Luis Jaime Corredor

Adrien Deloro
email: adrien.deloro@imj-prg.fr

Joshua Wiscons
email: joshua.wiscons@csus.edu

Jaime Luis

Adrien Corredor

AND Joshua Deloro

Wiscons

Sym(n)-AND Alt(n)-MODULES WITH AN ADDITIVE DIMENSION (PARIS ALBUM NO. 2)

Keywords: Mathematics Subject Classification. Primary 20C30; Secondary 03C60, 20F11 symmetric groups, finite-dimensional algebra, first-order representation theory

published or not. The documents may come

Introduction

This work belongs to the topic of first-order representation theory, i.e. representation theory viewed through an elementary lens. Here the focus in on the category of explicitly constructible objects, or what mathematical logic calls the definable category; a consequence is that we avoid any reference to characters. This is not motivated by the mere 'purity of methods' but by questions in model theory. The topic is naturally emerging out of several recent works, rooted in the 2008 article of Alexandre Borovik and Gregory Cherlin [START_REF] Borovik | Model theory with applications to algebra and analysis[END_REF] and pushed further by papers such as [Del09, BD16, BB18, Bor, BB21]. (Another road to the effective understanding of geometric algebra is black box algebra as in [START_REF] Borovik | Adjoint representations of black box groups PSL2(Fq)[END_REF] and ongoing work.) We stress that, though inspired by model theory, the present article is likely to be of broad interest; no exposure to model theory is required to understand our work.

We give a significant expansion and clarification of a classical result by Leonard Dickson from 1908 on the minimal linear representations of symmetric groups. Our Theorem identifies the minimal faithful representations of Sym(n) and Alt(n) on finite or infinite abelian groups in the presence of a rudimentary notion of dimension but no a priori-and often no a posteriorivector space structure. This may be viewed as a natural evolution of linear representation theory, one which focuses more on 'elementary' properties (e.g. generators and relations) and less on higher structure. In the case of algebraic groups, which we keep in mind for the future, our point of view would be quite in the spirit of the Chevalley-Steinberg approach. We stress that our context does not allow for character theory nor even Maschke's Theorem, making matters non-trivial though basic.

All that remains from the usual linear theory is a loose form of dimensionality. The study of structures whose definable sets are equipped with one of various notions of dimension is a central theme in model theory, and our work here treats numerous classes at once-including groups of finite Morley rank and o-minimal groups, but also finite groups-in a common, natural, and new setting.

Our original motivation was a set of concrete model-theoretic problems. One such is an application to permutation groups possessing a high degree of generic transitivity. A study of these was initiated in the setting of groups of finite Morley rank in work by Borovik and Cherlin where they posed the problem of showing that generic (n + 2)-transitivity on a set of Morley rank n implies that the group is PGL n+1 (K) in its natural action on P(K). More information and explicit connections to the present work can be found in [START_REF] Borovik | Model theory with applications to algebra and analysis[END_REF] as well as in the 2018 paper of Tuna Altınel and the third author [START_REF] Altınel | Recognizing PGL3 via generic 4-transitivity[END_REF], which solves the n = 2 case. The present work grew out of the third author's desire to generalise the n = 2 approach to n ≥ 3, but the topic turned out extremely interesting in its own right.

1.1. The result. Our main result generalizes a century-old theorem by Dickson [START_REF] Leonard | Representations of the general symmetric group as linear groups in finite and infinite fields[END_REF] and its more recent companions [START_REF] Wagner | The faithful linear representation of least degree of Sn and An over a field of characteristic 2[END_REF][START_REF] Wagner | The faithful linear representations of least degree of Sn and An over a field of odd characteristic[END_REF]; it also corrects and expands on [START_REF] Borovik | Model theory with applications to algebra and analysis[END_REF]Lemma 4.6]. In doing so, our treatment handles simultaneously the finite and the 'tame infinite', in a sense that model theory seeks to carve out.

We study Sym(n) and Alt(n)-modules V that carry a basic notion of dimension on certain groups (and quotients) definable-in the logical sensefrom V and the acting group. To the logician we must stress that our dimension need not apply to all definable sets; we only require it to be defined on the intersection of the 'definable universe' in the sense of logic and the 'variety generated by V ' in the sense of universal algebra. This intuition is axiomatised in Section 2 via the definitions of a modular universe and an additive dimension; the relevant notions of connectedness (dim-connectedness) and irreducibility (dc-irreducibility) are also key.

As one expects, the 'characteristic' of a module is an important parameter: V is said to have prime characteristic p if it has exponent p and characteristic 0 if it is divisible. This definition allows for modules without a well-defined characteristic such as Z/12Z; it also allows for torsion modules of characteristic 0 such as the Prüfer quasi-cyclic groups C p ∞ .

In the classical setting, the minimal faithful representations for Sym(n) and Alt(n) are canonical and easy to construct, assuming n is large enough. Among other places, they appear in [START_REF] Leonard | Representations of the general symmetric group as linear groups in finite and infinite fields[END_REF], but we briefly describe them here. This also gives us the opportunity to introduce notation. Notation (standard module). Let S := Sym(n).

(1) Let perm(n, Z) = Ze 1 ⊕ • • • ⊕ Ze n be the Z[S]-module with S permuting the e i naturally. There are two obvious submodules:

• std(n, Z) := [S, perm(n, Z)] = { i c i e i : c i ∈ Z and c i = 0}; • Z(perm(n, Z)) := C perm(n,Z) (S) = { i ce i : c ∈ Z}, using usual notation for commutators and centralisers. Over Z these are disjoint but not so in general.

(2) For any abelian group L (considered as a trivial S-module), define:

• perm(n, L) := perm(n, Z) ⊗ Z L;

• std(n, L) := [S, perm(n, L)] = std(n, Z) ⊗ Z L;

• Z(std(n, L)) := C std(n,L) (S). We arrive at the canonical subquotient:

• std(n, L) = std(n, L)/Z(std(n, L)), which we refer to as the (reduced) standard module over L.

(3) When L = C k is cyclic of order k, we simply write perm(n, k), std(n, k), and std(n, k).

Remarks.

• Notice how Z(std(n, L)) = { e i ⊗ a : a ∈ Ω n (L)}, so std(n, L) differs from std(n, L) only when Ω n (L) = 0. (Here Ω n (L) denotes the subgroup of elements of order dividing n.) • The module perm(n, L) may be realised as L n under σ(a 1 , . . . , a n ) = (a σ -1 (1) , . . . , a σ -1 (n) ); it is easily constructed from L together with the action of each element of S. Model theorists will recognize this as an interpretable object, which we simply call definable.

The classical setting is std(n, p) for p a prime. Here, std(n, p) is irreducible of dimension n -1 whenever p n (with the same true for std(n, Q)). However, when p | n and n ≥ 5, the module std(n, p) is faithful and irreducible of dimension n -2, a point which fails of std(4, 2).

Less classical is the following example regarding actions on tori.

Example. Notice how a maximal torus of GL n (C) is a Sym(n)-module via the action of the Weyl group. As Sym(n)-modules, one finds that

• perm(n, C * ) is isomorphic to a maximal torus of GL n (C); • std(n, C * ) is isomorphic to a maximal torus of SL n (C); • std(n, C * ) is isomorphic to a maximal torus of PSL n (C).
The various finite subgroups of n th -roots of unity yield finite submodules C ≤ Z(std(n, C * )), each naturally isomorphic to some Z ≤ Z(SL n (C * )). In each case, std(n, C * )/C is isomorphic to the maximal torus of SL n (C)/Z. The modules std(n, C * )/C all satisfy the relevant notion of irreducibility here (dc-irreducibility) and must be accounted for in our classification.

We now state our main result. In what follows, Mod(G, d, q) denotes the class of all G-modules ( § 2.1) that carry an additive dimension ( § 2.2) and are dim-connected ( § 2.3) of dimension d and characteristic q ( § 2.4). Also, V ∈ Mod(G, d, q) is dc-irreducible ( § 2.5) if V contains no non-trivial, proper, dim-connected G-submodule. Details are in § 2; the proof is in § 3.

Theorem. Let G = Alt(n) or Sym(n). Suppose V ∈ Mod(G, d, q) is faith- ful and dc-irreducible with d < n. Assume n ≥ 7; if G = Alt(n) and q = 2, further assume that n ≥ 10.
Then there is a dim-connected submodule L ≤ V such that the structure of V falls into one of the following cases:

q d Structure of V q > 0 and q | n n -2 isomorphic to std(n, L) or sgn ⊗ std(n, L)
q > 0 and q n n -1 isomorphic to std(n, L) or sgn ⊗ std(n, L) q = 0 n -1 covered by std(n, L) or sgn ⊗ std(n, L)

Moreover, when q = 0, the kernel of the covering map is

n-1 i=1 (e i -e n ) ⊗ K, in usual notation, for some K ≤ Ω n (L).

Remarks.

• Note that q = char L, and if q > 0, std(n, L) is completely reducible as X std(n, q) for X an F q -basis of L. The situation when q = 0 is complicated by tori like those given in the example above. In our more general setting, one can still establish the lower bound of 8 on the dimension of a faithful Alt(9)-module in characteristic 2, but we do not achieve (nor even try for) identification. (See the remark following the proof of the Geometrisation Lemma.) 1.2. Lingering questions. The Theorem places natural restrictions on n, but in fact, the minimal dimension of a faithful Sym(n)-module is indeed as expected for all n as a consequence of our First Geometrisation Lemma, where we also identify those of dimension (n -2). However, identification of the dc-irreducible modules in Mod(Sym(5), 4) and Mod(Sym(6), 4) remains open. (Do note that Mod(Sym(5), 4, 2) contains irreducible modules coming from the so-called Specht module for the partition (3, 2). See for example [Jam78, 5.2 Example].)

Identification of the minimal faithful Alt(n)-modules for small n is also open. Reconstructing the adjoint action of Alt(6) PSL 2 (F 9 ) is a problem of particular interest. One also has Alt(8) SL 4 (F 2 ) with the natural action as well as the particularly exceptional Alt(5): it appears as SL 2 (F 4 ) in characteristic 2, as PSL 2 (5) in characteristic 5, and as the symmetries of a regular icosahedron in all other characteristics (over a field where 5 is a square). The following table summarizes the conjectural lower bounds. Conjectural minimal dimension for faithful Alt(n)-modules in characteristic p with small n.

Problem. Identify the minimal faithful Sym(n)-Alt(n)-modules for all n.

Of course, one could target higher-dimensional dc-irreducible Sym(n)-Alt(n)-modules, but this appears to be out of reach at present. However, simply identifying a reasonable lower bound for the dimension of the 'second smallest' dc-irreducible module would be welcome. Following the classical case, we expect something along the lines of n(n -5)/2 (see [START_REF] James | On the minimal dimensions of irreducible representations of symmetric groups[END_REF]).

Problem. Let G = Alt(n) or Sym(n) with n sufficiently large. Prove that if V ∈ Mod(G, d, q
) is faithful and dc-irreducible with d < n(n -5)/2, then up to tensoring with the signature, V is standard (i.e. the structure of V is as in the Theorem) with (n -2) dim L ≤ d ≤ (n -1) dim L and intermediate values are possible only when q = 0.

As should be clear, we are operating under the conjectural principle that although our context is quite general, the minimal objects still fall into the familiar linear-algebraic setting, a principle well-aligned with the recent work of Borovik [Bor]. We quite believe in this and would thus love to see a counter-example to shatter our illusion.

The problem of determining the minimal dimension of a group carrying a faithful action of Sym(n) or Alt(n) seems both interesting and relevant in the nonabelian case as well. With additional definability/compatibility hypotheses, the soluble case can easily be controlled. For nonsoluble groups, we propose the following crude bound, which is likely far from optimal.

Problem. Let H be a nonsoluble group on which Alt(n) acts faithfully and definably by automorphisms. Suppose there is a nonabelian notion of dimension, say Morley rank, making H dim-connected. Show dim H ≥ n for sufficiently large n.

Notice that low values of n will complicate the picture even for Sym(n): for example, Sym(5) PGL 2 (5), which can be construed as 3-dimensional.

For the present article we however stick to the abelian case. The proof of the Theorem will be in § 3; we first turn to the general setting.

The context

We now give the setting for our study of Sym(n)-and Alt(n)-modules. In addition to defining modules equipped with an additive dimension, we also present notions of connectedness, irreducibility, and the characteristic. In short, the goal of this section is to fully explain the phrase 'let V ∈ Mod(G, d, q) be dc-irreducible'.

The landscape will likely be both familiar and surprising to the reader versed in model theory. We seek a notion of dimension that encompasses simultaneously the linear dimension over F p for finite representations as well as model-theoretic dimensions (e.g. Morley rank) for infinite representations. The context we present is extremely natural, yet looks new to us. (Unfortunately, somewhat conflicting terminology with Frank Wagner's recent 'dimensional groups' was unavoidable [START_REF] Frank | Dimensional groups and fields[END_REF].)

We first define modules ( § 2.1) with an additive dimension ( § 2.2), and the notion of dim-connectedness ( § 2.3). We discuss the characteristic of a module ( § 2.4) and then introduce classes Mod(G, d, q) as well as the relevant notion of irreducibility ( § 2.5). The overview concludes with our key tool: an expected Coprimality Lemma ( § 2.6).

2.1. Modular universes and modules. A balance is difficult to strike between categorical generality and model-theoretic care for elementary constructions. We opted for a categorical vision but avoided any specialised language. We believe the categorist will instantly grasp the context, and the logician will readily check that it generalises definable universes. Do note that we use ker f and im f to refer to the kernel and image of f in the algebraic sense. Definition. A modular universe is a subcategory U of the category Ab of abelian groups satisfying the following closure properties.

• [inverses] If f ∈ Ar(U) is an isomorphism, then f -1 ∈ Ar(U). • [products] If V 1 , V 2 ∈ Ob(U) and f 1 , f 2 ∈ Ar(U), then V 1 × V 2 ∈
Ob(U), and Ar(U) contains f 1 ×f 2 , the projections

π i : V 1 ×V 2 → V i , and the diagonal embeddings ∆ k : V 1 → V k 1 . • [sections] If W ≤ V are in Ob(U), then V /W ∈ Ob(U) and Ar(U)
contains the inclusion ι : W → V and quotient p :

V → V /W maps. • [kernels/images] If f : V 1 → V 2 is in Ar(U), then ker f, im f ∈ Ob(U), and for all W 1 , W 2 ∈ Ob(U), -if W 1 ≤ ker f , the induced map f : V 1 /W 1 → V 2 is in Ar(U); -if im f ≤ W 2 ≤ V 2 , the induced map f : V 1 → W 2 is in Ar(U). • [module structure] If V ∈ Ob(U)
, then Ar(U) contains the sum map σ : V × V → V and the multiplication-by-n maps µ n : V → V .

The objects of a modular universe U are called its modules and the arrows its compatible morphisms.

Remarks.

• We could not find an official categorical name for our setting. 'Topologising, abelian subcategory' does not suffice; the axiom of inverses is of importance to us, but we won't go as far as assuming that the subcategory is replete. • The axioms immediately allow for restrictions of compatible maps and the computing of inverse images. Indeed, if Examples.

f : V 1 → V 2 is in Ar(U) with W 1 ≤ V 1 in Ob(U), then the restriction of f to W 1 is f •ι for ι : W 1 → V 1 the inclusion. And for W 2 ≤ V 2 in Ob(U), f -1 (W 2 ) is the kernel of p • f for p : V 2 → V 2 /W 2 the
• From algebra: the category of all abelian groups or of all abelian p-groups (equipped with all group morphisms) forms a modular universe; both are full subcategories of Ab. An important variation is the category of all abelian p-groups of finite Prüfer p-rank. (Such groups can contain only finite powers of the quasi-cyclic group C p ∞ ; the maximal such power is called the p-rank.)

For a given ring R, the category of R-modules (equipped with all R-morphisms) is a modular universe.

• From Lie theory: the collection of all abelian Lie groups (with Lie morphisms) forms a modular universe after adjusting the notion of ≤ (which has to be closed). • From universal algebra: the variety generated by a given abelian group V (with group morphisms) forms a modular universe; this can be computed as HSP(V ), which is the collection of all Homomorphic images of Subgroups of Products of V . • From model theory: the abelian part of the 'interpretable universe' (which we call definable) forms a modular universe; specifically, this is looking at the category of all abelian groups definable in some first-order theory equipped with the definable group morphisms. • If V is a module of a universe U, then there is a smallest subuniverse containing V . This is contained in both HSP(V ) and the abeliandefinable universe U def from model theory. Notice that the intersection HSP def (V ) = HSP(V ) ∩ U def (V ) might be substantially smaller than U def if we add a highly extrinsic, secondary abelian structure on some non-definable subsets of V . (Typically, if K is one of the pathological fields model theorists adore, then HSP def (K + ) will miss all the exotic structure.)

Modular universes allow for a variety of additional constructions. We highlight several important ones after giving the relevant notion of a Gmodule for our setting.

Definition. Suppose V is a module in the universe U. If a group G acts on V by compatible morphisms of U, we say that V is a G-module in U.

In model-theoretic terms, we are assuming that G acts by definable automorphisms, but we are not assuming definability of the action, viz. definability of the triple (G, V, •). For instance, G itself need not be definable.

Universe Properties. Let U be a modular universe.

(i)

[meet/join] If V 1 , V 2 ≤ V are modules in U, then so are V 1 ∩ V 2 and V 1 + V 2 . (ii) [extensions] Suppose V 1 , V 2 ≤ V are modules in U with V 1 + V 2 = V . If f i : V i → W are compatible morphisms that agree on V 1 ∩ V 2 , then there is a compatible f : V → W extending both. (iii) [permutation modules] If V is a module in U, then perm(n, V ) is a Sym(n)-module in U. (iv) [enveloping algebras] If V is a G-module in U for some group G,
then each map in the subring of End(V ) generated by G is compatible.

Proof. Assume V 1 , V 2 ≤ V are modules in U. Let σ : V × V → V be the addition map and ∆ k : V → V k the diagonal embedding. (i) Note that V 1 ∩ V 2 = ∆ -1 2 (V 1 × V 2 ) and V 1 + V 2 = σ(V 1 × V 2 ). (ii) Let g be the restriction of σ to V 1 × V 2 , and set I := {(a, -a) : a ∈ V 1 ∩V 2 } = ker g. Then the induced isomorphism g : V 1 ×V 2 /I → V is in Ar(U), so g -1 is as well. Define h : V 1 × V 2 → V via restriction of σ •(f 1 ×(f 2 •µ -1 )) to V 1 ×V 2 , where µ -1 : V → V is inversion; then h computes f 1 -f 2 . Since I ≤ ker h, the induced map h : V 1 ×V 2 /I → V is in Ar(U)
, and f := h • g -1 ∈ Ar(U) extends both f 1 and f 2 . (iii) As a set, we may identify perm(n, V ) with V n ∈ Ob(U) and the canonical summands V e i with j =i ker π k (for π k : V n → V the k thprojection). Now, for α ∈ Sym(n), viewed as the automorphism of V n permuting the coordinates naturally, we must show α ∈ Ar(U).

Let ∆ be the diagonal embedding of

V n into (V n ) n . Then α = (π α -1 (1) × • • • × π α -1 (n) ) • ∆. (iv) The (extended) sum map σ k : V k → V may be inductively defined as σ • (σ k-1 × Id V ) so is in Ar(U). Further, Ar(U) contains the multiplication-by-n maps µ n : V → V for each n ∈ Z. Thus, for g 1 , . . . g k ∈ G and n 1 , . . . , n k ∈ Z, the image of n i g i in End(V ) is σ k • [(µ n 1 • g 1 ) × • • • × (µ n k • g k )] • ∆ k ∈ Ar(U).

Additive dimensions.

Definition. Let U be a modular universe. An additive dimension on U is a function dim : Ob(U) → N such that for all f :

V → W in Ar(U), dim V = dim ker f + dim im f.
This property will be called additivity.

Examples. Each of the following has an additive dimension:

• the universe of finite-dimensional vector spaces over a fixed field, equipped with the linear dimension; • the universe of abelian p-groups of finite Prüfer p-rank, with dimension the p-rank; • the universe of all abelian groups definable in a theory of finite Morley rank, with dimension the Morley rank; • the universe of all abelian groups definable in an o-minimal structure, equipped with o-minimal dimension; • the universe of abelian Lie groups, equipped with Lie (manifold) dimension.

Remarks. The following remarks are better understood in relation to say, behaviour of Morley rank, but can be read independently.

• No assumption is made on when the dimension increases nor how; as a matter of fact, any multiple of a dimension function is again one. • There need not be a descending chain condition ('dcc') on objects.

This calls for a modified notion of connectedness in § 2.3. • No relationship between finiteness and 0-dimensionality is implied.

Thus, we handle in the same operational setting F n p (n-dimensional, in finite group theory) and Z (0-dimensional, in Lie theory).

• It is unclear whether the non-abelian case is restrictive enough for a general theory to emerge. Typically, handling commutators requires the dimension to be defined on subsets, not only subgroups.

For example, we do not know about free groups or Tarski monsters; the question seems to be of interest but beyond our expertise.

• We briefly mused on the possibility of determining natural criteria for a pre-dimension-defined only on submodules of some V -to extend to a genuine dimension on subquotients as well, but this remains mostly unexplored. Such topics are nontrivial in model theory.

Remarks (for model theorists). Our work stems from model theory but we wish to stress a couple of differences.

• The focus is on group subquotients instead of general definable sets.

(One could define dim on cosets but we will not need that.) • We work with a dimension on a universe containing a fixed group structure and make no demands on its behaviour in elementary extensions. Thus dimension is not required to be a 'strong' invariant (viz. a property of the theory); as a matter of fact, an elementary extension need not bear a dimension function.

Morley rank and o-minimal dimension are strong invariants. We do not have an example of (the definable universe of) an abelian group V carrying an additive dimension and an elementary extension V * not admitting one.

Arguably we touch here the difference between first-order (definability in one structure) and model-theoretic (definability in family, viz. in elementary extensions) properties; or between model-theoretic algebra and model-theory properly speaking.

Dimension Properties. Let (U, dim) be a modular universe with an additive dimension, from which we take modules.

(i

) dim{0 V } = 0. (ii) If W ≤ V , then dim W ≤ dim V . (iii) dim(V 1 × V 2 ) = dim V 1 + dim V 2 . (iv) If V 1 , V 2 ≤ V , then dim(V 1 + V 2 ) = dim V 1 + dim V 2 -dim(V 1 ∩ V 2 ).
Proof. For the first, apply additivity to the identity map ι : V → V . The second follows from additivity of the quotient map p : V → V /W (and that dim is non-negative). For the third, additivity of the projection

π 1 : V 1 × V 2 → V 1 shows dim(V 1 × V 2 ) = dim V 1 + dim(ker π 1 )
, so the result then follows from the fact that {0 V 1 } × V 2 and V 2 are compatibly isomorphic. For the final point, apply additivity to the restriction of

σ to V 1 × V 2 (using that the kernel is compatibly isomorphic to V 1 ∩ V 2 ).
Definition. If (U, dim) is a modular universe with an additive dimension and V ∈ Ob(U), we call V a module with an additive dimension, leaving U implicit from context.

In practice, we often have in mind HSP def (V ) (viz. those homomorphic images of submodules of powers of V which are model-theoretically definable) when we say V is a module with an additive dimension.

2.3. dim-connectedness. We now present a weak form of connectedness, which is not the classical one in mathematical logic.

Definition. A module V with an additive dimension is called dim-connected (for dimension-connected), or simply a dc-module, if every proper submodule

W < V in U satisfies dim W < dim V .
Examples.

• The only dc-module of dimension 0 is {0}.

• Every finite-dimensional vector space over F p is dim-connected with respect to the linear dimension. • If the dimension function satisfies 'dim A = 0 ⇐⇒ A is finite', then V is dim-connected if and only if V is connected in the usual model-theoretic sense of having no proper subobjects of finite index. But this is not so in general as we assume neither implication.

Remarks.

• If V contains a dc-submodule V 1 ≤ V with dim V 1 = dim V , then V 1 is unique (hence invariant under U-automorphisms of V ) and called the dc-component of V . • If W ≤ V and both have dc-components, then W 1 ≤ V 1 . Indeed, additivity implies dim((W 1 +V 1 )/V 1 ) = 0, so dim(W 1 /W 1 ∩V 1 ) = 0 also. Then dim-connectedness of W 1 forces W 1 = W 1 ∩ V 1 . • If U
satisfies the descending chain condition on objects, then dccomponents exist. The converse is false (0-dimensional Z). • In the universe of divisible abelian p-groups of finite Prüfer rank, the dc-component of V is the largest subtorus entirely contained in V . • The dc-component of a group of finite Morley rank or an o-minimal group is its connected component.

• In general, even if dc-components exist, the index [V : V 1 ] need not be finite (0-dimensional V ).
We now highlight various operations preserving dim-connectedness.

Connectedness Properties. Let (U, dim) be a modular universe with an additive dimension, from which we take modules.

(i) If V is dim-connected and f : V 1 → V 2 is compatible, then im f is dim-connected. (ii) If V 1 and V 2 are dim-connected, then so is V 1 × V 2 . (iii) If V 1 , V 2 ≤ V and V 1 and V 2 are dim-connected, then so is V 1 + V 2 . Proof. (i) Let W 2 ≤ im f be a submodule of maximal dimension. Set W 1 = f -1 (W 2 ) ≥ ker f , and let g : W 1 → W 2 be the restriction of f . Then dim W 1 = dim ker g + dim im g = dim ker f + dim im f = dim V 1 . By dim-connectedness, W 1 = V 1 , so W 2 = im f . (ii) Suppose Z ≤ V 1 × V 2 is a submodule of maximal dimension. Let π i :
V → V i be the projections and π Z i the restrictions to Z. Notice that ker π i is compatibly isomorphic with V j , hence dim-connected of the same dimension; of course ker

π Z i ≤ ker π i . Then dim V 1 + dim V 2 = dim(V 1 × V 2 ) = dim Z = dim im π Z i + dim ker π Z i ≤ dim im π i + dim ker π i = dim V 1 + dim V 2 . By dim-connectedness, one finds ker π Z 1 = {0} × V 2 ≤ Z and V 1 × {0} ≤ Z, so Z ≥ (V 1 × {0}) • ({0} × V 2 ) = V 1 × V 2 . (iii) Apply the first two points to the sum map V 1 × V 2 → V 1 + V 2 .
2.4. The characteristic of a module.

Definition. Let V be a module with an additive dimension. We say that:

• V has characteristic p, for p a prime, if it is of exponent p; • V has characteristic 0 if it is divisible.

Remarks.

• A module need not have a well-defined characteristic: consider Z/6Z.

• Modules of characteristic 0 may well contain torsion: consider C p ∞ .

Characteristic Lemma. Let V be a module with an additive dimension.

If V is dim-connected, then V has a finite-length, Aut U (V )-invariant, dim- connected composition series 0 = V 0 < • • • < V n = V in U with n ≤ dim V
and each factor either of prime exponent or divisible.

Proof. We proceed by induction on dim V ; the only dc-module of dimension 0 is {0}. Let p be a prime; consider the multiplication by p morphism. The image pV ≤ V is dim-connected, so either pV < V and we apply induction, or pV = V . In the latter case, V is p-divisible, and we resume with another prime.

Examples.

• Not all interesting modules have a characteristic. Any embedding Alt(4) → GL 2 (Z/4Z) makes V = (Z/4Z) 2 a faithful, 2-dimensional Alt(4)-module of exponent 4. It is not minimal, but Alt(4) is faithful on neither 2V nor V /2V . (This of course relates to solubility; one should always be careful with std(4, 4).) • Let I be infinite and T = I C p ∞ as a pure group. We suspect there is a non-trivial additive dimension on HSP def (T ); however Morley rank is infinite. Hence our setting seems to allow for tori of infinite Prüfer rank.

Divisibility Properties. If V is a dc-module with an additive dimension and p is a prime, then V is p-divisible iff Ω p (V ) := {v ∈ V : pv = 0} has dimension 0. In particular, if W ≤ V is a dc-submodule and V has a characteristic, then V and W have the same characteristic.

Proof. The multiplication by p morphism has kernel Ω p (V ); since V is dimconnected, the map is onto if and only if the kernel has dimension 0. When restricted to W , the kernel is Ω p (W ) = W ∩ Ω p (V ).

Remark. In the case where V = (C p ∞ ) n (which has characteristic 0) with p > 2 a prime, it is well-known (for instance [START_REF] Deloro | p-rank and p-groups in algebraic groups[END_REF]) that the restriction morphism: ρ : Aut(V ) → Aut(Ω p (V )) kills no element of finite order. (There is a kernel {±1} n if p = 2.) We could however not make profit of this remark in our present work.

Dc-irreducibility and classes Mod(G, d, q).

Definition. A dim-connected G-module V is dc-irreducible (as a G-module), if it has no non-trivial, proper, dim-connected G-submodule.

Notation. Mod(G, d) is the class of all dim-connected G-modules of dimension exactly d. (They are not required to all live in a common universe; here one browses through universes and dimension functions.) Subclasses Mod(G, d, q) specify the characteristic.

By the Characteristic Lemma ( § 2.4), dc-irreducible modules always have a characteristic. 2.6. Coprimality results. We now let finite cyclic groups act on our modules; as one imagines, the characteristic plays a crucial role. Bear in mind that we do not assume the existence of dc-components for submodules (see remarks in § 2.3); this explains why we avoid centralisers and prefer to work in terms of ad and tr.

Notation. Let V be a g -module, where g has order p.

• Let ad g = 1 -g and tr

g = 1 + g + • • • + g p-1 .
• Let B g = im ad g and C g = im tr g .

When g acts compatibly on V (in some universe), the Universe Properties ensure that ad g and tr g are compatible endomorphisms, so B g and C g are submodules. Be careful that C g does not stand for the centraliser (though always C g ≤ C V (g)); for instance, if g is an involution acting in characteristic 2, then tr g = ad g and C g = B g . However, in characteristic not p, pathologies are confined as shown below.

Coprimality Lemma. Let p be a prime and V be a p-divisible, dimconnected g -module with an additive dimension, where g has order p. Then V = B g + C g and dim(B g ∩ C g ) = 0.

Proof. As B g and C g are images under compatible endomorphisms of V , they are dc-submodules. Notice how ad g • tr g = tr g • ad g = 1-g p = 0 in End(V ), so B g ≤ ker tr g and C g ≤ ker ad g . However one sees ker ad g ∩ ker tr g ≤ Ω p (V ), which is 0-dimensional by the Divisibility Properties. Thus, dim C g = dim im tr g = dim V -dim ker tr g ≥ dim ker ad g ≥ dim C g , so equality holds. Then:

dim(B g + C g ) = dim B g + dim C g = dim im ad g + dim ker ad g = dim V,
and V = B g + C g by dim-connectedness.

Remarks.

• The lemma proves that B g ∩ C g ≤ Ω p (V ), so if V has characteristic a prime different from p, then B g ∩ C g = 0. But in characteristic 0, B g ∩ C g need not be trivial, nor even finite.

For instance let T 1 , T 2 C 2 ∞ with central involutions i 1 , i 2 , and α an involution inverting T 1 while centralising

T 2 . Let S = (T 1 ⊕ T 2 )/ i 1 i 2 . Notice how B α = (1 -α)S = T 1 and C α = (1 + α)S = T 2 intersect.
Finally take an infinite direct sum of copies of S and say it has dimension 1.

• The proof also shows C g ≤ ker ad g = C V (g) have the same dimension: hence C V (g) has a dc-component, which equals C g . (Acting on a p-torus of infinite Prüfer rank, one can produce examples with [C g : C V (g)] = ∞.) However, without assuming p-divisibility, C V (g) need no longer have a dc-component.

The conclusion of the Coprimality Lemma is a 'quasi-direct' decomposition for V . (This terminology has other meanings in the literature.)

Definition. If A 1 , . . . , A n are submodules of a module V , the sum A i is said to be quasi-direct if dim A i = dim A i ,

in which case we write

A i = A 1 (+) • • • (+) A n .
Remark. Note that n 1 A i is quasi-direct if and only if both:

n-1 1 A i is quasi-direct, and dim A n ∩ n-1 1 A i = 0.
The following lemma highlights an important application of the Coprimality Lemma-it will be used often in the sequel.

Weight Lemma. Let E be a finite elementary abelian 2-group and V a 2-divisible, dim-connected E-module with an additive dimension. Then V decomposes into a quasi-direct sum of dim-connected weight submodules V λ where λ : V → {±1} is a group morphism and each e ∈ E acts on V λ as λ(e). Each λ is called a weight of E and V λ the corresponding (dim-connected) weight space.

Proof. Write E = E 0 ⊕ e . By the Coprimality Lemma, V = B e (+) C e with e inverting the first factor and centralizing the latter. Applying induction to the action of E 0 on each of B e and C e then yields the desired result.

Remark. If V is a 2-divisible, dim-connected G-module and K ≤ G is a Klein four-subgroup whose nontrivial elements are conjugate in G, then the spaces attached to non-trivial weights have constant dimension , and dim V = dim C V (K) + 3 . This will be used repeatedly.

The proof

The proof of the Theorem will be assembled from three components: the Recognition Lemma ( § 3.1) provides an elementary geometric condition sufficient to identify the standard Sym(n)-module (and its quotients in relevant characteristics); the Extension Lemma ( § 3.2) uses an analogous geometric condition to identify when an Alt(n)-module extends to Sym(n) in such a way that Recognition applies; and the Geometrisation Lemma ( § 3.3) details how Alt(n)-modules of low dimension naturally possess the geometric condition needed to invoke Extension (and thus Recognition). Notably, the first two of these results are quite general with hypotheses only on the existence of a dimension function, and no restriction on its values.

Notation.

1. Elements. We reserve i, j, k, . . . for elements of {1, . . . , n}. Permutations are typically denoted by lower-case greek letters, reserving:

• τ, τ for transpositions;

• α, β for bi-transpositions;

• γ, δ for 3-cycles. We indicate that permutations σ 1 and σ 2 have disjoint supports |σ 1 | and |σ 2 | by writing σ 1 ⊥ σ 2 . 2. Subgroups.

• We avoid stabiliser notation G i,j and G {i,j} . Instead, if S = Sym(n) and I ⊆ {1, . . . , n}, we let S I ≤ S be the subgroup of permutations with support contained in I. For σ ∈ S, we let S σ = S |σ| and S σ ⊥ = S |σ| c . Likewise in Alt(n).

• We often consider subgroups of A = Alt(n) isomorphic to Sym(k). Typically, for |I| ≤ n -2 and symbols k, / ∈ I, we let:

Σ (k ) I = A I ((S I \ A I ) • (k )) = σ if ε(σ) = 1 σ(k ) if ε(σ) = -1 : σ ∈ Sym(I) ,
a subgroup of Alt(n) isomorphic to Sym(I). When there is no ambiguity we simply write Σ I . • We use K for Klein four-groups of bitranspositions, with K ijk the Klein four-group having support {i, j, k, }.

Also, recall:

• from § 2.5, that Mod(G, d, q) stands for the class of dim-connected G-modules of dimension d and characteristic q; • from § 2.5 as well, the notion of dc-irreducibility meaning irreducibility in the class of dim-connected G-modules; • from § 2.6, that for V a G-module and g ∈ G, we define B g := [g, V ] = im ad g where ad g = 1 -g ∈ End(V ).

3.1.

Recognising the standard module. The Recognition Lemma constructs a natural covering module under assumptions of an elementary geometric nature. In prime characteristic (or in the torsion-free case), the kernel is known, and the isomorphism type fully determined.

Recall from § 1.1 that std(n, Z) = f 1 , . . . , f n-1 is the Sym(n)-submodule of the permutation module perm(n, Z) = e 1 , . . . , e n generated by f i = e i -e n . For any abelian group L, std(n, L) := std(n, Z)⊗ Z L, and std(n, L) = std(n, L)/C std(n,L) (Sym(n)).

Recognition Lemma. Let n ≥ 1, S := Sym(n), and V ∈ Mod(S, d, q) be faithful and dc-irreducible. Suppose that for any transposition τ , one has

[S τ ⊥ , B τ ] = 0.
Then for some abelian group L and arrow ϕ in U, there is a surjective morphism ϕ : std(n, L)

V of S-modules. Moreover:

• if 0 = q | n, then ker ϕ = C std(n,L) (S) and V std(n, L); • if 0 = q n, then ker ϕ = 0 and V std(n, L) std(n, L); • if q = 0, then ker ϕ = c ⊗ K where c = n-1 i=1 f i and K ≤ Ω n (L) is 0-dimensional.
Remark. In the first two cases, V is completely reducible into a direct sum of isotypical summands std(n, F q ); when V is torsion-free, the same is true with summands of the form std(n, Q). However, actions on tori could give rise to non-trivial quotients (in characteristic 0).

Proof. The case of n ≤ 2 is clear, so we suppose n ≥ 3. By dc-irreducibility, V = [S, V ] = n-1 i=1 B (in) ; this will be used several times below. Claim 1 (local equations). Let i = j. Then ad

(ij) = 1 -(ij) acts as:    2 on B (ij) (jk) on B (ik) for k / ∈ {i, j} 0 on B (k ) for {k, } ⊥ {i, j}.
Proof of Claim. This is obvious on B (ij) ; of course 2 = 0 is a possibility. We turn to the middle case. Let A := S . First, note that B (ij) ∩ B (jk) is S-invariant. Indeed, it is S {i,j,k} -invariant as it is inverted by both (ij) and (jk), and by our main assumption, it is also centralised by A (ij) ⊥ , A (jk) ⊥ = A j ⊥ , implying S-invariance. Now let a ∈ B (ik) , and write:

[1 -(ij)]a ∈B (ij) -(jk)a ∈B (ij) = [1 -(ij) -(jk)]a = -[1 -(ij) -(jk)](ik)a = -[(ik) -(jk)(ik)]a ∈B (jk) + (ikj)a ∈B (jk) . Thus, X := [1 -(ij) -(jk)]B (ik) is a dim-connected subgroup of B (ij) ∩ B (jk) . As noted above, B (ij) ∩ B (jk) is S-invariant,
so σ∈S σX is a dimconnected and S-invariant subgroup of B (ij) ∩ B (jk) < V , hence trivial by dc-irreducibility. Hence X = 0, as desired.

It remains to verify the third equation, for which we may assume n ≥ 4. As [A τ ⊥ , B τ ] = 0, each τ ⊥ τ gives rise to the same B + τ := tr τ (B τ ), which is centralised by S τ ⊥ . We aim to show

B + τ = B τ . For s ∈ S, s(B + (ij) ) = B + (s(i)s(j)) , so V + := i =j B + (ij) is S-invariant. Notice how ad (ij) B + (k ) = ad (ij) tr (ij) (B (k ) ) = 0
, and (by the second local equation)

ad (ij) (B + (ik) ) = (jk)B + (ik) = (jk) tr (j ) (B (ik) ) = (jk) tr (j ) (jk)B (ij) = tr (k ) B (ij) = B + (ij) .
Thus, ad (ij) V + = B + (ij) . By dc-irreducibility, V + equals 0 or V , so either

B + (ij) = ad (ij) V + = 0 or B + (ij) = ad (ij) V + = ad (ij) V = B (ij)
. The latter is our goal, so it remains to consider B + (ij) = 0. Assume B + (ij) = 0. If q = 2, we have the desired result since 0 We construct a covering S-module as follows:

= B + (ij) = tr (k ) (B (ij) ) = ad (k ) (B (ij) ). If q = 2,
• let L := B (1n) as a trivial S-module;

• let V := std(n, L) as an S-module, and define f i = e i -e n as usual;

• let ϕ : V → V be the additive map such that ϕ(f i ⊗ ) = (1i) • , where (11) is interpreted as the identity.

Every element of V has a unique decomposition as n-1 i=1 f i ⊗ i for i ∈ L, so ϕ is indeed well-defined and additive. Note that in the image of ϕ, we use the non-trivial action of S on B (1n) ≤ V .

By the Universe Properties of § 2.1, perm(n, L) is an S-module in U, so each f i ⊗ L, being the image of perm(n, L) under ad (in) , is also in U. Taking the sum, we find that V is an S-module in U. Now let ϕ i : f i ⊗ L → V be (1i) • ι • π i where ι : L → V is the inclusion map and π i is the restriction to f i ⊗ L of the i th -projection of perm(n, L) = L n . Thus, each ϕ i is compatible so their common extension to V (which is ϕ) is as well.

Claim 2 (covering). ϕ is a surjective morphism of S-modules.

Proof of Claim. Notice that ϕ(f i ⊗ ) ∈ (1i) • B (1n) = B (in) . Therefore im ϕ ≥ n-1 i=1 B (in) = V ,
so ϕ is surjective. It remains to prove covariance. We use Claim 1 freely.

Since {(jn) : 1 ≤ j ≤ n -1} generates S and {f i ⊗ :

1 ≤ i ≤ n -1, ∈ L} generates V , it is enough to treat the basic cases. If i = j, ϕ((in) • (f i ⊗ )) = -ϕ(f i ⊗ ) = (in) • ϕ(f i ⊗ ). If i = j, then ϕ((jn) • (f i ⊗ )) = ϕ((f i -f j ) ⊗ ) = (1i) -(1j) . To compute (jn) • ϕ(f i ⊗ ), we consider separately j = 1 or not. If j = 1, (1n)(1i) = (1i)(in) = (1i)[(in) -1 + 1] = -(1i) 2 + (1i) , completing this case. And if j = 1, (jn)(1i) = (1i)(jn) = (1i)[(jn) -1 + 1] = -(1i)(1j) + (1i) ,
which establishes this case since (1j) ∈ B (jn) is centralised by (1i). ♦ Claim 3 (kernel control).

• If 0 = q | n, then ker ϕ = C V (S) and V std(n, L); • if 0 = q n, then ker ϕ = 0 and V std(n, L) std(n, L);

• if q = 0, then ker ϕ = c ⊗ K where c = n-1 i=1 f i and K ≤ Ω n (L) is 0-dimensional. Proof of Claim. Let c = n-1 i=1 f i ∈ std(n, Z). We first contend that: ϕ -1 (C V (S)) = C V (S) = {c ⊗ : ∈ L : n = 0} = c ⊗ Z Ω n (L) Ω n (L).
Notice at once that c generates the centraliser of Sym(1, . . . , n -1), in symbols

C std(n,Z) (S n ⊥ ) = c ; the same holds in std(n, L), viz. C V (S n ⊥ ) = c ⊗ L. Also, in std(n, Z) one sees (1n)c = c -nf 1 , so C V (S) = {c ⊗ : n = 0} = c ⊗ Ω n (L) Z ⊗ Z Ω n (L) Ω n (L). It remains to prove ϕ -1 (C V (S)) = C V (S). The latter is clearly contained in the former; we now show that v ∈ C V (S) is in the image of C V (S). As noted after Claim 1, v = n-1 i=1 v i with v i ∈ B (in)
. Applying ad (1i) and using the local computations, we see that (in

)v 1 + (1n)v i = 0, implying that v i = (1i)v 1 . Thus, v = n-1 i=1 (1i)v 1 = ϕ(c ⊗ v 1
). We may now finish the proof. Clearly ker ϕ ≤ ϕ -1 (C V (S)) = C V (S).

• First suppose q = 0 and q n. Then Ω n (L) and C V (S) ≥ ker ϕ are trivial, so V V = std(n, L). (The same holds if V is torsion-free.)

• Next suppose q = 0 and q|n.

Then L = Ω n (L) so C V (S) = c ⊗ L L; the image ϕ(C V (S)) = C V (S) is now a quotient module of L, hence dim-connected. By dc-irreducibility of V , C V (S) is trivial, so C V (S) = ker ϕ. Hence V V / ker ϕ = std(n, L).
(The last is seen by recalling that q is prime and std(n, F q ) is irreducible.) • Finally suppose q = 0. Recall that Ω n (V ) is 0-dimensional by the Divisibility Properties. Subgroups of c ⊗ Ω n (L) Ω n (L) are of the form c ⊗ K for K ≤ Ω n (L), and ker ϕ is one such. ♦

This completes the proof of the Recognition Lemma.

On can also rephrase Claim 2 of the Recognition Lemma as follows, with no reference to dimensionality.

Corollary. Let V be an abelian group equipped with an irreducible, faithful action of Sym(n). Suppose that for any two distinct transpositions τ, τ one has g∈ τ,τ ε(g)g = 0. Then V is a homomorphic image of some std(n, L).

Proof. The 'integration formula' implies [1-(ij)][1-(k )] = 0 (when τ ⊥ τ ) and [1 -(ij)][1 -(ik)] = (jk)[1 -(ik)] (when τ ⊥ τ ) in End(V ). Hence V satisfies the conclusion of Claim 1, which is enough to produce a covering map ϕ : std(n, Z) ⊗ Z B (1n) → V .
3.2. Extending an Alt(n)-module to Sym(n). We now turn to Alt(n)modules, giving a geometric condition (analogous to that for the Recognition Lemma) under which an Alt(n)-module extends to a Sym(n)-module subject to Recognition. But do note that the two lemmas, Recognition and Extension, are independent.

Extension Lemma (cf. [START_REF] Wagner | The faithful linear representation of least degree of Sn and An over a field of characteristic 2[END_REF][START_REF] Wagner | The faithful linear representations of least degree of Sn and An over a field of odd characteristic[END_REF]). Let n ≥ 7, A := Alt(n), and V ∈ Mod(A, d, q) be faithful and dc-irreducible. Suppose that for any bitransposition α, one has [A α ⊥ , B α ] = 0. Then:

• if q = 2 there is a unique compatible action of Sym(n) extending the Alt(n)-structure; • if q = 2 there are exactly two such, obtained from each other by tensoring with the signature. Moreover, up to tensoring with the signature, the extension satisfies that for any transposition τ ∈ S := Sym(n), one has [S τ ⊥ , B τ ] = 0.

Remark. If n ≥ 8 the main assumption is equivalent to: for any 3-cycle γ, one has [A γ ⊥ , B γ ] = 0. The 'γ'-version is however stronger if n = 7.

Proof. The bulk of the proof is devoted to existence; uniqueness will result afterwards. We aim to extend the action of Alt(n) to Sym(n); to that end, we first identify what should be B (ij) = [(ij), V ], which (up to tensoring with the sign representation) should be thought of as a line that we will call L (ij) . In the standard module, L (ij) can be computed as B α ∩ B γ using any bi-transposition α that swaps i and j and any 3-cycle γ satisfying |α| ∩ |γ| = {i, j}. (There are certainly other ways to isolate B (ij) such as by intersecting B α and B β for α and β distinct bi-transpositions that both swap i and j; this was in fact our original point of view.) Claim 1. Let i = j be given. For distinct a, b, k / ∈ {i, j},

L (ij) := im ad (ij)(ab) • ad (ijk) = (B (ij)(ab) ∩ B (ijk) ) 1
is nontrivial and independent of the choice of a, b, k. Also, [A {i,j} ⊥ , L (ij) ] = 0.

Proof of Claim. Consider L (ij) := ad (ij)(ab) B (ijk) for distinct a, b, k / ∈ {i, j}; set α := (ij)(ab) and γ := (ijk).

We first prove centralisation by A {i,j} ⊥ and independence from a, b, k. Note that the group (A γ ⊥ ) is generated by its bi-transpositions β (when n = 7, this fails of A γ ⊥ itself), which all commute with γ and satisfy:

ad β (B γ ) = ad β • ad γ (V ) = ad γ • ad β (V ) = ad γ (B β ) ≤ [A β ⊥ , B β ] = 0,
implying that [(A γ ⊥ ) , B γ ] = 0. Also notice that α inverts γ. In particular ad α leaves im ad γ invariant; hence L (ij) ≤ B α ∩ B γ , so by assumption and what we just noted, L (ij) is centralised by A α ⊥ , (A γ ⊥ ) = A {i,j} ⊥ (even if n = 7). And as A {i,j} ⊥ ≥ Alt(5) is 3-transitive off of {i, j}, we also find that L (ij) is independent of the choice of a, b, k.

We now show L (ij) = (B α ∩ B γ ) 1 , which will follow readily from Coprimality and dim-connectedness of

L (ij) = ad α • ad γ (V ). If char V = 2, then letting α act on B γ we find B γ = L (ij) (+) tr α (B γ ). Now B α ∩ tr α (B γ ) ≤ B α ∩ C α has dimension 0, so dim(B α ∩ B γ ) = dim(B α ∩ (L (ij) (+) tr α (B γ )) = dim L (ij) + dim(B α ∩ tr α (B γ )) = dim L (ij) . And if char V = 3, write V = B γ (+) C γ , an α-invariant decomposition. Applying ad α , we find that B α = L (ij) + ad α (C γ ) with B γ ∩ ad α (C γ ) being 0-dimensional. As before, we find that dim(B α ∩ B γ ) = dim L (ij) .
It remains to show that L (ij) is nontrivial, which is equivalent to showing that α does not centralise B γ . Suppose it does. Conjugating, α centralises B (ij ) for all / ∈ {i, j, a, b} and also B (ab ) for all / ∈ {i, j, a, b}. Thus, C V (α) contains both [A {a,b} ⊥ , V ] and [A {ij} ⊥ , V ], hence all of [A, V ]. This contradicts our assumptions of dc-irreducibility and faithfulness. ♦

Remark. There is a counterexample to Claim 1 when n = 6. In the case of the adjoint representation of Alt(6) PSL 2 (F 9 ), [A α ⊥ , B α ] = 0 and L (ij) has positive dimension, but L (ij) is not independent of the choice of a, b, k.

The next claim establishes various expected properties of the

L (ij) ; recall that L (ij) is a proxy for B (ij) .
Claim 2 (Geometry of lines).

(1)

V = i =j L (ij) . (2) If {i, j} = {k, } are distinct pairs, then L (ij) ∩ L (k ) = 0.
(3) If i, j, k, , x are distinct symbols, then:

• ad (ijk) (L (ix) ) = L (ij) ; • ad (ij)(k ) (L (ix) ) = L (ij) . (4) L (ij) ≤ L (ik) + L (jk) .

Proof of Claim. Clearly

i =j L (ij) is A-invariant so must be equal to V , establishing (1).

We now handle (2) and will prove that distinct lines are disjoint. We first consider disjoint index sets; by conjugacy, we may take i, j, k, to be 1, 2, 3, 4. Using Claim 1,

L (12) ∩ L (34) ≤ B (125) ∩ C V ((125)) ≤ Ω 3 (V ),
but on the other hand:

L (12) ∩ L (34) ≤ B (12)(56) ∩ C V ((12)(56)) ≤ Ω 2 (V ).
Thus, L (12) ∩ L (34) = 0, so lines with disjoint index sets are disjoint; we turn to intersecting sets. Notice that 123)). Hence:

L (12) ∩ L (23) ≤ B (12)(45) ∩ B (23)(45) ≤ C V ((
L (12) ∩ L (23) ≤ B (123) ∩ C V ((123)) ≤ Ω 3 (V ).
On the other hand L (12) ∩ L (23) ≤ B (12)(34) ∩ B (23)(14) ≤ C V ((13)(24)), so:

L (12) ∩ L (23) ≤ B (12)(34) ∩ [C V ((13)(24)) ∩ C V ((123))] ≤ B (12)(34) ∩ C V ((12)(34)) ≤ Ω 2 (V ).
Therefore L (12) ∩ L (23) = 0; distinct lines are disjoint.

We now prove (3). Here we take i, j, k, x to be 1, 2, 3, 4. Let α = (12)(56), β = (23)(56), and γ = αβ = (123). We show ad γ (L (14) ) = L (12) . Observe:

ad γ = 1 -αβ = 1 -β + (1 -α)β = ad β + ad α •β. ( * ) If v ∈ L (14) , then v ∈ C V (β), so applying ( * ) we find ad γ (v) = ad α (v) ∈ B α . Thus ad γ (L (14) ) ≤ (B γ ∩ B α ) 1 = L (12) , by Claim 1. And as γ • L (14) = L (24) , we have L (14) ∩ C V (γ) ≤ L (14) ∩ L (24) = 0 by (2), so L (14) ∩ ker ad γ is trivial. Thus, dim ad γ (L (14) ) = dim L (14) = dim L (12) , forcing ad γ (L (14) ) = L (12) .
For the second part of (3), we keep α, β, γ as before and show ad β L (24) = L (23) . Now, β = αγ, so here ad β = ad γ + ad α •γ. But ad α •γ(L (24) ) = ad α (L (34) ) = 0, so by our previous work ad β (L (24) ) = ad γ (L (24) ) = L (23) .

For (4), consider H := Σ

V I := (i,j)⊆I L (ij) .
Claim 3. Let I, J ⊆ {1, . . . , n} with |I|, |J| ≥ 2. The following hold:

(1) V {1,...,n} = V ;

(2

) if |I ∩ J| ≥ 1, then V I∪J = V I + V J ; (3) if |I| ≤ n -3, then V I ∩ L (ab) = 0 for a, b / ∈ I; (4) the sum L (12) + L (23) + • • • + L (n-2,n-1) is direct; (5) [A I ⊥ , V I ] = 0; (6) [A I , V ] = V I provided |I| ≥ 3.
Proof of Claim. Part (1) is merely Claim 2(1), and (2) follows readily from Claim 2(4). Parts (5) and (6) are also fairly immediate. Indeed, [A I ⊥ , V I ] = (i,j)⊆I [A I ⊥ , L (ij) ] = 0 by Claim 1, and if |I| ≥ 3, then

[A I , V ] = |γ|⊆I B γ = (i,j,k)⊆I L (ij) + L (jk) = (i,j)⊆I L (ij) .
For (3), choose distinct a, b, k / ∈ I; set X := L (ab) ∩ V I . Let i ∈ I \ {a, b, k} and γ := (iak). Then using all of Claim 2 we find ad γ (X) ≤ L (ak) ∩L (ia) = 0, so X ≤ L (ab) ∩ γ(L (ab) ) = L (ab) ∩ L (kb) = 0. This establishes (3), and (4) now follows readily by induction. ♦

For fixed i = j, let E i,j := {α | α is a bi-transposition exchanging i and j}.

The final ingredient we need to define the action of a transposition τ is the hyperplane representing C V (τ ), which we now define as H τ . (Regarding our definition below, recall that C α = tr α V , and though always contained in C V (α), it may be significantly smaller.)

Claim 4. Let H (ij) := α∈E i,j C α . For distinct k, a, b / ∈ {i, j}, we have V = V {i,j,k} + H (ij) with V {i,j,k} ∩ H (ij) ≤ C V ((ij)(ab)). Further, V {i,j} ⊥ ≤ H (ij) .
Proof of Claim. We first show V {i,j} ⊥ ≤ H (ij) . Let (k ) ⊥ (ij). If q = 2, then since ad α = tr α for every bi-transposition α, one has

L (k ) ≤ B (ij)(k ) = C (ij)(k ) ≤ H (ij) . If q = 2, then taking (x, y) ⊥ {i, j, k, }, one has by 2-divisibility L (k ) ≤ C (ij)(xy) ≤ H (ij) . In either case, V {i,j} ⊥ ≤ H (ij)
. Moreover, any two α, β ∈ E i,j agree modulo [A {i,j} ⊥ , V ], so using Claim 3(6),

C β ≤ C α + [A {i,j} ⊥ , V ] = C α + V {i,j} ⊥ .

Thus, we have in fact shown

H (ij) = C α + V {i,j} ⊥ for any α ∈ E i,j .
Fix distinct k, a, b / ∈ {i, j}. Since V {i,j} ⊥ ≤ H (ij) , Claim 3(1,2) imply V = V {i,j,k} + H (ij) . Also, for α = (ij)(ab), by Claim 2(3) one has ad α (V {i,j,k} ) ≤ L (ij) , and

ad α H (ij) = ad α (C α + V {i,j} ⊥ ) = ad α (C α + V {i,j,a,b} ⊥ ≤C V (α)
+V {k,a,b} ) ≤ V {k,a,b} .

By Claim 3(3), L (ij) ∩ V {k,a,b} = 0, so ad α (V {i,j,k} ∩ H (ij) = 0, meaning

V {i,j,k} ∩ H (ij) ≤ C V (α). ♦ Remark. If q = 2, then one even has V = H (ij) + L (ij) , while if q = 2 then L (ij) ≤ H (ij)
. We however give a characteristic-independent endgame, treating reflections and transvections at once.

Claim 5. There are compatible involutive operators {τ (ij) : i = j} ⊂ Aut U (V ) such that for Σ := τ (ij) : i = j and S := Sym(n) we have:

• the map (ij) → τ (ij) extends to an isomorphism S Σ with A = Σ ; • the image of S (ij) ⊥ centralises B τ (ij) .
Proof of Claim. Fix any k / ∈ {i, j}. By Claim 3(5) V {i,j,k} is centralised by A (ijk) ⊥ , so every choice of α = (ij)(ab) with a, b / ∈ {i, j, k} yields the same action on V {i,j,k} . Using Claim 4, define τ (ij) (currently depending on k) to agree with any such α on V {i,j,k} while centralising H (ij) , which makes sense as the intersection lies in C V (α). Also, notice how . Disjointness of lines proves t (13) a = a 23 , and similarly we find t (23) a = a 13 . Thus, a 23 ∈ L + (23) and a 13 ∈ L + (13) , as desired. We claim that t (ij) inverts L (ij) and centralises V {i,j} ⊥ . Let V + :=

[τ (ij) , V ] = [τ (ij) , V {i,j,k} ] = [α, L (ik) + L (jk) ] = L (ij)
i =j L + (ij)
. It is Alt(n)-invariant so equals 0 or V by dc-irreducibility. And from our work above,

V + = L + (12) + L + (23) + • • • + L + (n-1,n) .
Since we are assuming 2 + ≤ , Claim 3(1,4) now yields:

dim V + ≤ (n -1) + ≤ (n -1) 2 < (n -2) ≤ dim V, so V + < V , implying V + = 0. In particular, t (ij) inverts L (ij)
, and then by Claim 1, we see that for (k )

⊥ (ij), t (k ) = t (ij) (ij)(k ) centralises L (ij) .
We now prove that t (ij) and τ (ij) agree everywhere. By Claim 3(1,2), V = V {i,j,k} + V {i,j} ⊥ , with both maps centralising the latter term. By definition, τ (ij) acts on V {i,j,k} as (ij)(ab) for any (ab) ⊥ (ij), and as t (ab) centralises V {i,j,k} , t (ij) also acts on V {i,j,k} as (ij)(ab). We are done. ♦

This completes the proof of the Extension Lemma.

Remark. Uniqueness does not hold without assuming compatibility of the action of S: one could break L = L + ⊕ L -into abstract pieces (e.g. viewing the complex field as a real vector space), producing a non-compatible decomposition V = V + ⊕ V -, one being the sign-tensored version of the other.

3.3. The Geometrisation Lemma. Here we establish that Sym(n)-and Alt(n)-modules of sufficiently low dimension necessarily carry the geometric structure of the standard module, leading to their identification by our previous work. We begin by treating the minimal case for Sym(n). The authors disagree on the significance and relevance in the present paper of isolating this situation; it is to the reader to decide. Nevertheless, the First Geometrisation Lemma will be used in the proof of the significantly more powerful Geometrisation Lemma. Do note that the special case addressed here holds for all n (trivially for n < 3). As a result, we need to account for Sym(6)-modules that are 'quasi-equivalent' to std(6, 2) via a twisting of the module by an outer automorphism of Sym(6).

First Geometrisation Lemma. Let n ≥ 3, S := Sym(n), and V ∈ Mod(S, d, q) be faithful. Then d ≥ n -2; if equality holds, then n ≥ 5, and for any transposition τ ∈ S, we are in one of the following cases:

• up to tensoring with the signature, V satisfies [S τ ⊥ , B τ ] = 0; • n = 6, q = 2, and up to composing the action with some σ ∈ Out(Sym(6)), V satisfies [S τ ⊥ , B τ ] = 0.

Remarks.

• The (reduced) standard module is not faithful when n = 3, 4.

• The proof requires going to quotients, so dealing with objects which are not subobjects. See Remarks on p. 6.

Proof. Let S = Sym(n) and A = S . We begin with the soluble cases.

Claim 1. If n ∈ {3, 4}, then d ≥ n -1.
Proof of Claim. Suppose d ≤ n -2. If n = 3, then the action cannot be faithful since involutions must either centralise or invert a 1-dimensional dc-module, forcing A to act trivially.

If n = 4, then for the same reason d = 2, and if q = 2, then the Weight Lemma yields a contradiction. So if n = 4, then q = 2, and involutions act quadratically (with (i -1) 2 = 0). By faithfulness, for bi-transpositions α = β, one has dim B α = dim B β = 1. Thus, B α = (C V (α)) 1 , and letting α act on B β , one finds

B β ≤ (C V (α)) 1 , forcing B α = B β . Hence, B α = [A , V ] is S-invariant, but then A centralises the 1-dimensional dc-modules [A , V ] and V /[A , V ],

contradicting faithfulness (and non-nilpotence of A). ♦

From now on n ≥ 5. We may thus also suppose that d is minimal such that V is faithful; consequently, V is now dc-irreducible.

Assume d ≤ n -2; we shall identify the module. Let b = dim B τ , which does not depend on the transposition τ .

Claim 2 ('up to tensoring'). We may assume 2b ≤ d.

Proof of Claim. If q = 2 then τ acts quadratically, viz. (τ -1) 2 = 0. Hence B τ = im(1 -τ ) ≤ ker(1 -τ ). On the other hand, dim ker(1 -τ ) + dim im(1τ ) = d; hence 2b ≤ d. (This argument already appeared in the Extension Lemma, Claim 6; it will be used again in the Geometrisation Lemma, Claim 2.) If q = 2 then by the Coprimality Lemma, V = B τ (+) C τ . Tensoring with ε exchanges τ with -τ , hence B τ with C τ , so up to tensoring, we may assume dim B τ ≤ dim C τ and 2b ≤ d. Moreover, by Claim 1, S τ ⊥ Sym(4) is faithful on neither B τ nor V /B τ , so bi-transpositions in S τ ⊥ are quadratic. Thus q = 2, and it remains to deal with this exotic configuration. We claim that for any 3-cycle γ one has B γ = V . We have seen that K τ ⊥ = A τ ⊥ is trivial on each of B τ and V /B τ , so on each factor, all 3-cycles from S τ ⊥ have the same action. Let γ be one such 3-cycle. Notice how B τ = ad γ (B τ ) (+) tr γ (B τ ) is an S τ ⊥ -invariant decomposition, so if either factor is 1-dimensional, then γ ∈ A τ ⊥ centralises both, a contradiction. So, ad γ (B τ ) has dimension 0 or 2, and the analogous statement holds (with analogous proof) for V /B τ . Thus, dim B γ is 2 or 4. If γ centralises V /B τ then B γ ≤ B τ and equality holds. Then B γ = B τ = B γ for any other 3-cycle in S τ ⊥ ; conjugating, we contradict dc-irreducibility. If γ centralises B τ we find C γ = B τ = C γ , again a contradiction. This shows B γ = V . Now let γ, δ be disjoint 3-cycles and ε = γδ. If B ε = V as well, then tr γ = tr δ = tr ε = 0, forcing: We now present our main geometrisation result.

0 = 1 + ε + ε 2 = 1 + γδ + (-1 -γ)(-1 -δ) = γ + δ,
Geometrisation Lemma. Let A = Alt(n) and V ∈ Mod(A, d, q). Suppose d < n and that either • q = 2 and n ≥ 10; or • q = 2 and n ≥ 7. Then for any bi-transposition α ∈ A one has [A α ⊥ , B α ] = 0.

Proof. We want to show that A α ⊥ acts trivially on B α ; by conjugacy the desired property does not depend on α. The proof will by induction on n, but methods depend on the value of q. Let b = dim B α .

Claim 1 (odd case). If q = 2, then we are done.

Proof of Claim. Let K = {1, α i , α j , α k } be a four-group of bitranspositions on the same support. Let Λ = {λ 0 , λ i , λ j , λ k } be the weights with λ 0 = 1 and λ i = α ∨ i , viz. λ i (α i ) = 1 while λ i (α j ) = λ i (α k ) = -1. By the Weight Lemma there is a decomposition, in obvious notation,

V = + λ∈Λ V λ .
Notice at once B α i = V λ j + V λ k . Of course, A α ⊥ Alt(n -4) normalises all weight spaces. Actually, we have more. There is A α ⊥ < Σ i < C A (α i ) with Σ i Sym(n -4); the group Σ i normalises V λ 0 and V λ i while swapping V λ j and V λ k . (Typically, if α i = (12)(34), we take Σ i = Σ (12) α ⊥ , which exchanges α j = (13)(24) and α k = (14)(23), hence also the relevant weights.) Now, suppose that A α ⊥ does not centralise B α . Then there is i such that A α ⊥ does not centralise V λ i . The action extends to one of Σ i .

Let a = dim V λ i , which does not depend on i = 0. As we are assuming [A α ⊥ , V λ i ] = 0, we find 1 < a ≤ d 3 .

• The case n = 7 requires scrutiny. The only nontrivial subcase is when d = 6, a = 2, and V = V λ i + V λ j + V λ k . Also Σ i Sym(3) acts on V λ i , and the action of Σ i = A α ⊥ is nontrivial. Therefore an involution τ ∈ Σ i will satisfy dim[τ, V λ i ] = 1. As Σ i swaps V λ j and V λ k , which have dimension 2, dim[τ, V λ j + V λ k ] = 2. As a conclusion, dim B τ (V ) = 3, but τ is a bitransposition of A, hence dim B τ = 3 = 2a, a contradiction.

• The same argument removes n = 8. (A possible 1-dimensional V λ 0 causes no problem.) • If n ≥ 9, the action of Σ i on V λ i is faithful, and the First Geometrisation Lemma yields n -6 ≤ a ≤ d 3 ≤ 1 3 (n -1), a contradiction. Hence A α ⊥ centralises B α , as claimed. ♦

Remark. Assuming V is faithful, one can even show a = 1 fairly directly; however it also is a consequence of applying Extension and Recognition.

Claim 2 (even case). If q = 2 then we are done.

Proof of Claim. Suppose the action of A α ⊥ on B α is not trivial. As in Claim 1, this action extends to a faithful action of some Σ Sym(n -4) with A α ⊥ < Σ < C A (α ⊥ ).

Let b := dim B α , for α a bitransposition; by quadraticity, b ≤ d 2 . (See the proof of the First Geometrisation Lemma, Claim 2).

• The case n = 10 requires a close look. Here, b ≤ 4.

Observe that Σ is also faithful on V /B α . If not, then any disjoint bi-transposition β ⊥ α will satisfy B β ≤ B α , whence equality; conjugating, we find that B α is A-invariant and centralised by all bi-transpositions, a contradiction. Hence B α , V /B α ∈ Mod(Σ) are both faithful.

By the First Geometrisation Lemma, the Σ-module B α is known (be careful that dim(V /B α ) could be 5). Now choose γ = (ijk) ∈ A α ⊥ . Inspection of the possible structures for B α ∈ Mod(Σ, 4, 2) gives that [γ, B α ] has dimension 2 if B α is the standard module; for the exotic twist, the dimension is 3. Looking at V /B α , we certainly have dim[γ, V /B α ] ≥ 2; otherwise, for any (ij)(ab) ∈ A α ⊥ inverting γ, the group γ, (ij)(ab) Sym(3) does not act faithfully on the 1dimensional [γ, V /B α ], a contradiction since q = 3. Using dim B γ = dim[γ, V ] = dim[γ, B α ] + dim[γ, V /B α ] by the Coprimality Lemma, we find that 4 ≤ dim B γ < d.

Turn to the action of Σ (ij) γ ⊥ Sym(7) on both B γ and C γ . It must be faithful on at least one, so by the First Geometrisation Lemma, one has dimension at least 6 (and the other at most 3). All this shows dim B γ ≥ 6 and dim C γ ≤ 3. In particular A γ ⊥ centralises C γ , so C γ = C γ for any 3-cycle γ ⊥ γ. Conjugating, we find a contradiction.

• Assume n ≥ 11. The action of Σ on B α is faithful, so the First Geometrisation Lemma yields n -6 ≤ b ≤ d 2 ≤ 1 2 (n -1). This is an immediate contradiction when n ≥ 12, and when n = 11, the First also thanks the Hausdorff Institute for Mathematics (HIM) in Bonn, and Fromagerie Quatrehomme for providing cheese during all Paris stays and trips abroad.

The third author also benefited immensely from the generous support of HIM and Fromagerie Quatrehomme. Additionally, he is grateful to the Oberwolfach Research Institute for Mathematics (MFO) for his stay in January 2020 during the program on Model Theory: Groups, Geometries and Combinatorics; it was there that the Recognition Lemma was born. The work of the third author was partially supported by the National Science Foundation under grant No. DMS-1954127 as well as by the Creative Activity Faculty Awards Program and the College of Natural Science and Mathematics at California State University, Sacramento.

All authors wish to thank Tuna Altınel, the first colleague with whom we discussed the new 'dimensional' setting. A Turkish mathematician and maître de conférences at Université Claude Bernard Lyon 1, Altınel remained a de facto prisoner of the Republic of Turkey for over two years. His passport was locked up by the Turkish authorities from April 2019 to June 2021; for 81 of those days, he was as well. The second author met with Altınel in İstanbul in January 2020; Gregory Cherlin was also there. The occasion was a hearing in Altınel's trial: but high spirits always welcome research seminars, despite political prosecution and judicial harassment. Thanks to international protest, including uprise from various scientific societies, Altınel was acquitted in two distinct cases, all appeals were settled by September 2020, and, with incredible delay, his passport was then returned over nine months later. However, his freedom of travel is still under under judicial threat. Details on math.univ-lyon1.fr/SoutienTunaAltinel. Paris Album No. 2 is dedicated to those laboring for peace and freedom of speech.

  then Coprimality implies that B (ij) = B (k ) ; conjugating, this quickly contradicts dc-irreducibility when n ≥ 5. If n = 4, then V = B (12) + B (23) + B (34) ; sinceB (12) = B (34) , one has V = B (12) + B (23) . Moreover, V = B (12) + B (23) is a K-invariant decomposition (for K the Klein four-group) with the first factor centralised by (12)(34) and the second by (23)(14). A weight space decomposition of V for K must then result in the trivial weight space having positive dimension (by the Weight Lemma), against dc-irreducibility and faithfulness. ♦

  , and observe that [H, B (123) ] = ad (13)(45) (B (123) ) + ad (23)(45) (B (123) ) = L (13) + L (23) . Of course, [H, B (123) ] also contains ad (12)(45) (B (123) ) = L (12) . ♦ For I ⊆ {1, . . . , n} with |I| ≥ 2, define

♦

  Claim 3. If n ∈ {5, 6}, we are done. Proof of Claim. Let n ≤ 6 and τ be a transposition. If b = 1, then S τ ⊥ must centralise or invert B τ , forcing A τ ⊥ to centralises B τ , so we may assume b ≥ 2. Since d ≥ 2b, we have reduced to the case of n = 6, d = 4, and b = 2.

a

  clear contradiction. So B ε < V . Since the outer (class of) automorphism swaps (the class of) γ with (that of) ε, we deduce that twisting by σ, we no longer have b = 2. Hence, in this exotic case as well, we reduced to b = 1, implying that A τ ⊥ centralises B τ . ♦ Claim 4. If n ≥ 7, we are done. Proof of Claim. Let τ be a transposition. By induction, A τ ⊥ centralises B τ , as the alternative implies n -4 ≤ b ≤ 1 2 (n -2). ♦ This completes the first geometrisation argument.

  by Claim 2(3,4). and t (23) a = a 13 ; then, as t (13) (L + (12) ) = L + (23) and t (23) (L + (12) ) = L + (13) , we will be done. Now, using a 23 ∈ L (23) ≤ B (123) ≤ ker tr (123) from Claim 2(3): 0 = a 23 + (123)a 23 + (123) 2 a 23 = a 23 + t (13) t (12) a 23 + (132)a 23 = a 23 + t (13) a 13 + (132)a 23

	= a 23	+ t (13) a	-t (13) a 23	+ (132)a 23
	∈L (23)	∈L (23)	∈L (12)	∈L (12)

Acknowledgements

This is: Paris Album No. 2. Research began in early Summer 2018 in Paris by the last two authors who treated the lower bound (but not recognition) for Sym(n); in Fall 2018, the first two met in Bogotá and the last two in Bonn to attack Alt(n). All three authors met in Paris in early Summer 2019 to clarify the approach; Recognition and Extension (as they now appear) evolved throughout 2020. The general context was only finalized recently.

The second author wishes to thank both the cimpa network (Centre international de mathématiques pures et appliquées) and the ecos-Nord programme for sponsoring visits to Universitad de los Andes in Bogotá. He

We claim that our definition of τ (ij) does not depend on the choice of k. Fix distinct k, k / ∈ {i, j}, with corresponding τ (ij) and τ (ij) . Consider the decomposition given by Claim 3(1,2):

Note that V {i,j,k,k } = V {i,j,k} +L (kk ) = V {i,j,k } +L (kk ) . For a, b / ∈ {i, j, k, k }, τ (ij) and (ij)(ab) agree on V {i,j,k} (by definition) and on L (kk ) , which they both centralise since L (kk ) ≤ V {i,j} ⊥ ≤ H (ij) by Claim 4. Hence τ (ij) and (ij)(ab) agree on V {i,j,k,k } . The same holds for τ (ij) , so

, they agree globally. This completes the definition of the transpositions; they are in U by the existence of compatible extensions from the Universe Properties.

We now verify that τ (ij) τ (jk) = (ijk). Fix distinct , a, b / ∈ {i, j, k}. Similar to as above (with replacing k ), consider

Since we may define τ (ij) and τ (jk) using the common parameter , we find as before that τ (ij) acts on V {i,j,k, } = V {i,j, } + L (k ) = V {j,k, } + L (i ) as (ij)(ab) and τ (jk) as (jk)(ab). Since each of τ (ij) , τ (jk) , and (ijk) centralise V {i,j,k} ⊥ , we are done.

These relations (which also imply that disjoint transpositions commute) guarantee Σ Sym(n) with Σ = A. As noted, B τ (ij) = L (ij) , so distinct transpositions give rise to disjoint brackets, which clearly implies that the image of

This proves existence and it remains to deal with uniqueness.

Claim 6. If S ≤ Aut U (V ) represents any compatible action of Sym(n) extending Alt(n), then up to tensoring with the signature, S = Σ.

Proof of Claim. Say that S is generated by transpositions t (ij) . We prove that, up to tensoring, each transposition t (ij) coincides with τ (ij) as defined in Claim 5. Clearly for any s ∈ S one has s(L (ij) ) = L (s(i)s(j)) . Consider the action of t = t (ij) on L t = L (ij) ; write L + t = tr t (L t ). Of course = dim L t and + = dim L + t do not depend on t; as a matter of fact for s ∈ S we still have

. We shall prove that up to tensoring with the signature, L + t = 0 for any transposition t ∈ S. First, we show that, up to tensoring, we may assume 2 + ≤ . Here is an argument we shall repeat in the proof of the First Geometrisation Lemma, Claim 2. In characteristic 2, one has (t + 1) 2 = 0, so im(1 + t) ≤ ker(1 + t). Also, restricting t to L t , we have that dim ker(1 + t) + dim im(1 + t) = dim L t = ; hence 2 + ≤ . In characteristic not 2, up to tensoring, we may exchange tr t with ad t , hence tr t (L t ) with ad t (L t ), and therefore assume the same. So up to tensoring (to no effect if q = 2), 2 + ≤ . Under this assumption, we prove t (ij) = τ (ij) , viz. uniqueness.

We first contend L + (12) ≤ L + (13) + L + (23) . Let a ∈ L + (12) ; by Claim 2(4) there is a decomposition a = a 13 +a 23 with a 13 ∈ L (13) and a 23 ∈ L (23) . Notice how a = t (12) a = t (12) a 23 +t (12) a 13 , so as L (13) ∩L (23) = 0, we find that t (12) swaps a 13 and a 23 . Our approach is to show that we similarly have t (13) a = a 23 Geometrisation Lemma further implies that q divides n -4 = 7, again a contradiction. ♦

This completes the proof of the Geometrisation Lemma.

Remark. Although we are not able to identify the minimal faithful V ∈ Mod(Alt(9), d, 2) (bearing in mind that there are three such in the classical setting), we can still easily show that the minimal dimension is n-1. Indeed, with notation as above, notice first that if the action of A α ⊥ on B α is trivial, then Extension followed by Recognition shows that d ≥ n -1 (since 2 9). And if the action of A α ⊥ on B α is not trivial, the First Geometrisation Lemma (applied to Σ ∼ = Sym(5)) shows that b ≥ 4 (since 2 5), so by quadraicity, 4 ≤ b ≤ d 2 , as desired. 3.4. Assembling the Theorem.

Proof of the Theorem. Let S := Sym(n) and A := Alt(n). Suppose G = A or S and V ∈ Mod(G, d, q) is faithful and dc-irreducible with d < n. We always assume n ≥ 7; if G = Alt(n) and q = 2, we further assume n ≥ 10.

We first assume that V is dc-irreducible as an A-module. In this case, we are done except when V ∈ Mod(S, d, 2) with 7 ≤ n ≤ 9. This follows immediately from applying (in order) the Geometrisation Lemma, the Extension Lemma, and the Recognition Lemma. We are also using that d < n forces dim L = 1 (with notation as in the Recognition Lemma) and d = n -2 or n -1, according to the value of q.

We next address when V ∈ Mod(S, d, τ centralises B τ , and the Recognition Lemma applies. So it remains to dispose of the case when V is dc-irreducible as an Smodule but not as an A-module. Assume this is the case, and consider a dcirreducible A-series for V . By our work above, each factor is either a trivial module or is standard, and by simplicity of A, some factor is nontrivial, hence of dimension at least n -2. Thus d = n -1, and there must be an A-submodule W ≤ V such that either dim W = n -2 and dim V /W = 1 or vice versa. The former implies W = [A, V ], against dc-irreduciblity of V as an S-module. The latter implies W ≤ C V (A), hence equality by the structure of V /W , again violating dc-irreduciblity as an S-module.