The q -Weyl group of a q -Schur algebra

The q -Weyl group of a q -Schur algebra

Pierre Baumann

Abstract

Theq-Schur algebras of Dipper and James are quotients of the quantized enveloping algebras U_q(gl_m) of Drinfeld and Jimbo. Theq-Weyl group of U_q(gl_m) (also known as Lusztig’s automorphisms braid group) induces a group of inner automorphisms of theq- Schur algebras. We describe precisely elements in theq-Schur algebras that define these inner automorphisms. This description allows us to recover certain known properties of the q-Weyl group.

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

, . . . , s

}. Let W f be the braid group of W , that is the group with generators {¯ s

, . . . , s ¯

} and relations

¯

s

s ¯

s ¯

· · · = ¯ s

s ¯

s ¯

· · · , with m

factors in each side, (m

) being the Coxeter matrix. There are several morphisms of W f into G (see [25]), through which f W acts on the integrable g- modules, 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 f on U (g) become the restrictions of inner automorphisms of U [ (g).

Let U

(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

). Lusztig, Levendorskii and Soibelman have defined invertible elements ¯ s

, . . . , s ¯

in some completion U [

(g) of U

(g). These elements define a morphism f W →

U [

(g)

, so that f W acts on the integrable U

(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 f W on the zero- weight 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 [22, § 1][20, Sect. 4].

Another way to see that is the following. The Hecke algebra admits a geometric re- alization in terms of double B-cosets in G, hence in terms of G-orbits of pairs of flags

1991 Mathematics Subject Classification: 16S50, 17B37.

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

(f, f

) ∈ (G/B)

. In the same vein, Beilinson, Lusztig and MacPherson [1] have constructed certain finite-dimensional algebras S

(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

(gl

). (Du [8] has observed that the algebra S

(m) was iso- morphic 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

(m) is a quotient not only of U

(gl

) but even of U \

(gl

).

It therefore makes sense to consider the images in S

(m) of the generators ¯ s

, . . . , s ¯

of the q-Weyl group of U

(gl

). Despite the complicated original definition of the ¯ s

, 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 concern- ing the quantized enveloping algebras U

(gl

), the q-Schur algebras S

(m), and the spaces S

( M

) 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 [11]) the standard basis theorems for S

( M

) and S

(m) (proved algorithmically in [15] and [12]). In Section 2, we recall the definition of the q-Weyl group of U

(gl

) and explain how to compute the images of the generators ¯ s

, . . . , s ¯

under the epimorphism U

(gl

) S

(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

, gl

)-duality and how all this is related to the aforementioned result of Lusztig, Levendorskii and Soibelman.

1 q-Deformations

1.1 Basic notations

Let m be a positive integer. We denote by P

the free Z-module with basis (ε

, . . . , ε

).

Elements of P

are called weights. A weight λ = λ

ε

+ · · · + λ

ε

is said to be polynomial (respectively dominant) if all the λ

are non-negative (respectively if λ

≥ · · · ≥ λ

). The degree of a weight λ = λ

ε

+ · · · + λ

ε

is the integer |λ| = λ

+ · · · + λ

. The simple roots are the elements α

= ε

− ε

(for 1 ≤ i ≤ m − 1). We endow P

with the bi-additive form P

× P

→ Z defined by (ε

|ε

) = δ

(Kronecker’s symbol). The symmetric group on m letters S

acts on the set {ε

, . . . , ε

}, hence acts Z-linearly on P

. The image W of S

in Aut(P

) is generated by the simple reflections s

: P

→ P

, λ 7→ λ − (α

|λ)α

(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

. The q-numbers, q-factorials and q-binomial coefficients are defined as in [23]:

[n] =

, [n]! = Q

i=1

[i],

k

=

.

1.2 Quantized enveloping algebra

Let m be a positive integer and let (a

)

be the Cartan matrix: a

= (α

|α

).

Definition 1 [17] The quantized enveloping algebra U

(gl

) is the Q(v)-algebra presented by the generators K

(for λ ∈ P

), E

, . . . , E

and F

, . . . , F

, with the relations:

K

K

= K

(for λ, µ ∈ P

),

K

E

= v

E

K

and K

F

= v

F

K

(for λ ∈ P

and 1 ≤ i ≤ m − 1), P

r=0

(−1)

r

E

E

E

= 0 (for 1 ≤ i 6= j ≤ m − 1), P

r=0

(−1)

F

F

F

= 0 (for 1 ≤ i 6= j ≤ m − 1), [E

, F

] = δ

K

− K

v − v

(for 1 ≤ i, j ≤ m − 1).

It is a Hopf algebra when endowed with the coproduct ∆ and the counit ε given by:

∆(K

) = K

⊗ K

, ε(K

) = 1 (for λ ∈ P

),

∆(E

) = E

⊗ K

+ 1 ⊗ E

, ε(E

) = 0 (for 1 ≤ i ≤ m − 1),

∆(F

) = F

⊗ 1 + K

⊗ F

, ε(F

) = 0 (for 1 ≤ i ≤ m − 1).

The natural U

(gl

)-module is the vector space V

with basis (e

, . . . , e

), and the action of U

(gl

) is given by:

K

· e

= v

e

(for λ ∈ P

and 1 ≤ k ≤ m),

E

· e

= δ

e

and F

· e

= δ

e

(for 1 ≤ i ≤ m − 1 and 1 ≤ k ≤ m).

The dual of V

is a right U

(gl

)-module. We will denote it by V

and we will denote by (f

, . . . , f

) the basis dual to (e

, . . . , e

).

The comultiplication of U

(gl

) allows to endow the tensor power (V

)

with the struc- ture of a U

(gl

)-module. We will denote by e

the element e

⊗ · · · ⊗ e

in (V

)

. Analogously, (V

)

is a right U

(gl

)-module, with basis f

.

The subalgebra U

generated in U

(gl

) by the elements K

is commutative. A vector e in a U

(gl

)-module is said to be of weight λ ∈ P

if the action of U

on e is scalar and given by the character K

7→ v

. A left U

(gl

)-module M will be called a weight module if it is the sum of its weight subspaces:

M = L

λ∈P_m

{e ∈ M | ∀µ ∈ P

, K

· e = v

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

(gl

) → U

(gl

)

given on the generators by K

7→ K

, E

7→ F

, F

7→ E

. With it, one has the notion of

contravariant duality.

1.3 Hecke algebras of type A and the quantum Schur–Weyl dual- ity

Let d be a positive integer, S

be the symmetric group on d letters, and S = {s

, . . . , s

} be the set of usual generators of S

.

Definition 2 The Hecke algebra H

(S

) of the Coxeter system (S

, S) is the Q(v)-algebra presented by the generators (T

)

, with the relations:

T

T

= T

T

(for |i − j| ≥ 2), T

T

T

= T

T

T

(for 1 ≤ i ≤ d − 2), (T

− q)(T

+ 1) = 0 (for 1 ≤ i ≤ d − 1).

For each w ∈ S

, one can define T

as the product T

· · · T

, where s

· · · s

is a reduced decomposition of w. The (T

)

form a basis of H

(S

).

The algebra H

(S

) acts on the spaces (V

)

and (V

)

, by means of the following formulas:

e

· T

=



 

 

v e

if j

< j

,

q e

if j

= j

,

v e

+ (q − 1)e

if j

> j

,

T

· f

=



 

 

v f

if j

< j

,

q f

if j

= j

,

v f

+ (q − 1)f

if j

> j

.

In this way, (V

)

becomes a U

(gl

)-H

(S

)-bimodule and (V

)

is its dual H

(S

)- U

(gl

)-bimodule. These assertions are part of the theory of quantized Schur–Weyl duality, due to Jimbo (see Section 3.2).

1.4 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 [1].

Let m, n and d be three positive integers, and denote by Θ

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

⊆ · · · ⊆ V

= V ) and f

= (0 = V

⊆ · · · ⊆ V

= 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

= dim(V

∩ V

) − dim(V

∩ V

) − dim(V

∩ V

) + dim(V

∩ V

), so that dim(V

∩ V

) = P

r≤i,s≤j

a

. Let X

be the set of all m-step filtrations in V . The group GL(V ) acts on all the X

, hence acts also (diagonally) on X

× X

. The map (f, f

) 7→ c(f, f

) defines a bijection between the set of GL(V )-orbits in X

× X

and the set of matrices Θ

.

If A ∈ Θ

, we denote by O

the corresponding GL(V )-orbit in X

× X

. To a matrix A ∈ Θ

, we associate the weights ro(A) = P

i,j

a

ε

and co(A) = P

i,j

a

ε

belonging to

P

and P

respectively. If (f, f

) ∈ O

, with f = (0 = V

⊆ · · · ⊆ V

= V ) and f

= (0 = V

⊆ · · · ⊆ V

= V ), then ro(A) = P

i=1

dim(V

/V

)ε

and co(A) = P

j=1

dim(V

/V

)ε

. Let m, n, p be three positive integers and let A ∈ Θ

, B ∈ Θ

, C ∈ Θ

. There exists a polynomial g

∈ Z[v

] satisfying the following property: if F is a finite field with q elements, if one chooses filtrations f ∈ X

and f

∈ X

such that (f, f

) belongs to the orbit O

, then the value at v

= q of polynomial g

is equal to the number of filtrations f

∈ X

such that (f, f

) ∈ O

and (f

, f

) ∈ O

. One has g

= 0 except if ro(A) = ro(C), co(A) = ro(B ), and co(B) = co(C). If m = p and A ∈ Θ

is a diagonal matrix, then g

is δ

or 0 according as ro(B) is equal or different from co(A). If p = n and B ∈ Θ

is a diagonal matrix, then g

is δ

or 0 according as co(A) is equal or different from ro(B).

We denote by S

(m, n) the Q(v)-vector space with basis the family of symbols (e

)

m,n

. The coefficients g

afford a bilinear map S

(m, p) × S

(p, n) → S

(m, n), (e

, e

) 7→ P

c∈Θ^d_m,n

g

e

, and we have an associativity property for these “prod- ucts”. Letting m = n = p, we see that S

(m, m) is an algebra; the unit is the element P

A

e

, where the sum runs over the set of diagonal matrices in Θ

. Similarly, S

(m, n) is a S

(m, m)- S

(n, n)-bimodule. We will simplify the notation and denote S

(m, m) by S

(m).

Finally if A ∈ Θ

, one puts [A] = v

e

in S

(m, n). From the above, it follows that if A ∈ Θ

is a diagonal matrix, then [A] ∈ S

(m) is an idempotent and the left multiplication by [A] in S

(m, n) is the projection on the subspace spanned by the set {[B] | B ∈ Θ

, ro(B) = co(A)}. An analogous property holds on the right. The unit in S

(m) is the element P

A

[A], where the sum runs over the set of diagonal matrices in Θ

. 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.

To a polynomial weight λ = λ

ε

+ · · · + λ

ε

in P

of degree d, one can associate the parabolic subgroup S

= S

× · · · × S

of S

and the element x

= P

w∈S_λ

T

in H

(S

). In the U

(gl

)-H

(S

)-bimodule (V

)

, the weight space (for the action of U

(gl

)) for the weight λ is a right H

(S

)-submodule isomorphic to the induced module x

H

(S

). Hence the commutant of the image of the algebra H

(S

) inside End

(V

)

is isomorphic to the algebra End

L

λ

x

H

(S

)

, where the sum runs over all polynomial weights λ ∈ P

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

(gl

) to S

(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

be the field with

q elements, let G be the group GL

(F

), let B be its standard Borel subgroup, and

identify the symmetric group S

with the Weyl group of G in the usual way. To each

weight λ ∈ P

of degree d is associated the standard parabolic subgroup P

= B S

B

of G. Let M (respectively N

) be the right RG-module obtained by inducing the trivial

representation of B (respectively P

) to G. The module M (respectively N

) can be identified with the set of functions from G/B (respectively G/P

) to R, where G acts by left translation.

The endomorphism algebra End

(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

(S

)

. The subspace x

H

(S

)

may be identified with the set of functions from B\G/P

to R.

Now M is an H

(S

)

-RG-bimodule, so we have the functor Hom

(M, −) from the category of right RG-modules to the category of right H

(S

)

-modules. It sends the object N

to the object x

H

(S

)

, and if µ ∈ P

is another weight of degree d, it defines a bijection between the spaces of homomorphisms Hom

(N

, N

) and Hom

x

H

(S

)

, x

H

(S

)

. (This is Theorem 2.24 of [4]. Both spaces are naturally in bijection with the space of functions from P

\G/P

to R, which can be identified with the subspace H

(S

)

x

∩ x

H

(S

)

; 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

( L

λ

N

). This correspondence, traced at the level of “generic” algebras, explains the aforementioned result of [8].

(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

, → Θ

, 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

(m, n) , → S

(N ), defined on the basic vectors. If N is also greater than p and if A ∈ Θ

, B ∈ Θ

, then the product [A][B] is the same computed in S

(N ) or by the map S

(m, p) × S

(p, n) → S

(m, n). In particular, we have a map S

(m) , → S

(N ), and we denote by η

the image in S

(N ) of the unit element in S

(m). Thus η

is the sum of elements [A], for A ∈ Θ

running over the set of diagonal matrices with co(A) ∈ P

. Then η

is an idempotent in S

(N), the algebra S

(m) may be identified with the algebra η

S

(N ) η

, and the S

(m)- S

(n)-bimodule S

(m, n) may be identified with the subspace η

S

(N ) η

.

There is an algebra homomorphism from U

(gl

) to S

(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

(gl

) to S

(m) which:

• sends K

to P

v

[j

E

+· · ·+j

E

], the sum running over all the m-uples of non-negative integers (j

, . . . , j

) of sum d;

• sends E

to P

[j

E

+ · · · + j

E

+ E

], the sum running over all the m-uples of non-negative integers (j

, . . . , j

) of sum d − 1;

• sends F

to P

[j

E

+ · · · + j

E

+ E

], the sum running over all the m-uples of

non-negative integers (j

, . . . , j

) of sum d − 1.

So S

(m) is a quotient of U

(gl

) and S

(m, n) is a U

(gl

)-U

(gl

)-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

(gl

)-U

(gl

)-bimodule structure on S

(m, n), one has for all A ∈ Θ

:

K

· [A] = v

[A], E

· [A] =

n

P

p=1 ah+1,p≥1

[a

+ 1] v

[A + E

− E

],

F

· [A] =

n

P

p=1 ahp≥1

[a

+ 1] v

[A − E

+ E

],

[A] · K

= v

[A], [A] · E

=

m

P

p=1 aph≥1

[a

+ 1] v

[A − E

+ E

],

[A] · F

=

m

P

p=1 ap,h+1≥1

[a

+ 1] v

[A + E

− E

].

As a consequence, the left U

(gl

)-module S

(m) is a finite-dimensional polynomial module of degree d.

1.5 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 6= n also appears in numerous places.

Definition 3 The Q(v)-algebra presented by generators X

(1 ≤ i ≤ m and 1 ≤ j ≤ n) and relations:

X

X

=



 



 



X

X

if r > s and t < u or if r < s and t > u, v X

X

if r > s and t = u or if r = s and t > u, X

X

+ (v − v

)X

X

if r > s and t > u.

is called the algebra of functions on the quantum matrix space and is denoted by S

( M

).

S

( M

) is in a natural way a N-graded algebra, whose d-th homogeneous components will be denoted by S

( M

).

Fix the natural integer d, and let I

and J

be the sets of all the d-tuples of pairs of integers ((i

, j

), . . . , (i

, j

)) ∈ ({1, . . . , m} × {1, . . . , n})

defined by the conditions of being lexicographically non-decreasing:

I

= {((i

, j

), . . . , (i

, j

)) | i

≤ i

, i

= i

⇒ j

≤ j

},

J

= {((i

, j

), . . . , (i

, j

)) | j

≤ j

, j

= j

⇒ i

≤ i

}.

Theorem 2 [24, § 3.5]

(a) The set consisting of the vectors X

· · · X

for ((i

, j

), . . . , (i

, j

)) running over I

(or running over J

) is a Q(v)-basis in S

( M

), which is thus of dimension

mn+d−1 d

.

(b) S

( M

) is an algebra without zero-divisor.

We will need later another parametrization of the basis given by the above theorem.

Let E

be the set of all d-uples of pairs of integers ((i

, j

), . . . , (i

, j

)) ∈ ({1, . . . , m} × {1, . . . , n})

such that for all 1 ≤ a < b ≤ d, one has i

< i

, j

< j

or (i

, j

) = (i

, j

).

Recall that Θ

is the set of all matrices of size m ×n, whose coefficients are non-negative integers of sum d. Let E

∈ Θ

denotes the elementary matrix, with a 1 in position (i, j) and with 0 elsewhere. To the element ((i

, j

), . . . , (i

, j

)) of E

, one associates the matrix ϕ((i

, j

), . . . , (i

, j

)) = P

a=1

E

∈ Θ

.

On E

, we put the equivalence relation ≈ generated by the following elementary moves:

((i

, j

), . . . , (i

, j

)) ∼ ((i

, j

), . . . , (i

, j

)) if and only if there exists 1 ≤ a < d such that (i

− i

)(j

− j

) < 0 and ((i

, j

), . . . , (i

, j

)) is obtained from ((i

, j

), . . . , (i

, j

)) by exchanging (i

, j

) with (i

, j

).

Proposition 1 The equivalence classes in E

are the fibers of ϕ : E

→ Θ

. The subsets I

and J

of E

contain each one exactly one element of each equivalence class for ≈. If ((i

, j

), . . . , (i

, j

)) and ((i

, j

), . . . , (i

, j

)) are two ≈-equivalent elements in E

, then X

· · · X

= X

1,j₁⁰

· · · X

d,j_d⁰

.

Proof. One first remark that ϕ induces bijections between I

, J

and Θ

. If (i

, . . . , j

)

∈ E

, then (i

> i

) ⇒ (j

< j

). By induction on the number of inversions in the sequence (i

, . . . , i

), one then shows that ((i

, j

), . . . , (i

, j

)) is equivalent for the relation

≈ to at least one element of I

. Since ϕ : E

→ Θ

is surjective and constant on the equivalence classes, the relation ≈ has at least Card Θ

= Card I

equivalence classes.

One deduces from this that I

is a set of representatives for the equivalence classes of ≈ in E

. The same is true for J

. From the fact that ϕ : I

→ Θ

is a bijection, one deduces that ϕ induces a bijection between E

/ ≈ and Θ

. The last assertion of the lemma follows from the first relation in Definition 3.

If B ∈ Θ

, we will denote by X(B) the element X

· · · X

∈ S

( M

) with ((i

, j

), . . . , (i

, j

)) ∈ E

∩ ϕ

(B). The interesting structure on S

( M

) is given in the following proposition.

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

( M

) to S

( M

) ⊗ S

( M

) which sends X

to P

k=1

X

⊗ X

.

Letting m = n = p in this proposition, we see that the algebra S

( M

) may be endowed

with the structure of a bialgebra. The counit is given by ε(X

) = δ

for 1 ≤ i, j ≤ m. Letting

p = m (respectively p = n) in the proposition, we see that S

( M

) is a left S

( M

)- comodule (respectively a right S

( M

)-comodule). It is in fact a S

( M

)-S

( M

)- bicomodule. Each homogeneous component S

( M

) is a subcoalgebra of S

( M

) and S

( M

) is a S

( M

)-S

( M

)-bicomodule.

Remark. There is a stabilization procedure here. Let N be an integer greater than m, n and p. Let us denote by K

the two-sided ideal generated in the algebra S

( M

) by the elements X

with i > m or j > n. The quotient algebra S

( M

)/ K

iden- tifies naturally with the algebra S

( M

). If N is also greater than p, then the coprod- uct given in Proposition 2 is compatible with this construction since it sends K

into

K

⊗ S

( M

)

+ S

( M

) ⊗ K

.

Let us denote by t

∈ (U

(gl

))

the matrix coefficients of the natural U

(gl

)-module V

: t

: x 7→ hf

, x · e

i. They belong to the restricted (Hopf) dual (U

(gl

))

of U

(gl

).

Proposition 3 There is a bialgebra morphism κ from S

( M

) to (U

(gl

))

which sends X

to t

.

Proof. The R-matrix of U

(gl

) defines the U

(gl

)-linear map:





 (V

)

→ (V

)

, e

7→



 

 

e

if j

< j

, v e

if j

= j

, e

+ (v − v

)e

if j

> j





 .

The U

(gl

)-linearity of this map implies that the elements t

satisfy the relations of Def- inition 3. We thus have an algebra homomorphism from S

( M

) to (U

(gl

))

, that clearly respects the coproduct and the counit.

The S

( M

)-S

( M

)-bicomodule structure on S

( M

) gives rise to a (U

(gl

))

- (U

(gl

))

-bicomodule structure on S

( M

), and thus to a U

(gl

)-U

(gl

)-bimodule structure on S

( M

). Another way to describe that is to look at the U

(gl

)-U

(gl

)- bimodule (V

)

⊗ (V

)

, with basis e

⊗ f

= (e

⊗ · · · ⊗ e

) ⊗ (f

⊗ · · · ⊗ f

).

Proposition 4 The linear map

π : (V

)

⊗ (V

)

→ S

( M

), e

⊗ f

7→ X

· · · X

factorizes through (V

)

⊗

(V

)

and defines an isomorphism of U

(gl

)-U

(gl

)- bimodules between this latter space and S

( M

).

Proof. The relations defining the algebra S

( M

) imply that the kernel of π is exactly the subspace spanned by the vectors e · T

⊗ f − e ⊗ T

· f for e ∈ (V

)

, f ∈ (V

)

and 1 ≤ k ≤ d − 1. Hence π defines an injection from (V

)

⊗

(V

)

to S

( M

).

Further, π is surjective.

We now fix (j

, . . . , j

) ∈ {1, . . . , n}

and consider the map

u : (V

)

→ S

( M

), f

7→ X

· · · X

.