Pierre Baumann

The q-Weyl group of a q-Schur algebra

Keywords:

come

The q-Weyl group of a q-Schur algebra

Introduction

Let G be a complex reductive Lie group, g its Lie algebra, and W the Weyl group (relative to some torus and some Borel subgroup B), with its standard generators {s 1 , . . . , s ℓ }. Let W be the braid group of W , that is the group with generators {s 1 , . . . , sℓ } and relations si sj si • • • = sj si sj • • • , with m ij factors in each side, (m ij ) being the Coxeter matrix. There are several morphisms of W into G (see [START_REF] Tits | Normalisateurs de tores : I. Groupes de Coxeter étendus[END_REF]), through which W acts on the integrable gmodules, and hence (by the adjoint action) on the enveloping algebra U(g). One can imbed U(g) and the group algebra CG into a bigger algebra U(g), so that these actions of W on U(g) become the restrictions of inner automorphisms of U(g).

Let U q (g) be the Drinfeld-Jimbo quantization of U(g) : this is a Hopf algebra over the field Q(v) of rational functions (with q = v 2 ). Lusztig, Levendorskii and Soibelman have defined invertible elements s1 , . . . , sℓ in some completion U q (g) of U q (g). These elements define a morphism W → U q (g) × , so that W acts on the integrable U q (g)-modules, this construction being a deformation of the classical case.

One of the results of Lusztig, Levendorskii and Soibelman is that if g is of type A, D or E, and if M is the q-deformation of the adjoint g-module, then the action of W on the zeroweight subspace of M satisfies quadratic relations and factorizes through the Hecke algebra of W (at the value q of the parameter). This result may be proved by a simple computation [START_REF] Lusztig | On quantum groups[END_REF]§ 1][20,Sect. 4].

Another way to see that is the following. The Hecke algebra admits a geometric realization in terms of double B-cosets in G, hence in terms of G-orbits of pairs of flags 1 (f, f ′ ) ∈ (G/B) 2 . In the same vein, Beilinson, Lusztig and MacPherson [START_REF] Beilinson | A geometric setting for the quantum deformation of GL n[END_REF] have constructed certain finite-dimensional algebras S d q (m) in terms of the geometry of relative positions of two m-step filtrations in a d-dimensional vector space, and have shown that these algebras were epimorphic images of U q (gl m ). (Du [8] has observed that the algebra S d q (m) was isomorphic to the q-Schur algebra of Dipper and James, the construction of Beilinson, Lusztig and MacPherson being the translation in a geometric framework of the original algebraic definition.) It is easy to see that S d q (m) is a quotient not only of U q (gl m ) but even of U q (gl m ). It therefore makes sense to consider the images in S d q (m) of the generators s1 , . . . , sm-1 of the q-Weyl group of U q (gl m ). Despite the complicated original definition of the si , these images have a very simple description, that enables us to give a perhaps more conceptual proof (in case A) of the result of Lusztig, Levendorskii and Soibelman recalled above.

Section 1 of this paper is aimed at giving a unified treatment of the basic facts concerning the quantized enveloping algebras U q (gl m ), the q-Schur algebras S d q (m), and the spaces S d q (M * m,n ) of homogeneous functions of degree d on the quantum spaces of matrices of size m × n. Section 1 is supplemented by an appendix explaining how to deduce from a simple argument (given in the non-quantum case by J. A. Green [START_REF] Green | Combinatorics and the Schur algebra[END_REF]) the standard basis theorems for S d q (M * m,n ) and S d q (m) (proved algorithmically in [START_REF] Huang | Standard basis theorem for quantum linear groups[END_REF] and [START_REF] Green | A straightening formula for quantized codeterminants[END_REF]). In Section 2, we recall the definition of the q-Weyl group of U q (gl m ) and explain how to compute the images of the generators s1 , . . . , sm-1 under the epimorphism U q (gl m ) ։ S d q (m). Our main result there is Theorem 3. Section 3 deals with applications of this result. We explain how to recover the quantum Schur-Weyl duality from a quantum (gl m , gl n )-duality and how all this is related to the aforementioned result of Lusztig, Levendorskii and Soibelman.

q-Deformations

Basic notations

Let m be a positive integer. We denote by P m the free Z-module with basis (ε 1 , . . . , ε m ). Elements of P m are called weights. A weight λ = λ 1 ε 1 + • • • + λ m ε m is said to be polynomial (respectively dominant) if all the λ i are non-negative (respectively if

λ 1 ≥ • • • ≥ λ m ). The degree of a weight λ = λ 1 ε 1 + • • • + λ m ε m is the integer |λ| = λ 1 + • • • + λ m .
The simple roots are the elements α i = ε i -ε i+1 (for 1 ≤ i ≤ m -1). We endow P m with the bi-additive form P m × P m → Z defined by (ε i |ε j ) = δ ij (Kronecker's symbol). The symmetric group on m letters S m acts on the set {ε 1 , . . . , ε m }, hence acts Z-linearly on P m . The image W of S m in Aut(P m ) is generated by the simple reflections s i :

P m → P m , λ → λ -(α i |λ)α i (for 1 ≤ i ≤ m -1).
Our ground ring will be the field Q(v) of rational functions in v over the rational numbers, although most of our constructions are valid over an arbitrary commutative base ring. We put q = v 2 . The q-numbers, q-factorials and q-binomial coefficients are defined as in [START_REF] Lusztig | Introduction to quantum groups[END_REF]:

[n] = v n -v -n v-v -1 , [n]! = n i=1 [i], n k = [n]! [k]![n-k]! .

Quantized enveloping algebra

Let m be a positive integer and let (a ij ) 1≤i,j≤m-1 be the Cartan matrix: a ij = (α i |α j ).

Definition 1 [START_REF] Jimbo | A q-analogue of U(gl(N + 1)), Hecke algebra and the Yang-Baxter equation[END_REF] The quantized enveloping algebra U q (gl m ) is the Q(v)-algebra presented by the generators K λ (for λ ∈ P m ), E 1 , . . . , E m-1 and F 1 , . . . , F m-1 , with the relations:

K λ K µ = K λ+µ (for λ, µ ∈ P m ), K λ E i = v (λ|α i ) E i K λ and K λ F i = v -(λ|α i ) F i K λ (for λ ∈ P m and 1 ≤ i ≤ m -1), 1-a ij r=0 (-1) r 1-a ij r E r i E j E 1-a ij -r i = 0 (for 1 ≤ i = j ≤ m -1), 1-a ij r=0 (-1) r 1-a ij r F r i F j F 1-a ij -r i = 0 (for 1 ≤ i = j ≤ m -1), [E i , F j ] = δ ij K α i -K -α i v -v -1 (for 1 ≤ i, j ≤ m -1).
It is a Hopf algebra when endowed with the coproduct ∆ and the counit ε given by:

∆(K λ ) = K λ ⊗ K λ , ε(K λ ) = 1 (for λ ∈ P m ), ∆(E i ) = E i ⊗ K α i + 1 ⊗ E i , ε(E i ) = 0 (for 1 ≤ i ≤ m -1), ∆(F i ) = F i ⊗ 1 + K -α i ⊗ F i , ε(F i ) = 0 (for 1 ≤ i ≤ m -1).
The natural U q (gl m )-module is the vector space V m with basis (e 1 , . . . , e m ), and the action of U q (gl m ) is given by:

K λ • e k = v (λ|ε k ) e k (for λ ∈ P m and 1 ≤ k ≤ m), E i • e k = δ i+1,k e i and F i • e k = δ i,k e i+1 (for 1 ≤ i ≤ m -1 and 1 ≤ k ≤ m).
The dual of V m is a right U q (gl m )-module. We will denote it by V * m and we will denote by (f 1 , . . . , f m ) the basis dual to (e 1 , . . . , e m ).

The comultiplication of U q (gl m ) allows to endow the tensor power (V m ) ⊗d with the structure of a U q (gl m )-module. We will denote by e j 1 ,...,j d the element e j 1 ⊗ • • • ⊗ e j d in (V m ) ⊗d . Analogously, (V * m ) ⊗d is a right U q (gl m )-module, with basis f i 1 ,...,i d . The subalgebra U 0 generated in U q (gl m ) by the elements K λ is commutative. A vector e in a U q (gl m )-module is said to be of weight λ ∈ P m if the action of U 0 on e is scalar and given by the character K µ → v (λ|µ) . A left U q (gl m )-module M will be called a weight module if it is the sum of its weight subspaces:

M = λ∈Pm {e ∈ M | ∀µ ∈ P m , K µ • e = v (λ|µ) e}.
The module M is said to be polynomial (respectively polynomial of degree d) if all the weights occurring in the sum are polynomial (respectively polynomial of degree d).

Finally, there is an involutive antiautomorphism of algebras Φ : U q (gl m ) → U q (gl m ) given on the generators by

K λ → K λ , E i → F i , F i → E i .
With it, one has the notion of contravariant duality.

Hecke algebras of type A and the quantum Schur-Weyl duality

Let d be a positive integer, S d be the symmetric group on d letters, and S = {s 1 , . . . , s d-1 } be the set of usual generators of S d .

Definition 2

The Hecke algebra H q (S d ) of the Coxeter system (S d , S) is the Q(v)-algebra presented by the generators (T s i ) 1≤i≤d-1 , with the relations:

T s i T s j = T s j T s i (for |i -j| ≥ 2), T s i T s i+1 T s i = T s i+1 T s i T s i+1 (for 1 ≤ i ≤ d -2), (T s i -q)(T s i + 1) = 0 (for 1 ≤ i ≤ d -1).
For each w ∈ S d , one can define T w as the product

T s i 1 • • • T s i k , where s i 1 • • • s i k is a reduced decomposition of w. The (T w ) w∈S d form a basis of H q (S d ).
The algebra H q (S d ) acts on the spaces (V m ) ⊗d and (V * m ) ⊗d , by means of the following formulas:

e j 1 ,...,j d • T s i =      v e j 1 ,...,j i+1 ,j i ,...,j d if j i < j i+1 , q e j 1 ,...,j d if j i = j i+1 , v e j 1 ,...,j i+1 ,j i ,...,j d + (q -1)e j 1 ,...,j d if j i > j i+1 , T s i • f i 1 ,...,i d =      v f j 1 ,...,j i+1 ,j i ,...,j d if j i < j i+1 , q f j 1 ,...,j d if j i = j i+1 , v f j 1 ,...,j i+1 ,j i ,...,j d + (q -1)f j 1 ,...,j d if j i > j i+1 .
In this way, (V m ) ⊗d becomes a U q (gl m )-H q (S d )-bimodule and (V * m ) ⊗d is its dual H q (S d )-U q (gl m )-bimodule. These assertions are part of the theory of quantized Schur-Weyl duality, due to Jimbo (see Section 3.2).

q-Schur algebras and the BLM construction

In this section, I present a construction of the q-Schur algebras due to Beilinson, Lusztig and MacPherson. One can find detailed proofs for the assertions below in the article [START_REF] Beilinson | A geometric setting for the quantum deformation of GL n[END_REF].

Let m, n and d be three positive integers, and denote by Θ d m,n the set of all matrices of size m × n, whose coefficients are non-negative integers of sum d.

Let V be a vector space of finite dimension d over a field

F. Let f = (0 = V 0 ⊆ • • • ⊆ V m = V ) and f ′ = (0 = V ′ 0 ⊆ • • • ⊆ V ′ n = V ) be two filtrations of V . To the pair (f, f ′ ), one associates the matrix A = c(f, f ′ ) of size m × n with coefficients a ij = dim(V i ∩ V ′ j ) - dim(V i ∩ V ′ j-1 ) -dim(V i-1 ∩ V ′ j ) + dim(V i-1 ∩ V ′ j-1
), so that dim(V i ∩ V ′ j ) = r≤i,s≤j a rs . Let X m be the set of all m-step filtrations in V . The group GL(V ) acts on all the X m , hence acts also (diagonally) on X m × X n . The map (f, f ′ ) → c(f, f ′ ) defines a bijection between the set of GL(V )-orbits in X m × X n and the set of matrices Θ d m,n . If A ∈ Θ d m,n , we denote by O A the corresponding GL(V )-orbit in X m × X n . To a matrix A ∈ Θ d m,n , we associate the weights ro(A) = i,j a ij ε i and co(A) = i,j a ij ε j belonging to

P m and P n respectively. If (f, f ′ ) ∈ O A , with f = (0 = V 0 ⊆ • • • ⊆ V m = V ) and f ′ = (0 = V ′ 0 ⊆ • • • ⊆ V ′ n = V ), then ro(A) = m i=1 dim(V i /V i-1 )ε i and co(A) = n j=1 dim(V ′ j /V ′ j-1 )ε j . Let m, n, p be three positive integers and let A ∈ Θ d m,p , B ∈ Θ d p,n , C ∈ Θ d m,n
. There exists a polynomial g A,B,C ∈ Z[v 2 ] satisfying the following property: if F is a finite field with q elements, if one chooses filtrations f ∈ X m and f ′ ∈ X n such that (f, f ′ ) belongs to the orbit O C , then the value at v 2 = q of polynomial g A,B,C is equal to the number of filtrations f ′′ ∈ X p such that (f, f ′′ ) ∈ O A and (f ′′ , f ′ ) ∈ O B . One has g A,B,C = 0 except if ro(A) = ro(C), co(A) = ro(B), and co(B) = co(C). If m = p and A ∈ Θ d m,m is a diagonal matrix, then g A,B,C is δ B,C or 0 according as ro(B) is equal or different from co(A). If p = n and B ∈ Θ d n,n is a diagonal matrix, then g A,B,C is δ A,C or 0 according as co(A) is equal or different from ro(B).

We denote by S d q (m, n) the Q(v)-vector space with basis the family of symbols (e A ) A∈Θ d m,n . The coefficients g A,B,C afford a bilinear map

S d q (m, p) × S d q (p, n) → S d q (m, n), (e A , e B ) → c∈Θ d m,n g A,B
,C e C , and we have an associativity property for these "products". Letting m = n = p, we see that S d q (m, m) is an algebra; the unit is the element A e A , where the sum runs over the set of diagonal matrices in Θ d m,m . Similarly, S d q (m, n) is a S d q (m, m)-S d q (n, n)-bimodule. We will simplify the notation and denote S d q (m, m) by S d q (m). Remarks. (a) The original definition of the q-Schur algebra goes back to Dipper and James. The fact that both constructions define the same object has been noticed by Du [8, § 1.4]. Let me explain here briefly and without proofs this correspondence.

Finally if A ∈ Θ d m,n , one puts [A] = v -i≥k,j<l a ij a kl e A in S d q (m, n). From the above, it follows that if A ∈ Θ d m,m is a diagonal matrix, then [A] ∈ S d q (m)
To a polynomial weight λ = λ 1 ε 1 + • • • + λ m ε m in P m of degree d, one can associate the parabolic subgroup S λ = S λ 1 × • • • × S λm of S d and the element x λ = w∈S λ T w in H q (S d ). In the U q (gl m )-H q (S d )-bimodule (V m ) ⊗d , the weight space (for the action of U q (gl m )) for the weight λ is a right H q (S d )-submodule isomorphic to the induced module x λ H q (S d ). Hence the commutant of the image of the algebra H q (S d ) inside End Q(v) (V m ) ⊗d is isomorphic to the algebra End Hq(S d ) λ x λ H q (S d ) , where the sum runs over all polynomial weights λ ∈ P m of degree d. This latter object is the q-Schur algebra of Dipper and James (see [5, § 1.2]). The Schur-Weyl duality (see the end of Section 1.3) implies the existence of an algebra morphism from U q (gl m ) to S d q (m). Dipper and James' definition for the q-Schur algebra is valid over an arbitrary base ring R. Let q be a power of a prime number and take R so that the image of q in the prime subring of R is invertible and has a square root v. Let F q be the field with q elements, let G be the group GL d (F q ), let B be its standard Borel subgroup, and identify the symmetric group S d with the Weyl group of G in the usual way. To each weight λ ∈ P m of degree d is associated the standard parabolic subgroup P λ = BS λ B of G. Let M (respectively N λ ) be the right RG-module obtained by inducing the trivial representation of B (respectively P λ ) to G. The module M (respectively N λ ) can be identified with the set of functions from G/B (respectively G/P λ ) to R, where G acts by left translation.

The endomorphism algebra End G (M) is classically identified with the set of functions from B\G/B to R, endowed with the convolution product (see [2, Chap. IV, § 2, exerc. 22]), that is, with the "specialized" Hecke algebra H q (S d ) R . The subspace x λ H q (S d ) R may be identified with the set of functions from B\G/P λ to R. Now M is an H q (S d ) R -RG-bimodule, so we have the functor Hom G (M, -) from the category of right RG-modules to the category of right H q (S d ) R -modules. It sends the object N λ to the object x λ H q (S d ) R , and if µ ∈ P m is another weight of degree d, it defines a bijection between the spaces of homomorphisms Hom G (N λ , N µ ) and Hom [START_REF] Dipper | The q-Schur algebra[END_REF]. Both spaces are naturally in bijection with the space of functions from P λ \G/P µ to R, which can be identified with the subspace

Hq(S d ) R x λ H q (S d ) R , x µ H q (S d ) R . (This is Theorem 2.24 of
H q (S d ) R x λ ∩ x µ H q (S d ) R ; see [7, Lemma 1.1 (e)].)
Thus the "specialized" q-Schur algebra in the definition of Dipper and James identifies with the endomorphism ring End G ( λ N λ ). This correspondence, traced at the level of "generic" algebras, explains the aforementioned result of [START_REF] Du | A note on quantized Weyl reciprocity at roots of unity[END_REF].

(b) There is a stabilization procedure in this framework. Let N be an integer greater that two given positive integers m and n. There is a canonical injection Θ d m,n ֒→ Θ d N,N , obtained by putting a matrix A of size m × n in the top left corner of an N × N matrix and by padding with zeros elsewhere. This gives rise to a map S d q (m, n) ֒→ S d q (N), defined on the basic vectors. If N is also greater than p and if

A ∈ Θ d m,p , B ∈ Θ d p,n , then the product [A][B] is the same computed in S d q (N) or by the map S d q (m, p) × S d q (p, n) → S d q (m, n).
In particular, we have a map S d q (m) ֒→ S d q (N), and we denote by η m the image in S d q (N) of the unit element in S d q (m). Thus η m is the sum of elements [A], for A ∈ Θ d N,N running over the set of diagonal matrices with co(A) ∈ P m . Then η m is an idempotent in S d q (N), the algebra S d q (m) may be identified with the algebra η m S d q (N) η m , and the S d q (m)-S d q (n)-bimodule S d q (m, n) may be identified with the subspace η m S d q (N) η n .

There is an algebra homomorphism from U q (gl m ) to S d q (m). The main result obtained by Beilinson, Lusztig and MacPherson asserts its surjectivity and provides formulas for it:

Theorem 1 [1, § 5.7] There exists a (unique) surjective homomorphism ζ of Q(v)-algebras from U q (gl m ) to S d q (m) which: • sends K λ to v λ 1 j 1 +•••+λmjm [j 1 E 11 +• • •+j m E mm ],
the sum running over all the m-uples of non-negative integers (j 1 , . . . , j m ) of sum d;

• sends E i to [j 1 E 11 + • • • + j m E mm + E i,i+1
], the sum running over all the m-uples of non-negative integers (j 1 , . . . , j m ) of sum d -1;

• sends F i to [j 1 E 11 + • • • + j m E mm + E i+1,i ],
the sum running over all the m-uples of non-negative integers (j 1 , . . . , j m ) of sum d -1.

So S d q (m) is a quotient of U q (gl m ) and S d q (m, n) is a U q (gl m )-U q (gl n )-bimodule. The laws for this latter structure are given by the formulas in [1, Lemmas 3.2 and 3.4]:

Lemma 1 For the U q (gl m )-U q (gl n )-bimodule structure on S d q (m, n), one has for all A ∈ Θ d m,n : K λ • [A] = v (λ| ro(A)) [A], E h • [A] = n p=1 a h+1,p ≥1 [a hp + 1] v j>p (a hj -a h+1,j ) [A + E hp -E h+1,p ], F h • [A] = n p=1 a hp ≥1 [a h+1,p + 1] v j<p (a h+1,j -a hj ) [A -E hp + E h+1,p ], [A] • K λ = v (λ| co(A)) [A], [A] • E h = m p=1 a ph ≥1 [a p,h+1 + 1] v i<p (a i,h+1 -a ih ) [A -E ph + E p,h+1 ], [A] • F h = m p=1 a p,h+1 ≥1 [a ph + 1] v i>p (a ih -a i,h+1 ) [A + E ph -E p,h+1 ].
As a consequence, the left U q (gl m )-module S d q (m) is a finite-dimensional polynomial module of degree d.

The algebra of functions on quantum matrix spaces

Let m and n be two positive integers. The following definition can be traced back at least to Manin in the case m = n, and the immediate generalization to the case m = n also appears in numerous places.

Definition 3

The Q(v)-algebra presented by generators X ij (1 ≤ i ≤ m and 1 ≤ j ≤ n) and relations:

X rt X su =        X su X rt if r > s and t < u or if r < s and t > u, v X su X rt if r > s and t = u or if r = s and t > u, X su X rt + (v -v -1 )X ru X st if r > s and t > u.
is called the algebra of functions on the quantum matrix space and is denoted by

S q (M * m,n ). S q (M * m,n
) is in a natural way a N-graded algebra, whose d-th homogeneous components will be denoted by S d q (M * m,n ). Fix the natural integer d, and let I d m,n and J d m,n be the sets of all the d-tuples of pairs of integers ((i 1 , j 1 ), . . . , (i d , j d )) ∈ ({1, . . . , m} × {1, . . . , n}) d defined by the conditions of being lexicographically non-decreasing:

I d m,n = {((i 1 , j 1 ), . . . , (i d , j d )) | i a ≤ i a+1 , i a = i a+1 ⇒ j a ≤ j a+1 }, J d m,n = {((i 1 , j 1 ), . . . , (i d , j d )) | j a ≤ j a+1 , j a = j a+1 ⇒ i a ≤ i a+1 }. Theorem 2 [24, § 3.5] (a)
The set consisting of the vectors

X i 1 ,j 1 • • • X i d ,j d for ((i 1 , j 1 ), . . . , (i d , j d )) running over I d m,n (or running over J d m,n ) is a Q(v)-basis in S d q (M * m,n ), which is thus of dimension mn+d-1 d . (b) S q (M * m,n
) is an algebra without zero-divisor.

We will need later another parametrization of the basis given by the above theorem. Let E d m,n be the set of all d-uples of pairs of integers ((i 1 , j 1 ), . . . ,

(i d , j d )) ∈ ({1, . . . , m} × {1, . . . , n}) d such that for all 1 ≤ a < b ≤ d, one has i a < i b , j a < j b or (i a , j a ) = (i b , j b ).
Recall that Θ d m,n is the set of all matrices of size m×n, whose coefficients are non-negative integers of sum d. Let E ij ∈ Θ 1 m,n denotes the elementary matrix, with a 1 in position (i, j) and with 0 elsewhere. To the element ((i 1 , j 1 ), . . . , (i d , j d )) of E d m,n , one associates the matrix ϕ((i 1 , j 1 ), . . . ,

(i d , j d )) = d a=1 E ia,ja ∈ Θ d m,n . On E d m,n
, we put the equivalence relation ≈ generated by the following elementary moves:

((i 1 , j 1 ), . . . , (i d , j d )) ∼ ((i ′ 1 , j ′ 1 ), . . . , (i ′ d , j ′ d )) if and only if there exists 1 ≤ a < d such that (i a -i a+1 )(j a -j a+1 ) < 0 and ((i ′ 1 , j ′ 1 ), . . . , (i ′ d , j ′ d )
) is obtained from ((i 1 , j 1 ), . . . , (i d , j d )) by exchanging (i a , j a ) with (i a+1 , j a+1 ).

≈. If ((i 1 , j 1 ), . . . , (i d , j d )) and ((i ′ 1 , j ′ 1 ), . . . , (i ′ d , j ′ d )) are two ≈-equivalent elements in E d m,n , then X i 1 ,j 1 • • • X i d ,j d = X i ′ 1 ,j ′ 1 • • • X i ′ d ,j ′ d .
Proof. One first remark that ϕ induces bijections between I d m,n , J d m,n and Θ d m,n . If (i 1 , . . . , j d ) ∈ E d m,n , then (i a > i a+1 ) ⇒ (j a < j a+1 ). By induction on the number of inversions in the sequence (i 1 , . . . , i d ), one then shows that ((i 1 , j

If B ∈ Θ d m,n , we will denote by X(B) the element X i 1 ,j 1 • • • X i d ,j d ∈ S d q (M * m,n ) with ((i 1 , j 1 ), . . . , (i d , j d )) ∈ E d m,n ∩ ϕ -1 (B).
The interesting structure on S q (M * m,n ) is given in the following proposition.

Proposition 2 [15, p. 210] Let m, n, p be three natural numbers. There is a unique homomorphism of algebras from

S q (M * m,n ) to S q (M * m,p ) ⊗ S q (M * p,n ) which sends X ij to p k=1 X ik ⊗ X kj .
Letting m = n = p in this proposition, we see that the algebra S q (M * m,m ) may be endowed with the structure of a bialgebra. The counit is given by ε

(X ij ) = δ ij for 1 ≤ i, j ≤ m. Letting p = m (respectively p = n) in the proposition, we see that S q (M * m,n ) is a left S q (M * m,m )- comodule (respectively a right S q (M * n,n )-comodule). It is in fact a S q (M * m,m )-S q (M * n,n )- bicomodule. Each homogeneous component S d q (M * m,m ) is a subcoalgebra of S q (M * m,m ) and S d q (M * m,n ) is a S d q (M * m,m )-S d q (M * n,n )-bicomodule.
Remark. There is a stabilization procedure here. Let N be an integer greater than m, n and p. Let us denote by K m,n the two-sided ideal generated in the algebra S q (M * N,N ) by the elements X ij with i > m or j > n. The quotient algebra S q (M * N,N )/K m,n identifies naturally with the algebra S q (M * m,n ). If N is also greater than p, then the coproduct given in Proposition 2 is compatible with this construction since it sends

K m,n into K m,p ⊗ S q (M * p,n ) + S q (M * m,p ) ⊗ K p,n .
Let us denote by t ij ∈ (U q (gl m )) * the matrix coefficients of the natural U q (gl m )-module V m :

t ij : x → f i , x • e j .
They belong to the restricted (Hopf) dual (U q (gl m )) * res of U q (gl m ).

Proposition 3 There is a bialgebra morphism κ from S q (M * m,m ) to (U q (gl m )) * res which sends

X ij to t ij .
Proof. The R-matrix of U q (gl m ) defines the U q (gl m )-linear map:

   (V m ) ⊗2 → (V m ) ⊗2 , e j 1 ,j 2 →      e j 2 ,j 1 if j 1 < j 2 , v e j 1 ,j 2 if j 1 = j 2 , e j 2 ,j 1 + (v -v -1 )e j 1 ,j 2 if j 1 > j 2    .
The U q (gl m )-linearity of this map implies that the elements t ij satisfy the relations of Definition 3. We thus have an algebra homomorphism from S q (M * m,m ) to (U q (gl m )) * res , that clearly respects the coproduct and the counit.

The S d q (M * m,m )-S d q (M * n,n )-bicomodule structure on S d q (M * m,n
) gives rise to a (U q (gl m )) * res -(U q (gl n )) * res -bicomodule structure on S d q (M * m,n ), and thus to a U q (gl n )-U q (gl m )-bimodule structure on S d q (M * m,n ). Another way to describe that is to look at the U q (gl n )-U q (gl m )bimodule (V n ) ⊗d ⊗ (V * m ) ⊗d , with basis e j 1 ,...,

j d ⊗ f i 1 ,...,i d = (e j 1 ⊗ • • • ⊗ e j d ) ⊗ (f i 1 ⊗ • • • ⊗ f i d ).
Proposition 4 The linear map

π : (V n ) ⊗d ⊗ (V * m ) ⊗d → S d q (M * m,n ), e j 1 ,...,j d ⊗ f i 1 ,...,i d → X i 1 ,j 1 • • • X i d ,j d factorizes through (V n ) ⊗d ⊗ Hq(S d ) (V * m )
⊗d and defines an isomorphism of U q (gl n )-U q (gl m )bimodules between this latter space and S d q (M * m,n ).

Proof. The relations defining the algebra S q (M * m,n ) imply that the kernel of π is exactly the subspace spanned by the vectors e

• T s k ⊗ f -e ⊗ T s k • f for e ∈ (V n ) ⊗d , f ∈ (V * m ) ⊗d and 1 ≤ k ≤ d -1. Hence π defines an injection from (V n ) ⊗d ⊗ Hq(S d ) (V * m ) ⊗d to S d q (M * m,n ). Further, π is surjective.
We now fix (j 1 , . . . , j d ) ∈ {1, . . . , n} d and consider the map

u : (V * m ) ⊗d → S d q (M * m,n ), f i 1 ,...,i d → X i 1 ,j 1 • • • X i d ,j d . Let x ∈ U q (gl
m ) and denote its iterated coproduct by the Sweedler notation

x (1) ⊗• • •⊗x (d) . Then u(e j 1 ,...,j d ) • x = (X i 1 ,j 1 • • • X i d ,j d ) • x = m k 1 ,...,k d =1 x, κ(X i 1 ,k 1 • • • X i d ,k d ) X k 1 ,j 1 • • • X k d ,j d = m k 1 ,...,k d =1 x, t i 1 ,k 1 • • • t i d ,k d X k 1 ,j 1 • • • X k d ,j d = m k 1 ,...,k d =1 x (1) , t i 1 ,k 1 • • • x (d) , t i d ,k d X k 1 ,j 1 • • • X k d ,j d = m k 1 ,...,k d =1 f i 1 , x (1) • e k 1 • • • f i d , x (d) • e k d X k 1 ,j 1 • • • X k d ,j d = m k 1 ,...,k d =1 f i 1 ,...,i d , x • e k 1 ,...,k d u(f k 1 ,...,k d ) = u(f i 1 ,...,i d • x).
This calculation shows that the linear map in the statement of the proposition is U q (gl m )linear. An analogous reasoning shows that it respects the left action of U q (gl n ).

In the case m = n, under the identification (through κ) of S d q (M * m,m ) with the subcoalgebra of (U q (gl m )) * res spanned by monomials of degree d in the elements t ij , the U q (gl m )-U q (gl m )linearity of the map π in Proposition 4 is of course a standard fact (see [START_REF] Dixmier | Enveloping algebras[END_REF]Proposition 2.7.11] for instance).

Remark. In [START_REF] Dipper | Quantum GL n[END_REF], Dipper and Donkin have introduced another version of the algebra of functions on the quantum matrix space, which has the advantage of being defined over the subring Z[q] ⊂ Q(v). Dipper and Donkin have pointed out that their bialgebra was not isomorphic to S d q (M * mm ). Several techniques of [loc. cit.] may however easily be adapted to our purposes. For instance, Theorem 2 corresponds to [loc. cit., Theorem 1.1.8 and Lemma 1.

Duality with the function algebras

We already asserted the existence of a pairing between U q (gl m ) and S q (M * m,m ). By the definition of κ, the orthogonal in U q (gl m ) of S d q (M * m,m ) for this pairing is the annihilator of the left U q (gl m )-module (V m ) ⊗d . And by [10, § 1], this latter is also the kernel of the homomorphism ζ defined in Theorem 1. Thus there is a pairing between S d q (m) and S d q (M * m,m ). The explicit formula giving this pairing does not seem to be widely known. More generally, let us define a pairing between S d q (m, n) and S d q (M * m,n ) by, for A and B in Θ d m,n :

[A], X(B) = v -i<j,k a ik a jk if A = B, 0 if A = B.
This is a non-degenerate bilinear form.

Proposition 5 (a) For the structures of U q (gl m )-U q (gl n )-bimodule on S d q (m, n) and of U q (gl n )-U q (gl m )-bimodule on S d q (M * m,n ), one has for all a ∈ S d q (m, n), b ∈ S d q (M * m,n ), x ∈ U q (gl m ) and y ∈ U q (gl n ):

x • a, b = a, b • x and a • y, b = a, y • b . (b) If m, n, p, d are positive integers, if a ∈ S d q (m, p), b ∈ S d q (p, n), c ∈ S d q (M * m,n
), and if we denote by ab ∈ S d q (m, n) the product of a and b and by

c (1) ⊗ c (2) ∈ S d q (M * m,p ) ⊗ S d q (M * p,n ) the coproduct of c (as in Proposition 2), then ab, c = a, c (1) b, c (2)
.

(c) For x ∈ U q (gl m ) and a ∈ S d q (M * m,m ), one has ζ(x), a = x, κ(a) .

Proof. If we fix (i 1 , . . . , i d ) ∈ {1, . . . , m} d , the map

(V n ) ⊗d → S d q (M * m,n ), e j 1 ,...,j d → X i 1 ,j 1 • • • X i d ,j d
is a morphism of U q (gl n )-modules (Proposition 4). This allows to compute the action of U q (gl n ) on the elements X(B) ∈ S d q (M * m,n ), for B ∈ Θ d m,n , by taking ((i 1 , j 1 ), . . . , (i d , j d )) ∈ I d m,n ∩ ϕ -1 (B) (notations from Section 1.5). Using the relations in Definition 3, one finds after some simple calculations

K λ • X(B) = v (λ| co(B)) X(B), E h • X(B) = m p=1 [b p,h+1 ] v i>p b ih -b i,h+1 X(B + E ph -E p,h+1 ), F h • X(B) = m p=1 [b ph ] v i<p b i,h+1 -b ih X(B -E ph + E p,h+1 ).
In the right hand side of the second (respectively the third) formula, it is understood that a summand corresponding to an index p such that b p,h+1 = 0 (respectively b ph = 0) vanishes. Comparing with the formulas in Lemma 1, this proves the second equality in statement (a).

The first equality in statement (a) follows as for it from the following relations describing the action of U q (gl m ) on S d q (M * m,n ):

X(B) • K λ = v (λ| ro(B)) X(B), X(B) • E h = n p=1 [b hp ] v ( j≥p b hj -b h+1,j )-1 X(B -E hp + E h+1,p ), X(B) • F h = n p=1 [b h+1,p ] v ( j≤p b h+1,j -b hj )-1 X(B + E hp -E h+1,p ).
Here again, in the right hand side of the second (respectively the third) formula, it is understood that a summand corresponding to an index p such that b hp = 0 (respectively b h+1,p = 0) vanishes.

The formula in statement (c) is true for x = 1 and any a. Since ζ (respectively κ) is a morphism of left (respectively right) U q (gl m )-modules and since 1 generates the left regular U q (gl m )-module, statement (a) shows that the formula is true in the general case.

Statement (c), the fact that ζ is a surjective morphism of algebras, and the fact that κ is a morphism of coalgebras imply that statement (b) holds in the case m = n = p. For the general case, we choose an integer N greater than m, n and p, and use a stabilization procedure, as explained in previous remarks.

One can understand the statement (b) by saying that the algebra S d q (m) is the dual of the coalgebra S d q (M * m,m ) (see [ To study combinatorially the theory of invariants in a way independent of the characteristic of the ground field, Doubilet, Rota and Stein have introduced some objects called bitableaux. These bitableaux give a "standard basis" of the space of polynomial functions on a space of matrices, and J. A. Green has defined in a dual way a "standard basis" for the classical Schur algebra. In turn, these bases allow one to define filtrations on both these spaces, that are linked with the recent notions of tilting theory and quasi-hereditary algebra. (See [START_REF] Green | Combinatorics and the Schur algebra[END_REF] and references therein for all that.) In [START_REF] Huang | Standard basis theorem for quantum linear groups[END_REF] and [START_REF] Green | A straightening formula for quantized codeterminants[END_REF], Huang, Zhang and R. M. Green defined quantum analogues of these constructions and devised straightening algorithms in order to prove standard basis theorems for S d q (M * m,n ) (in a supersymmetric case) and for S d q (m) respectively. Using (and abusing of) an idea of J. A. Green [START_REF] Green | Combinatorics and the Schur algebra[END_REF], we will show in this appendix how to deduce both standard basis theorems from the explicit formula given in Section 1.6 for the pairing between S d q (m, n) and S d q (M * m,n ). Besides being simpler, this approach has the advantage of giving a more precise information than the straightening algorithms can do.

The set of polynomial dominant weights of degree d in P m is in bijection with the set of Young diagrams with d boxes and at most m rows. Such a Young diagram, filled with integers between 1 and m, is called semistandard if the entries in each row, from the left to the right, are non-decreasing and if the entries in each column, from the top to the bottom, are increasing.

Let T d m,n be the set of pairs (P, Q) consisting of Young semistandard tableaux of the same shape, with d boxes, such that P and Q are filled with positive integers less or equal than m and n respectively. To each (P, Q) ∈ T d m,n , one associates a "quantum semistandard bitableau" (or "bideterminant")

X P,Q ∈ S d q (M * m,n ), as in [21, § 1
], and a "quantized semistandard codeterminant" Y P,Q ∈ S d q (m, n), as in [START_REF] Green | A straightening formula for quantized codeterminants[END_REF] (the precise definitions are recalled below). We aim to show that these elements afford bases for the spaces S d q (M * m,n ) and S d q (m, n). First, I introduce some notations:

• If λ and µ ∈ P m are weights, we write λ µ if one can express µ -λ as a linear combination with non-negative coefficients of the simple roots α i . To a Young tableau P with at most m lines and filled with positive numbers less or equal than m, we associate two polynomial weights in P m : the shape of P is sh(

P ) = λ 1 ε 1 + • • • + λ m ε m , where λ 1 ≥ • • • ≥ λ m
are the lengths of the rows of P , and the content of P is cont

(P ) = n 1 ε 1 + • • • + n m ε m ,
where n i is the number of boxes in P filled by i. We say that P is column-standard if the entries is each column of P increase from top to bottom. If P is column-standard, then cont(P ) sh(P ).

• If d is a positive integer, we let A d = w∈S d (-q) -ℓ(w) T w be the antisymmetrizer in H q (S d ), where ℓ(w) denotes the usual length of the permutation w.

If 1 ≤ i 1 < • • • < i d ≤ m and 1 ≤ j 1 < • • • < j d ≤ n, one defines the quantum determinant as the element of S d q (M * m,n ) given by X i 1 ,j 1 . . . X i 1 ,j d . . . . . . X i d ,j 1 . . . X i d ,j d q = π(e j 1 ,...,j d • A d ⊗ f i 1 ,...,i d ) = w∈S d (-v) -ℓ(w) π(e j w(1) ,...,j w(d) ⊗ f i 1 ,...,i d ) = w∈S d (-v) -ℓ(w) X i 1 ,j w(1) • • • X i d ,j w(d) . • Let (P, Q) ∈ T d m,n . The common shape of P and Q is a Young diagram with say t columns of lengths d 1 , . . . , d t . Let 1 ≤ i s,1 < • • • < i s,ds ≤ m and 1 ≤ j s,1 < • • • < j s,
ds ≤ n be the entries in the s-th columns of P and Q respectively, from top to bottom. One then puts .

X P,Q = t s=1 X i s,1 , j s,1 . . . X i s,
This definition extends to the case of a pair of Young tableaux of the same shape, with P and Q filled with positive integers less or equal than m and n respectively, but that are only column-standard. In this latter case, we refer to X P,Q as a "quantum column-standard bitableau" (this is a less general notion than the one of "quantum bitabloid" defined in [21, § 1]).

• Let (P, Q) ∈ T d m,n . The common shape of P and Q is a Young diagram with at most p = min(m, n) rows. Let A P ∈ Θ d m,p be the lower triangular matrix whose element in position (i, j) is the number of times i appears in the j-th line of

P . Let B Q ∈ Θ d p,n
be the upper triangular matrix whose element in position (i, j) is the number of times j appears in the i-th line of Q. Then co(A) = ro(B) = sh(P ) = sh(Q), and, following R. M. Green, we define Y P,Q ∈ S d q (m, n) to be the product

[A P ][B Q ].
• Let m, n and d be positive integers. We say that a matrix A ∈ Θ d m,n is less than or equal to a matrix B ∈ Θ d m,n if ro(A) = ro(B), co(A) = co(B), and if for all 1 ≤ µ ≤ m and 1 ≤ ν ≤ n, there holds (i,j)∈{1,...,µ}×{1,...,ν} a ij ≤ (i,j)∈{1,...,µ}×{1,...,ν} b ij .

This defines a partial order ≤ on Θ d m,n .

We will also need two lemmas.

Lemma 2 [24, Lemma 4.4.2][15, Proposition 6] Let m, n, p, d be positive integers, with d ≤ min(m, n, p), and take

1 ≤ i 1 < • • • < i d ≤ m and 1 ≤ j 1 < • • • < j d ≤ n.
Then the image of the quantum determinant

X i 1 ,j 1 . . . X i 1 ,j d . . . . . . X i d ,j 1 . . . X i d ,j d q ∈ S d q (M * m,n )
under the coproduct defined in Proposition 2 is equal to

1≤k 1 <•••<k d ≤p X i 1 ,k 1 . . . X i 1 ,k d . . . . . . X i d ,k 1 . . . X i d ,k d q ⊗ X k 1 ,j 1 . . . X k 1 ,j d . . . . . . X k d ,j 1 . . . X k d ,j d q ∈ S d q (M * m,p ) ⊗ S d q (M * p,n ).
Lemma 3 Let m, n, d be positive integers.

(a) Take ((i 1 , j 1 ), . . . ,

(i d , j d )) ∈ ({1, . . . , m} × {1, . . . , n}) d , put A = d a=1 E ia,ja ∈ Θ d m,n , and expand X i 1 ,j 1 • • • X i d ,j d = B∈Θ d m,n c B X(B)
in terms of our usual basis (see conventions after Proposition 1). Then the coefficients c B belong to andexpand X i 1 ,j 1 . . . X i 1 ,j d . . . . . .

Z[v, v -1 ], c B = 0 only if B ≤ A, and c A is a power of v. (b) Take 1 ≤ i 1 < • • • < i d ≤ m and 1 ≤ j 1 < • • • < j d ≤ n, put A = d a=1 E ia,ja ∈ Θ d m,n ,
X i d ,j 1 . . . X i d ,j d q = B∈Θ d m,n c B X(B)
in terms of our usual basis. Then the coefficients c B belong to

Z[v, v -1 ], c B = 0 only if B ≤ A, and c A = 1.
(c) Let (P, Q) be a pair of column-standard Young tableaux of the same shape, such that P and Q are filled with positive integers less or equal than m and n respectively. (For instance, take (P, Q) ∈ T d m,n .) Enumerate the entries in the boxes of P and Q by (i 1 , . . . , i d ) and (j 1 , . . . , j d ) (in the same order), put A = andc A is a power of v.

d a=1 E ia,ja ∈ Θ d m,n , and expand X P,Q = B∈Θ d m,n c B X(B) in terms of our usual basis. Then the coefficients c B belong to Z[v, v -1 ], c B = 0 only if B ≤ A,
Proof. (a) Order the set {1, . . . , m} × {1, . . . , n} by the lexicographic order: (i, j) ≤ (i ′ , j ′ ) if i < i ′ or if i = i ′ and j ≤ j ′ , and denote by N the number of inversions in the sequence ((i 1 , j 1 ), . . . , (i d , j d )). There is nothing to prove if N = 0, since in that case ((i 1 , j 1 ), . . . , (i d , j d )) ∈ I d m,n (see Section 1.5). So assume that N ≥ 1. There is then an a ∈ {1, . . . , d -1} such that (i a , j a ) > (i a+1 , j a+1 ). Thanks to the relations in Definition 3, we have

X i 1 ,j 1 • • • X i d ,j d =                    X i 1 ,j 1 • • • X i a+1 ,j a+1 X ia,ja • • • X i d ,j d if i a > i a+1 and j a < j a+1 , vX i 1 ,j 1 • • • X i a+1 ,j a+1 X ia,ja • • • X i d ,j d if i a > i a+1
and j a = j a+1 or if i a = i a+1 and j a > j a+1 ,

X i 1 ,j 1 • • • X i a+1 ,j a+1 X ia,ja • • • X i d ,j d + (v -v -1 ) X i 1 ,j 1 • • • X i a+1 ,ja X ia,j a+1 • • • X i d ,j d if i a > i a+1 and j a > j a+1 .
The sequences arising from each terms above have fewer than N inversions. The matrix associated to the sequence ((i 1 , j 1 ), . . . , (i a+1 , j a+1 ), (i a , j a ), . . . , (i d , j d )) is the matrix A and, in the third case, the matrix associated to the sequence ((i 1 , j 1 ), . . . , (i a+1 , j a ), (i a , j a+1 ), . . . , (i d , j d )) is less than A. An induction concludes the proof.

(b) Choose w ∈ S d and consider in the quantum determinant the term

X i 1 ,j w(1) • • • X i d ,j w(d) .
By statement (a), the expansion of this term involve monomials X(B) such that the matrix B is less or equal than d a=1 E ia,j w(a) . This latter matrix is equal to A if w = 1 and is less than A otherwise. Thus after having gathered the expansions of the different terms, only monomials X(B) with B ≤ A can occur, and X(A) occurs only for the term

X i 1 ,j 1 • • • X i d ,j d = X(A).
(c) The assertion (c) is a direct consequence of the statements (a) and (b).

Now we can prove:

Proposition 6 The matrix Y P,Q , X P ′ ,Q ′ (P,Q),(P ′ ,Q ′ )∈T d m,n has entries in Z[v, v -1 ]
and is invertible over this ring.

Proof. Since the structure constants of the q-Schur algebras belong to

Z[v, v -1 ], Lemma 3 (c) implies that our matrix is Z[v, v -1 ]-valued. If (P, Q) and (P ′ , Q ′ ) ∈ T d
m,n , we say that (P, Q) is less or equal than (P ′ , Q ′ ) if either • the common shape of P and Q is less than the common shape of P ′ and Q ′ : sh(P ) ⊳ sh(P ′ ) ;

• P, Q, P ′ , Q ′ have the same shape, A P ≤ A P ′ and B Q ≤ B Q ′ .
This definition endows T d m,n with a partial order and it is enough to prove that the matrix

Y P,Q , X P ′ ,Q ′ (P,Q),(P ′ ,Q ′ )∈T d m,n
is upper triangular w.r.t. with invertible diagonal elements. The image of X P ′ ,Q ′ under the coproduct (defined in Proposition 2) can be computed using Lemma 2. The result is R X P ′ ,R ⊗ X R,Q ′ where the sum runs over the set of columnstandard Young tableaux with the same shape than P ′ and Q ′ that are filled with positive integers less or equal than p. Using Proposition 5 (b), we can write

Y P,Q , X P ′ ,Q ′ = R [A P ], X P ′ ,R [B Q ], X R,Q ′ .
If Y P,Q , X P ′ ,Q ′ is non-zero, then there is an R such that [A P ], X P ′ ,R and [B Q ], X R,Q ′ are non-zero. These conditions imply that cont(R) is equal to co(A P ) = ro(B Q ) = sh(P ), and this is possible only if sh(P ) = cont(R) sh(R) = sh(P ′ ). Suppose further that P, Q, P ′ , Q ′ have the same shape. In the sum R [A P ], X P ′ ,R [B Q ], X R,Q ′ , the only possible nonzero term occurs for the so-called Yamanouchi tableau R, filled with 1 in the first line, 2 in the second, and so on. According to Lemma 3 (c), the bracket [A P ], X P ′ ,R (respectively [B Q ], X R,Q ′ ) can be non-zero only if A P ≤ A P ′ (respectively B Q ≤ B Q ′ ). All this proves that (P, Q) (P ′ , Q ′ ) is a necessary condition to have Y P,Q , X P ′ ,Q ′ = 0. Therefore the matrix Y P,Q , X P ′ ,Q ′ is upper triangular.

The same reasoning proves that, up to a power of v, Y P,Q , X P,Q equals [A P ], X(A P ) × [B Q ], X(B Q ) . Hence Y P,Q , X P,Q is a power of v. This proves our assertion about the diagonal entries of the matrix.

A direct consequence of this proposition is that the sets (X P,Q

) (P,Q)∈T d m,n and (Y P,Q ) (P,Q)∈T d m,n
are bases of the spaces S d q (M * m,n ) and S d q (m, n) respectively. More precisely, they are bases of the Z[v, v -1 ]-lattices spanned by the sets [START_REF] Huang | Standard basis theorem for quantum linear groups[END_REF]Theorem 9] and [START_REF] Green | A straightening formula for quantized codeterminants[END_REF]Corollary 3.7]).

{X(A) | A ∈ Θ d m,n } and {[A] | A ∈ Θ d m,n } respectively (compare with
All the other consequences that J. A. Green has obtained from its technique quantify easily. For instance, one can show that the straightening of a quantum column-standard bitableau only affords quantum semistandard bitableaux whose shapes are less or equal than the shape of the original column-standard bitableau w.r.t. the dominance order (this fact does not seem to be a direct consequence of the algorithm described by Huang and Zhang [START_REF] Huang | Standard basis theorem for quantum linear groups[END_REF] and Galdi [9]). One can therefore use the quantum bitableaux and the quantized codeterminants to exhibit bimodule filtrations for S d q (M * m,n ) and S d q (m, n). I must point out further that, despite J. A. Green's claim, details of the straightening algorithm are not needed to get the precise isomorphism in [START_REF] Green | Combinatorics and the Schur algebra[END_REF]Theorem 7.3 (iii)].

2 The q-Weyl group 2.1 The completion and the q-Weyl group of U q (gl m )

In the above presentation of the q-Weyl group, I will use Levendorskii and Soibelman's idea of seeing the q-Weyl group inside a completion of U q (gl m ) and not as a subgroup of Aut(U q (gl m )).

Let J be the set of the annihilators I ⊆ U q (gl m ) of the finite-dimensional weight U q (gl m )modules. Endowing J with the order relation given by the inclusion, one may define the inverse limit U q (gl m ) = lim ←-(U q (gl m )/I). The canonical map U q (gl m ) → U q (gl m ) is injective (see [START_REF] Joseph | Local finiteness of the adjoint action for quantized enveloping algebras[END_REF]Lemma 8.3]) and we identify U q (gl m ) with its image.

Here is an instance where this construction arises naturally. Let λ ∈ P m be a weight. For each I ∈ J, one may choose a finite-dimensional weight U q (gl m )-module M whose annihilator is I. This module M is the direct sum of its weight subspaces: M = λ∈Pm M λ ; in particular, it is completely reducible as a U 0 -module. The projection on a summand M λ with respect to this decomposition belongs to the centralizer of End U 0 (M) in End Q(v) (M), hence is the image of a (unique) element (1 λ ) I ∈ U q (gl m )/I, by the density theorem. Forming direct sums, one can easily see that (1 λ ) I does not depend on the choice of M and that the family ((1 λ ) I ) I∈J defines an element 1 λ ∈ U q (gl m ). (Here, we used the same notation as in [23, § 23.1].)

More generally, an element x ∈ U q (gl m ) gives rise, for every finite-dimensional weight module M, to an operator x M ∈ End Q(v) (M), with the property that for all morphism f : M → N, there holds

x N • f = f • x M .
Now the symmetric group S m identifies with the Weyl group of GL m . This is a Coxeter system on the set of simple reflections {s 1 , . . . , s m-1 } and it acts on the weight lattice P m . Lusztig [START_REF] Lusztig | Introduction to quantum groups[END_REF]Chap. 5], and independently Levendorskii and Soibelman [START_REF] Levendorskii | Some applications of the quantum Weyl groups[END_REF], have introduced elements s1 , . . . , sm-1 of the algebra U q (gl m ) such that:

• for each weight λ ∈ P m , one has 1 s i (λ) si = si 1 λ (see [START_REF] Lusztig | Introduction to quantum groups[END_REF]Proposition 5.2.7]);

• the si satisfy the braid relations: si sj = sj si if |i -j| ≥ 2, and si si+1 si = si+1 si si+1

(see [START_REF] Lusztig | Introduction to quantum groups[END_REF]Theorem 39.4.3]);

• the si are invertible in U q (gl m ) and the inner automorphisms they define in U q (gl m ) stabilize the subalgebra U q (gl m ) (see [START_REF] Lusztig | Introduction to quantum groups[END_REF]Proposition 37.1.2]).

By the definition of U q (gl m ), the elements si give rise, for each finite-dimensional weight U q (gl m )-module M, to an operator T i = (s i ) M : M → M. In fact, Lusztig has found several possible choices for the T i , which he denotes by T ′ i,±1 and T ′′ i,±1 . These operators are defined by the following formulas, in which e is a vector of weight λ, and

E (a) i = E a i [a]! and F (b) i = F b i [b]!
are the q-divided powers (see [23, § 5.2.1]):

T ′ i,±1 (e) = a-b+c=(α i |λ) (-1) b v ±(b-ac) F (a) i E (b) i F (c) i • e , T ′′ i,±1 (e) = -a+b-c=(α i |λ) (-1) b v ±(b-ac) E (a) i F (b) i E (c) i • e .
It is known that being then induced by (s ′′ i ) -1 and (s ′ i ) -1 . The formulas above define the elements (s ′ i ) ±1 1 λ and (s ′′ i ) ±1 1 λ as the sum of series which converge in U q (gl m ). Finally, recall the involutive antiautomorphism Φ of the algebra U q (gl m ) defined in Section 1.2. It preserves the set J of ideals, since the contravariant dual of a finite-dimensional weight module is a finite-dimensional weight module. Thus Φ extends to an involutive antiautomorphism of the algebra U q (gl m ), which we denote by Φ. If λ ∈ P m , then Φ(1 λ ) = 1 λ in U q (gl m ). One also verifies by the formulas that Φ(s

T ′′ i,±1 = (T ′ i,∓1 ) -1 [
′ i ) = s′′ i in U q (gl m ): Φ(s ′ i 1 λ ) = Φ a-b+c=(α i |λ) (-1) b v b-ac F (a) i E (b) i F (c) i Φ(1 s i (λ) ) = a-b+c=(α i |λ) (-1) b v b-ac E (c) i F (b) i E (a) i 1 s i (λ) = -a+b-c=(α i |s i (λ)) (-1) b v b-ac E (c) i F (b) i E (a) i 1 s i (λ) = s′′ i 1 s i (λ) = Φ(1 λ ) s′′ i .
2.2 The q-Weyl group of the q-Schur algebra

As noticed at the end of Section 1.4, the left U q (gl m )-module S d q (m) is finite-dimensional and is a weight module. Thus the kernel of ζ belongs to J, and so ζ factorizes canonically through an epimorphism

ζ : U q (gl m ) → S d q (m). Let λ = λ 1 ε 1 + • • • + λ m ε m ∈ P m be a weight. Since the left U q (gl m )-module S d q (m) is polynomial of degree d, the image under ζ of 1 λ in S d q (m) is zero except if λ is polynomial of degree d, in which case it is equal to [ m j=1 λ j E jj ].
Our aim is to describe similarly the images of s′ i and s′′ i in S d q (m). We first begin with a proposition giving a pairing between S d q (M * n,m ) and S d q (M * m,n ). (This pairing is the quantum version of the non-supersymmetric case of [START_REF] Grosshans | Invariant theory and superalgebras[END_REF]Proposition 11].) Let us denote by A T the transposed of a matrix A.

Proposition 7

The non-degenerate bilinear form between S d q (M * n,m ) and

S d q (M * m,n ) defined by, for A ∈ Θ d n,m and B ∈ Θ d m,n (X(A), X(B)) = i,j v a ij (a ij -1)/2 [a ij ]! if B = A T , 0 else, satisfies, for all a ∈ S d q (M * n,m ), b ∈ S d q (M * m,n ), x ∈ U q (gl m ) and y ∈ U q (gl n ): (x • a, b) = (a, b • x) and (a • y, b) = (a, y • b).
Proof. One checks the relations directly with the help of the formulas given in the proof of Proposition 5.

An immediate corollary of Propositions 5 and 7 is the following statement.

Proposition 8

The linear map defined in the standard bases by

S d q (M * n,m ) → S d q (m, n), X(A T ) → i,j v a ij (a ij -1)/2 [a ij ]! v i<j,k a ik a jk [A]
is an homomorphism of U q (gl m )-U q (gl n )-bimodules.

One may notice that the expression i,j v a ij (a ij -1)/2 [a ij ]! v i<j,k a ik a jk also appears, in a slightly different context, in [12, p. 2896]. The proof of the following technical lemma relies on a result due to Levendorskii and Soibelman.

Lemma 4 Let λ = λ 1 ε 1 + • • • + λ m ε m ∈ P m
be a polynomial weight of degree d. In the U q (gl m )-module (V m ) ⊗d , the action of T ′′ i,+1 satisfies the relation: ) i e i+1,...,i+1 = e i,...,i . One may compute the action of T ′′ i,+1 on a tensor product of modules thanks to the link, discovered by Levendorskii and Soibelman, between the action of the q-Weyl group and the action of the "partial R-matrices". Remembering that we are using from the beginning the opposite coproduct as the one in [START_REF] Lusztig | Introduction to quantum groups[END_REF], the lemma and the proposition in [23, § 5.3] imply the following equality in (V m ) ⊗λ i ⊗ (V m ) ⊗λ i+1 :

T ′′ i,+1 (
T ′′ i,+1
• (e i,...,i ⊗ e i+1,...,i+1 ) = (-v) λ i e i+1,...,i+1 ⊗ e i,...,i .

Our formula can be deduced immediately from that.

Now we can prove our result:

Theorem 3 The images of s′ i and s′′ i in S d q (m) are given by the following sums, running over all the m-uples λ = (λ 1 , . . . , λ m ) of non-negative integers of sum d: ).

ζ(s ′ i ) = λ (-v)
Thus in the left U q (gl m )-module S d q (M * n,m ), by Proposition 4: s′′ i • X( j λ j E jj ) = (-v) λ i X( j∈{1,...,m}\{i,i+1} λ j E jj + λ i E i,i+1 + λ i+1 E i+1,i ) and so in the left U q (gl m )-module S d q (m, n), using Proposition 8: s′′ i • j λ j E jj = (-v) λ i j∈{1,...,m}\{i,i+1} λ j E jj + λ i E i+1,i + λ i+1 E i,i+1 .

Using the fact that the unit in the algebra S d q (m) is the sum of the [ j λ j E jj ] for (λ 1 , . . . , λ m ) running over the m-uples of non-negative integers of sum d, we find the second formula.

The Q(v)-linear map from S d q (m) to itself defined by [A] → [A T ] is an algebra antiautomorphism (see [START_REF] Beilinson | A geometric setting for the quantum deformation of GL n[END_REF]Lemma 3.10]). Comparing with Theorem 1, we see that this automorphism is induced by Φ. Applying this automorphism to our formula for s′′ i , we obtain the first formula.

It would be now easy to describe the elements s′ i and s′′ i with the help of the basis φ w λµ introduced by Dipper and James [5, p. 253]. We leave the translation to the reader, who will use the explicit isomorphism of Du [8, § 1.4]. The simplicity of the result hide the fact that the present approach gives for free the invertibility of ζ(s ′ i ) and ζ(s ′′ i ). This can for instance be used to simplify to proof of [START_REF] Green | A straightening formula for quantized codeterminants[END_REF]Lemma 2.4 (ii)], rendering the analysis of [loc. cit., § 2] unnecessary.

Applications 3.1 A result of Lusztig, Levendorskii and Soibelman

Let d ≤ m be two positive integers. It is well-known that the Hecke algebra H q (S d ) can be identified with a subalgebra of the q-Schur algebra (see [5, p. 256]). Maybe the simplest way to see that is the following. We can now relate this fact with the quantum Weyl group of U q (gl m ): for i ≤ d -1, the images under ζ of -s ′ i 1 ω = -1 ω s′ i 1 ω and of -s ′′ i 1 ω = -1 ω s′′ i 1 ω belong to [I d ]S d q (m)[I d ] and are both equal to the image of T s i under the map of Proposition 9. Taking m = d, we can state the following variant of the result of Lusztig, Levendorskii and Soibelman:

= (0 = V 0 ⊆ • • • ⊆ V m = V ) and f ′ = (0 = V ′ 0 ⊆ • • • ⊆ V ′ m = V ), one has that V d = V ′ d = V and that (V 0 ⊆ • • • ⊆ V d ) and (V ′ 0 ⊆ • • • ⊆ V ′ d )

  is an idempotent and the left multiplication by [A] in S d q (m, n) is the projection on the subspace spanned by the set {[B] | B ∈ Θ d m,n , ro(B) = co(A)}. An analogous property holds on the right. The unit in S d q (m) is the element A [A], where the sum runs over the set of diagonal matrices in Θ d m,m .

Proposition 1

 1 The equivalence classes in E d m,n are the fibers of ϕ : E d m,n → Θ d m,n . The subsets I d m,n and J d m,n of E d m,n contain each one exactly one element of each equivalence class for

  2.1], Proposition 1 is reminiscent of [loc. cit., Definition 4.3.1], and Proposition 2 corresponds to [loc. cit., Theorem 1.4.2]. The forthcoming Proposition 5 is an analogue of [loc. cit., Theorem 3.2.5] (taking into account [8, § 1.4]) and Lemma 3 has to do with [loc. cit., Lemma 4.3.2].

Proposition 9

 9 Consider the weight ω = ε 1 + • • • + ε d ∈ P m and the matrixI d = E 11 + • • • + E dd ∈ Θ d m,m . Then ζ(1 ω ) = [I d ] ∈ S d q(m) is an idempotent in the algebra S d q (m), and the algebra [I d ]S d q (m)[I d ] admits as basis the set of elements [A], for the matrices A such that ro(A) = co(A) = ω, that is the matrices A consisting of a permutation matrix of order d in the upper left corner, padded with zeros elsewhere. The mapH q (S d ) → [I d ]S d q (m)[I d ], T s i → v j∈{1,...,d}\{i,i+1} E jj + E i,i+1 + E i+1,iis an isomorphism of algebras.Proof. A matrix A ∈ Θ d m,m satisfies ro(A) = co(A) = ω if and only if for (f, f ′ ) ∈ O A , writing f

  are complete flags in V . Hence the above isomorphism is the usual geometric realization of the Hecke algebra of type A in terms of the geometry of flag varieties (see [2, Chap. IV, § 2, exerc. 22]).

  e 1,...,1Proof. It is easy to compute in (V m ) ⊗λ i , from the definition ofT ′′ i,+1 , that T ′′ i,+1 • e i,...,i = (-v) λ i F (λ i ) i e i,...,i = (-v) λ i e i+1,...,i+1 . With the help of [23, Proposition 5.2.2], one can also compute, in (V m ) ⊗λ i+1 , that T ′′ i,+1 • e i+1,...,i+1 = E

		,...,m,...,m	) = (-v) λ i e 1,...,1	,...,i-1,i+1,...,i+1 , i,...,i	,i+2,...,m,...,m
	λ 1	λm	λ 1	λ i	λ i+1
				(λ i+1

λm

).

  λ i j∈{1,...,m}\{i,i+1} λ j E jj + λ i E i,i+1 + λ i+1 E i+1,i , ζ(s ′′ i ) = λ (-v) λ i j∈{1,...,m}\{i,i+1} λ j E jj + λ i E i+1,i + λ i+1 E i,i+1 .

	Proof. Proposition 4 and Lemma 4 imply that
	s′′ i • π(e 1,...,1	,...,m,...,m	⊗f 1,...,1	,...,m,...,m	)
	λ 1	λm	λ 1	λm
			= π T ′′ i,+1 (e 1,...,1	,...,m,...,m	,...,m,...,m
					λ 1		λ 1	λm
			= (-v) λ i π(e 1,...,1	,...,i-1,i+1,...,i+1 , i,...,i	,i+2,...,m,...,m	⊗ f 1,...,1	,...,m,...,m
					λ 1		λ i	λ i+1	λm	λ 1	λm

λm ) ⊗ f 1,...,1

Acknowledgements. The author thanks F. Gavarini for bibliographic references, L. Blasco, J. Cassaigne, D. Manchon, G. Papadopoulo, P. Polo and M. Rosso for helpful discussions, and the examinators of his Ph.D. thesis for criticisms and encouragement. An anonymous referee kindly supplied me with a long list of misprints contained in a previous version of this article.

The research reported here has been supported by the french Ministry of Education, Research and Technology and by the CNRS.

Proposition 10 Let M be a finite-dimensional weight U q (gl m )-module. The operators T ′ i,+1 and T ′′ i,+1 of the q-Weyl group of U q (gl m ) act on M. They stabilize the weight subspace of M of weight ε 1 + • • • + ε m and satisfy on it the relations (-T i,+1 -q)(-T i,+1 + 1) = 0. This single result would not justify the development of all our machinery. In fact, we could have followed Levendorskii and Soibelman's method, without any further computation.

3.2 Quantum Schur-Weyl duality and quantum (gl m , gl n ) duality

The space (V n ) ⊗d is a U q (gl n )-H q (S d )-bimodule (see Section 1.3). Let us recall the following result of Jimbo.

Theorem 4 [START_REF] Jimbo | A q-analogue of U(gl(N + 1)), Hecke algebra and the Yang-Baxter equation[END_REF] The images of the algebras U q (gl n ) and H q (S d ) in End Q(v) ((V n ) ⊗d ) are each one the full commutant of the other.

The following result is known to the experts (see [9]), but apparently has not yet appeared in printed form. It may be called quantum (gl m , gl n ) duality.

Proposition 11 (a) The images in End Q(v) (S d q (M * m,n )) of (U q (gl m )) op and U q (gl n ) are each one the full commutant of the other.

) and (U q (gl n )) op are each one the full commutant of the other.

Proof. We first prove statement (b). Suppose that m ≤ n and denote by η ∈ S d q (n) the sum A [A] over all diagonal matrices A ∈ Θ d n,n such that ro(A) ∈ P m . Being the sum of a family of orthogonal idempotents, η is an idempotent. There are canonical isomorphisms S d q (m) ≃ η S d q (n) η and S d q (m, n) ≃ η S d q (n), compatible with the structure of S d q (m)-S d q (n)bimodule over S d q (m, n) (see Remark (b) in Section 1.4). Thus we have isomorphisms

whence one half of our assertion. The other half follows from the density theorem [16, Chap. 4, § 3], since the choice of our ground ring makes all our algebras semisimple. Statement (a) is now a simple consequence of Proposition 5 (a).

Our final result presents how to recover the quantum Schur-Weyl duality from the quantum (gl m , gl n ) duality (see [START_REF] Howe | Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond[END_REF] for instance in the classical case). Let n and d be integers. We consider the U q (gl n )-H q (S d )-bimodule (V n ) ⊗d and the U q (gl n )-U q (gl d )-bimodule S d q (M * d,n ). By Proposition 10 (and at the expense of exchanging left and right), the U q (gl d )-weight subspace of

)-bimodule under the action of the q-Weyl group of U q (gl d ).

Proposition 12 With the data above, the map

Proof. By Proposition 4 and Theorem 2 (a), the map u is an isomorphism of U q (gl n )-modules between the two aforementioned spaces. Lemma 4 and induction on length show that if w = s i 1 • • • s i k is a reduced expression of a permutation w ∈ S d , then the relation

(where T i,+1 denotes either T ′ i,+1 or T ′′ i,+1 ) holds in the H q (S d )-U q (gl d )-bimodule (V * d ) ⊗d . It follows from Proposition 4 that u preserves the action of H q (S d ).

Remarks. (a) A version of Proposition 12 has been announced in [19, p. 249].

(b) Proposition 8 shows that the U q (gl n )-U q (gl d )-modules S d q (M * d,n ) and S d q (n, d) are isomorphic in a natural way. Under this isomorphism, the subspace u

Up to a multiplication by the scalar v d(d-1)/2 , this is the geometric construction of quantum Schur-Weyl duality given in [10, § 1.13]. This is also the realization of the "q-tensor space" in [5, § 2.6]. These facts were already noticed by Du and R. M. Green, the new thing here is the relation with the q-Weyl group.

Finally, let us point out the difficulties that arise upon base ring change. Nearly all the constructions are valid in an integral setting: one has to replace the quantized enveloping algebra by its Lusztig form, but then Theorem 1 is still valid (see [START_REF] Du | A note on quantized Weyl reciprocity at roots of unity[END_REF]Theorem 3.4]), the q-Weyl group is still defined (see [START_REF] Lusztig | Introduction to quantum groups[END_REF]Chap. 41]), and Theorem 3 still holds. One can then specialize the objects in any commutative Z[v, v -1 ]-algebra. Theorem 4 extends to this setting (see [START_REF] Du | Quantum Weyl reciprocity and tilting modules[END_REF]Theorem 6.3]). Proposition 11 does not hold in general, but its validity is restored if one adds the assumption that m = n or that d ≤ min(m, n). Although the map given in Proposition 8 is no longer an isomorphism, it maps isomorphically the subspace


Eric Benoit

Laurent Foulloy

Keywords: Symbolic sensors Smart Sensors, Symbolic Sensors, Intelligent Sensors

come

Introduction

The concept of smart sensors or intelligent sensors introduced several years ago, is not so easy to define (Giachino [START_REF] Giachino | Smart sensors[END_REF]). One of the principal reasons is that many sensors use silicon as a transducer and have data processing integrated on the same medium (Jordan [START_REF] Jordan | Sensor technologies of the future[END_REF], Wadley [START_REF] Wadley | Sensors for intelligent processing of materials[END_REF], Muller [START_REF] Muller | Integrated Sensors[END_REF]). Therefore, some authors consider these integrated sensors as intelligent ones (Middelhoeck [START_REF] Middelhoek | Smart sensors: when and where?[END_REF], Yamasaki [START_REF] Yamasaki | new structure for intelligent sensors[END_REF]) whereas for other authors a microprocessor based sensor should be considered as intelligent. As for us, we think that if one wishes to introduce intelligence into the sensors, it is legitimate to wonder about what kind of intelligence is to be thought of. We consider that an intelligent sensor should own four basic mechanisms: learning, faculty of reasoning, perception (the main part of the sensor) and communication. Such a sensor has not been realized yet. Each simplified implementation of these mechanisms leads to one possible "definition" of the intelligent sensor. Nowadays, it is generally admitted that the characteristic functionality of a smart sensor should be its ability to communicate with a communication bus or network, to verify the correctness of the measurement and to adapt itself when the environment is changing (Giachino [START_REF] Giachino | Smart sensors[END_REF], Burd [START_REF] Burd | Intelligent transducers[END_REF], Bois [START_REF] Bois | Optical fibre displacement: how fibres can benefit smart sensors[END_REF]). One possible comparison between an analog sensor and a smart sensor (see Fig. 1.) has been proposed by Benoit E., Foulloy L., "Symbolic sensors", Proc of the 8th International Symposium on Artificial Intelligence based Measurement and Control (AIMaC'91), Ritsumeikan University, Kyoto, Japan, september 1991. pp. 131-136.

Favenec [START_REF] Favenec | Smart sensors in industry[END_REF]. However, this comparison should be discussed since the sensor should be an analog one and be intelligent in the sense defined previously.

Fig. 1. Comparison between a smart sensor and an analog one

Symbolic sensor

Nowadays, control systems become more and more complex. Artificial intelligence technics have been introduced to handle this complexity by means of new types of controllers (expert controller, intelligent controller, fuzzy controller...). As a consequence, such controllers are using more and more the symbolic coding. Rule based coding is certainly the most popular due to numbers of expert systems. For example, one can write the following rule for a temperature control problem. if temperature is high then reduce the control

Transducer

Smart sensor

Analog sensor

Physical phenomenon

Conditioner

Computation device

Communication interface

Digital data

Communication bus

Conditioner

Transducer

Transmiter

Analog data

However, even if controllers become more abstract, the perception and the control actions are still performed on the real world. Most often measurements are numerical values (e.g. the temperature in the previous example) which are converted in the controller itself. This numerical to symbolic interface can obviously be transferred into a sensor, leading to a new kind of components that we propose to call: symbolic sensors.

Definition: a symbolic sensor is a smart sensor which can create and handle one or several symbolic informations relative to the measurement. Just as smart sensors are the logical evolution from the analogical ones due to the extensive use of microprocessors, symbolic sensors can be seen as a natural consequence of symbolic based technics. Transferring the symbolic conversion into the sensor introduces specific problems that are emphasized in the following.

The relativity of the "symbolic measurement".

The use of a symbolic information leads us to wonder about the symbol semantics. Let us take an example in the robotics field. Assume the robot owns a distance sensor which currently returns 10cm for the distance between its tool and a workpiece. Considering an insertion task, the measurement semantics could certainly be "the distance is long" while in a painting case it may become "the distance is short". Just as for human beings, the semantics depends on the context in which the task is to be realized. Therefore, the sensor cannot be considered any longer as an independent organ, but must be seen as being part of the perception mechanism in which we include decisions and actions. From this fundamental property, one can conclude that the symbolic sensor must be configurated (by a supervisor for instance) with respect to the measurement context. We exhibit here one of the major differences between a symbolic sensor and a symbolic conversion within a controller or an expert system: the symbolic sensor is a general component that can be reconfigured to adapt itself to the measurement context. Since symbolic sensors keep all smart sensor properties, a first functional scheme can be the following one: Fig. 2. Internal scheme of a symbolic sensor The symbolic sensor concept will be emphasized in the following by a formal description of the links between the symbolic domain and the numerical one (Luzeaux [START_REF] Luzeaux | Normalizing symbolic reasoning[END_REF]).

Translation, concepts and interpretation

Let L be the symbolic domain and E be the numerical one.

The meaning of a symbolic value will be called a translation and be defined as an injective application from the symbolic set to the set of the subsets of the numerical domain (injectivity insures that two identical symbols have the same translation).

τ: L → P(E)

The association of a symbolic value and its translation is called a concept. The symbolic measurement will be obtained by means of a new application from the numerical domain to the symbolic one, called an interpretation.

ι: E → L Relations between the translation and the interpretation are summarized in Fig. 3.

Fig. 3. Relations between a translation and an interpretation

These definitions have been used to implement symbolic sensors based on crisp sets coding (Benoit [START_REF] Benoit | Symbolic sensors : one solution to the numerical-symbolic interface[END_REF]). Several conditions on translations have been assumed:

• Intersections between translations are empty.

• The numerical domain is ordered with a relation noted < E .

• Any translation is an interval:

∀L ∈ L, ∀x, y ∈ τ(L) such that x < E y, ∀z ∈ E x < E z < E y ⇒ z ∈ τ(L)
The numerical to symbolic interface is realized by the interpretation function. The simplest way to interpret a measurement x is to return the symbol L such that x belongs to its translation.

L = ι(x) if x ∈τ(L)
This relationship is an application when τ(L) is a partition on E.

From concepts to fuzzy concepts

In the next sections, the problem of symbolic sensors dealing with imprecise or uncertain knowledge is addressed (for example, a zero detector should certainly work more around zero than at zero precisely). Zadeh's approximate reasoning theory provides a smart tool for implementing the numeric to symbolic interface. Therefore the material presented in the following is not really new in terms of symbolic manipulations based on fuzzy set (see Zadeh [START_REF] Zadeh | Outline of a new approach to the analysis of complex systems and decision processes[END_REF], [START_REF] Zadeh | Outline of a new approach to the analysis of complex systems and decision processes[END_REF]) but it provides a quite powerful way for implementing symbolic sensors. To extend the previous section to the fuzzy case, fuzzy intervals have to be defined. Let us recall that a fuzzy interval is a fuzzy subset on the set of real numbers such that (Zadeh [START_REF] Zadeh | Fuzzy sets[END_REF])

L E τ L A = τ(L) A ∈P(E) x ι • • ∀ u, v, ∀ x ∈ [u, v], µ Q (x) ≥ min(µ Q (u), µ Q (v))
A fuzzy concept can be defined as the association of a symbol L and a fuzzy translation (i.e. a fuzzy interval). Two kinds of fuzzy intervals are usually recognized (Eshragh [START_REF] Eshragh | A general approach to linguistic approximation -Int[END_REF]):

• Fuzzy intervals defined as non decreasing or non increasing functions ("S" curves).

• Fuzzy intervals defined as a non increasing then non decreasing function ("Π" curves). A general definition of fuzzy intervals by means of a parametrized class of functions (L-R functions) has been proposed by Dubois [START_REF] Dubois | Fuzzy sets and systems : theory and applications -1980[END_REF].

Creating new concepts

Obviously, it is a tedious task to specify each translation of the symbols. Moreover, in this case, one can say that the sensor is no more smart. The symbolic sensor works with several symbols which are in relation by means of the semantics. Let us take the example of a temperature measurement. A semantic relation links the symbolic values hot and very_hot due to the order relation on the numerical domain. This relationship between all the symbols should be managed by the sensor itself. Defining a new concept leads to give a new symbol and its translation. Modifiers, usually called linguistic hedges, can be introduced to perform such an operation. Each operation in the symbolic domain is linked to an operation on the numerical one. Let us begin with crisp sets, a function F operating in the symbolic domain is defined.

F: L ¡ L L 1 ¡ L 2 = F(L 1 )
Now a function f F operating on the numerical domain is in- f F: P(E) ¡ P(E)

τ(L 1 ) ¡ τ(L 2 ) = τ(F(L 1 )) = f F (τ(L 1 ))
Links between these functions are represented in Fig. 5.

Fig. 5. Creating a new concept

The crisp case can be extended easily to the fuzzy one. The definition domain of f F becomes the set of fuzzy subsets of E with the following relation on membership functions:

µ τ(L 2 ) (x) = µ τ(F(L 1 )) (x)= f F (µ τ(L 1 ) (x))
To simplify the reading, µ τ(L 2 ) will be written µ L 2 . A more general definition has been proposed by Novak [START_REF] Novak | Fuzzy sets and their applications[END_REF]:

µ τ(L 2 ) (x) = µ τ(F(L 1 )) (x)= m F (µ τ(L 1 ) (q F (x))) m F : [0,1] → [0,1
] is a modifier for the hedge F while q F is a translator. Since two types of intervals have been defined, it is legitimate to define transformations between them. Four transformations can be exhibited:

• "S" curve to "S" curve • "Π" curve to "Π" curve • "Π" curve to "S" curve • "S" curve to "Π" curve Only the first three transformations seem to be of some interest for symbolic sensors: they are presented in the following.

"S" curve to "S" curve operators.

These operators are well known, most of them have been introduced by Zadeh [START_REF] Zadeh | Outline of a new approach to the analysis of complex systems and decision processes[END_REF]. Let C be a fuzzy concept and µ C its associated translation, the new generated concept and its translation will be respectively noted C1 and µ C1 . Several classical operators are recalled below:

Based on Novak 's definition, the translator is the identity for all previous cases. Let us also recall that binary operators such as C1 and C2, C1 or C2 are commonly defined by means of T-norms and T-conorms (see for example Pedrycz [START_REF] Pedrycz | Fuzzy control and fuzzy systems[END_REF]).

A first example of a symbolic sensor can now be given. Several symbolic informations are sent to the sensor via the communication bus (see Fig. 6.). The first one, called the "generic concept", is used by the sensor to define all the other concepts. Here, it is defined by an arctangent function "centered" in 5m.

L E τ

L 1 A = τ(L 1 ) • L 2 • F f F B = τ(F(L 1 ))

Symbolic domain Numerical domain

C1 = very (C) C1 = not(C) C1 = more_or_less(C) µ C1 (x) = µ 2 C (x) µ C1 (x) = 1-µ C (x) µ C1 (x) µ 1/2 C (x)
Fig. 6. Configuration of a symbolic sensor Once these informations have been sent, the sensor is able to provide symbolic measurements. Therefore, it can return, for a given measurement, either the symbolic value (based on the interpretation function) or the grade of membership for each symbol in the lexical domain (the numerical value and many other informations such that identification, installation date, revision date... can also be obtained) (see Fig. 9.). Based on configuration informations, the sensor generates membership functions for all the measurement domain (see Fig. 7.). Fig. 7. New concepts generated from the generic one distance_is_long

"Π" curve to "Π" curve operators

Operators defined by Zadeh act through left composition of functions. Since "Π" curves are not injective, these operators cannot be used to generate a new "Π" curve from a "Π" curve. To define such a new concept, one solution is to apply a translation. Using Novak's definition, one will now have the identity as modifier and x ¡ x ± ∆ as translator. This operation can also be seen as the convolution of the initial concept by a delayed Dirac's distribution. Just as composition, convolution could be taken as a general mechanism to define new hedges. Using previous writing conventions, new operators can be introduced: two of them are given below (δ is the Dirac's distribution and * the convolution operation). Let us give an example for a symbolic sensor dealing with "Π" curves. The generic concept is defined by a normalized gaussian function.

Symbolic domain Numerical domain

C1 = more_than (C) C1 = less_than(C) µ C1 (x) =µ C (x) * δ(x-∆) µ C1 (x) = µ C (x) * δ(x+∆)
Obviously, operators more_than and less_than give new gaussian functions defined by:

µ more_than(C) (x) = gaussian(m c + ∆, σ c , x) µ less_than(C) (x) = gaussian(m c -∆, σ c , x)
The translation value can be chosen such that the initial and the new concepts have intersecting points where the grade of membership is 0.5. For gaussian functions we have: Fig. 8. shows the example of a simple symbolic sensor dealing with three concepts defined from the generic one: distance_is_correct. Once again, the sensor generates the concept translations from configuration informations. The sensor can then be queried as shown in Fig. 9.

µ C x ( ) gaussian m C σ C x , , ( ) e x m C - ( ) 2 2σ C 2 ---------------------- - = = ∆ 2 σ C 2 ln 2 ( ) × × × =

"Π" curve to "S" curve operators

To define the last type of operators, let us assume that two opposite notions exist (e.g. cold / warm, far / close, heavy / light...). They will be noted N and N in the following. Let C be a concept defined by a "Π" curve and m be a modal value of this concept (i.e. µ C (m) = 1). Two operators and their respective reciprocals can be defined as follows:

∃ x 1 ≥ m such that ∀ x > x 1 , µ more_N_than(C) (x) = µ less_N_than(C) (x) > µ C (x) and µ more_N_than(C) is a non decreasing function ∃ x 2 ≤ m such that ∀ x < x 2 , µ less_N_than(C) (x) = µ more_N_than(C) (x) < µ C (x) and µ less_N_than(C) is a non increasing function
Several definitions are possible for these operators, Eshragh and Mamdani [START_REF] Eshragh | A general approach to linguistic approximation -Int[END_REF] have proposed two operators above and below which verify the previous properties. Let us recall their definitions:

Once again, let us build a simple symbolic sensor dealing with "Π" to "S" operators from a generic concept defined as a gaussian function (see Fig. 10.). Fig. 10. Symbolic sensor using "Π" to "S" operators.

All the previous examples can be run with other generic functions (e.g. triangle, trapezoidal,....), assuming they are known by the sensor (i.e. they belong to the database). Only the

Symbolic domain

Numerical domain

C1 =above (C) C1 =below(C) µ C1 (x) = 0 if x < m = 1-µ C (x) else µ C1 (x) = 1-µ C (x) if x < m = 0 else

Taking environment into account

In the preceding, the generic concept has been taken as an a priori information given to the sensor (every other concepts is automatically worked with by means of operators). This approach preserves the semantics between concepts but does not guarantee the coherence of symbolic interpretations with respect to the measurement context (e.g. based on its configuration, the sensor returns distance_is_very_long while for the given context it should have returned distance_is_long). Therefore, taking the environment into account is nothing more than adapting the interpretation to the measurement context. Several approaches are possible, a first one based on qualitative learning with a teacher has been proposed in Benoit [START_REF] Benoit | Symbolic sensors : one solution to the numerical-symbolic interface[END_REF]. A functional approach which modifies interpretations through composition is presented here.

Assume the generic concept and the new generated ones are defined on the numerical domain E called the ideal universe. Let E' be the measurement domain and h be a function, called the adaptation function, from E' to E (see Fig. 11.) Fig. 11. Adapting the sensor to the environment The interpretation of a measurement for crisp set is now: L = ι(h(x)) if x ∈τ(L) To run this approach, the adaptation function should be sent to the symbolic sensor. However, all adaptation functions should have specific properties that preserve the coherence of configuration informations. Especially, the generic concept definition point should not be changed by the adaptation function, furthermore the interpretation should remain linear around this point. Let m c be the generic concept definition point, the two previous properties lead to: h(m c ) = m c and h'(m c ) = 1 Effects of the adaptation function on the measurement domain are called semantic properties (compressing or expanding the definition domain, expanding after a particular measurement...). The choice of a "good" adaptation function (i.e. having particular semantic properties and the previous mathematical properties) is not so easy. One solution is to choose an adaptation function having the desired semantics, then apply a transformation to get the mathematical properties. Let g be a function with the desired semantics at point x p , then the adaptation function can be defined as follows (it is assumed that g'(x p ) ≠ 0):

h(x) = a.g(x-m c + x p ) + b a = 1 / g'(x p ) et b = m c -g(x p ) / g'(x p ) L E τ L 1 ι • • E' x • h
From these considerations, let us give four examples of functions g with interesting semantic properties: • expand_before_expand_after g(x) = argsh(k.x) • compress_before_compress_after g(x) = sh(k.x) • compress_before_expand_after g(x) = sh(k.x) if x ≥ 0 g(x) =argsh(k.x) else • expand_before_compress_after g(x) =argsh(k.x) if x ≥ 0 g(x) = sh(k.x) else These functions have been chosen due to their asymptotic properties and their behavior around the origin (here x p = 0). Fig. 12. shows resulting adaptation functions h for the generic concept definition point in 5 m. Fig. 12. presents the membership functions for a symbolic sensor dealing with seven concepts defined from a gaussian generic one whose definition point is in 5m: the adaptation function is obtained from the g compress_before_expand_after function.

Conclusion

Implementing a part of the knowledge and of the decision process into symbolic sensors results from the same logical evolution that lead to integrate analog to digital conversion and signal processing into smart sensors several years ago. This decentralization effort is devoted to the development of more abstract control systems that are able to work with high level informations.

Fig. 4 .Fig. 4 .

 44 Fig. 4. Several possible definitions of the concept distance_is_long

Fig. 8 .Fig. 9 .

 89 Fig. 8. Symbolic sensor based on "Π" curves

  have to be changed in configurations.

Fig. 12 .

 12 Fig. 12. Adaptating the sensor to the context