, etc.).

As far as the test infrastructure is concerned, the XML representations of the datapaths are translated to the Hades input format, the XML representations of the behavioral descriptions of the FSMs to Java code representing their behavior, and the RTG to Java code that controls the execution of the simulation through the set of temporal partitions. XML representations can also be translated to graph descriptions to be visualized with Graphviz [START_REF]Graphviz -open source graph drawing software[END_REF].

Memory contents and I/O data are stored in files. Those files are used when executing the Java input algorithm. A specific Java class with methods to interface to those files is provided. After simulation, a simple comparison of data content is performed to verify results.

Experimental results and discussion

Table I shows results with a fast algorithm for the Discrete Cosine Transform (FDCT) and with a Hamming decoder. The simulations have been performed in a Pentium 4, at 2.8 GHz, under Windows XP, and with 512 MB of RAM. For each algorithm, the number of lines of the source code (loJava) and of the XML and Java descriptions is shown. The column "operators" shows the number of functional units used for each datapath.

The FDCT performs 8 8 DCT blocks over an input image. We show results for two implementations of the algorithm. The first one, FDCT1, with only one configuration (i.e., the algorithm is implemented in a single datapath and control unit) and the second one, FDCT2, with two con-figurations. In the later case, the implementation uses two separate designs that respect the functionality of the original application by executing the two configurations in sequence. Both implementations use three SRAMs to store input, output, and intermediate images.

The simulation time results for the FDCTs are related to the computation with an input image of 4,096 pixels (64 DCT blocks). With images of 65,536 and 345,600 pixels, FDCT1 is simulated in 1 and 6.5 minutes, respectively. The results show that the test environment is fast enough to our purposes. It can be used to verify, at high abstraction levels, compiler results over a complete test suite in feasible time. The infrastructure does not require commercial tools to accomplish its goals, which has been one of our pragmatic requirements. Also note that with Hades, Java GUI features can be easily included, e.g., to graphically show input/output data when dealing with image processing algorithms.

Further work will focus on functional simulation of a microprocessor tightly coupled to reconfigurable hardware components.

XSLTs Figure 1. Diagram of the test infrastructure.

		datapath.xml datapath.xml	fsm.xml fsm.xml	rtg.xml rtg.xml
		to dotty to hds	to dotty to java	to dotty to java
	GraphViz GraphViz	datapath.dot datapath.hds fsm.dot fsm.java	rtg.dot	rtg.java
	Library of Operators	HADES		fsm.class	rtg.class
	(JAVA)
					ANT build
				I/O data
				(RAMs and Stimulus)

Table I . Results using the test infrastructure.

 I

	Example loJava loXML	loXML	loJava	Operators Simulation
			FSM	datapath	FSM		time (s)
	FDCT1	138	512	1,708 1,175	169	6.9
	FDCT2	138	258	860	667	90	2.9
			256	891	606	90	2.9
	Hamming	45	38	322	134	37	1.5

Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE'05) 1530-1591/05 $ 20.00 IEEE

* This work is partially supported by the Portuguese Foundation for Science and Technology (FCT) -FEDER and POSI programs -under the CHIADO project. ** The authors are also affiliated with INESC-ID, Lisbon, Portugal.