GENERATORS AND RELATIONS FOR SCHUR ALGEBRAS

STEPHEN DOTY AND ANTHONY GIAQUINTO

Abstract. We obtain a presentation of Schur algebras (and q-Schur algebras) by generators and relations, one which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra 𝔤𝔩_{n} of n × n matrices. We also find several new bases of Schur algebras.

1. Introduction

The classical Schur algebra S(n, d) (over ℚ) may be defined as the algebra End_{Σ_{d}}(V^{⊗d}) of linear endomorphisms on the dth tensor power of an n-dimensional ℚ-vector space V commuting with the action of the symmetric group Σ_{d}, acting by permutation of the tensor places (see [6]). Schur algebras determine the polynomial representation theory of general linear groups, and they form an important class of quasi-hereditary algebras.

We identify V with ℚ^{n}. Then V_{ℤ} = ℤ^{n} is a lattice in V. One can define a ℤ-order S_{ℤ}(n, d) (the integral Schur algebra) in S(n, d) by setting S_{ℤ}(n, d) = End_{Σ_{d}}(V_{ℤ}^{⊗d}). For any field K, one then obtains the Schur algebra S_{K}(n, d) over K by setting S_{K}(n, d) = S_{ℤ}(n, d) ⊗_{ℤ} K. Moreover, S_{ℚ}(n, d) ≅ S(n, d).

In the quantum case one can replace ℚ by ℚ(v) (v an indeterminate), V by an n-dimensional ℚ(v)-vector space, and Σ_{d} by the corresponding Hecke algebra H(Σ_{d}). Then the resulting commuting algebra, S(n, d), is known as the q-Schur algebra, or quantum Schur algebra. It appeared first in work of Dipper and James, and, independently, Jimbo. Beilinson, Lusztig, and MacPherson [1] have given a geometric construction of S(n, d) in terms of orbits of flags in vector spaces. (See also [5].)

In this situation ℤ is replaced by the ring A = ℤ[v, v^{−1}], and there is a corresponding “integral” form S_{A}(n, d) in S(n, d). The above quantized objects specialize to their classical versions when v = 1. We have more detailed information about the Schur algebras in rank 1; see [2] and [3]. Proofs of the main results will appear in [4].

2. Serre’s presentation of U

Let Φ = {ε_{i} − ε_{j} | 1 ≤ i ≠ j ≤ n} be the root system of type A_{n−1}, with simple roots Δ = {ε_{i} − ε_{i+1} | i = 1, …, n − 1}, where {ε_{1}, …, ε_{n}} denotes the standard basis of ℤ^{n}. The corresponding set Φ^{+} of positive roots is the set Φ^{+} = {ε_{i} − ε_{j} | 1 ≤ i < j ≤ n}.

Date: April 2, 2001.

2000 Mathematics Subject Classification. Primary 16P10, 16S15; Secondary 17B35, 17B37.

Key words and phrases. Schur algebras, finite-dimensional algebras, enveloping algebras, quantized enveloping algebras.

The abelian group ℤ^{n} has a bilinear form ( , ) given by (ε_{i}, ε_{j}) = δ_{ij} (Kronecker delta). We write α_{j} := ε_{j} − ε_{j+1}.

The enveloping algebra U = U(𝔤𝔩_{n}) is the associative algebra (with 1) on generators e_{i}, f_{i} (i = 1, …, n − 1) and H_{i} (i = 1, …, n) with relations

(R1)  H_{i}H_{j} = H_{j}H_{i}
(R2)  e_{i}f_{j} − f_{j}e_{i} = δ_{ij}(H_{j} − H_{j+1})
(R3)  H_{i}e_{j} − e_{j}H_{i} = (ε_{i}, α_{j})e_{j}, H_{i}f_{j} − f_{j}H_{i} = −(ε_{i}, α_{j})f_{j}
e_{i}^{2}e_{j} − 2e_{i}e_{j}e_{i} + e_{j}e_{i}^{2} = 0 (|i − j| = 1)
e_{i}e_{j} − e_{j}e_{i} = 0 (otherwise)

(R4)

f_{i}^{2}f_{j} − 2f_{i}f_{j}f_{i} + f_{j}f_{i}^{2} = 0 (|i − j| = 1)
f_{i}f_{j} − f_{j}f_{i} = 0 (otherwise).

(R5)

For α ∈ Φ, let x_{α} denote the corresponding root vector of 𝔤𝔩_{n} viewed as an element of U. We have in particular e_{i} = x_{α_{i}} and f_{i} = x_{−α_{i}}.

3. The quantized enveloping algebra

The Drinfeld-Jimbo quantized enveloping algebra U = U_{v}(𝔤𝔩_{n}), by definition, is the ℚ(v)-algebra with generators E_{i}, F_{i} (1 ≤ i ≤ n − 1), K_{i}^{±1} (1 ≤ i ≤ n) and relations

K_{i}K_{j} = K_{j}K_{i}, K_{i}K_{i}^{−1} = K_{i}^{−1}K_{i} = 1 (Q1)

E_{i}F_{j} − F_{j}E_{i} = δ_{ij} \frac{K_{i}K_{i+1}^{−1} − K_{i}^{−1}K_{i+1}}{v − v^{−1}} (Q2)

K_{i}E_{j} = v^{(ε_{i},α_{j})}E_{j}K_{i}, K_{i}F_{j} = v^{−(ε_{i},α_{j})}F_{j}K_{i} (Q3)

E_{i}^{2}E_{j} − (v + v^{−1})E_{i}E_{j}E_{i} + E_{j}E_{i}^{2} = 0 (|i − j| = 1)
E_{i}E_{j} − E_{i}E_{j} = 0 (otherwise)

(Q4)

F_{i}^{2}F_{j} − (v + v^{−1})F_{i}F_{j}F_{i} + F_{j}F_{i}^{2} = 0 (|i − j| = 1)
F_{i}F_{j} − F_{j}F_{i} = 0 (otherwise).

(Q5)

For α ∈ Φ^{+}, let E_{α} and F_{α} be the positive and negative quantum root vectors of U as defined by Jimbo [8]. We have in particular E_{i} = E_{α_{i}} and F_{i} = F_{α_{i}}.

4. Main results: classical case

We now give a precise statement of our main results in the classical case. The first result describes a presentation by generators and relations of the Schur algebra over the rational field ℚ. This presentation is compatible with the usual presentation (see section 2) of U = U(𝔤𝔩_{n}).

Theorem 1. Over ℚ, the Schur algebra S(n, d) is isomorphic to the associative algebra (with 1) on the same generators as for U subject to the same relations (R1)–(R5) as for U, together with the additional relations:

H_{1} + H_{2} + ⋯ + H_{n} = d (R6)

H_{k}(H_{k} − 1) ⋯ (H_{k} − d) = 0, (k = 1, ⋯, n). (R7)

The next result gives a basis for the Schur algebra which is the analogue of the Poincare-Birkhoff-Witt (PBW) basis of U.

Theorem 2. The algebra S(n, d) has a “truncated PBW” basis (over ℚ) which can be described as follows. Fix an integer k_{0} with 1 ≤ k_{0} ≤ n and set

G = {x_{α} | α ∈ Φ} ∪ {H_{k} | k ∈ {1, …, n} − {k_{0}} }

and fix some ordering for this set. Then the set of all monomials in G (with specified order) of total degree at most d is a basis for S(n, d).

Our next result constructs the integral Schur algebra S_{ℤ}(n, d) in terms of the generators given above. We need some more notation. For B in ℕ^{n}, we write

H_{B} = ∏_{k=1}^{n} \binom{H_{k}}{b_{k}}.

Let Λ(n, d) be the subset of ℕ^{n} consisting of those λ ∈ ℕ^{n} satisfying |λ| = d; this is the set of n-part compositions of d. Given λ ∈ Λ(n, d) we set 1_{λ} := H_{λ}. One can show that the collection {1_{λ}} as λ varies over Λ(n, d) forms a set of pairwise orthogonal idempotents in S_{ℤ}(n, d) which sum to the identity element.

For m ∈ ℕ and α ∈ Φ, set x_{α}^{(m)} := x_{α}^{m}/(m!). Any product in U of elements of the form

x_{α}^{(m)}, \binom{H_{k}}{m} (m ∈ ℕ, α ∈ Φ, k ∈ \{1, …, n\}),

taken in any order, will be called a Kostant monomial. Note that the set of Kostant monomials is multiplicatively closed, and spans U_{ℤ}. We define a function χ (content function) on Kostant monomials by setting

χ(x_{α}^{(m)}) := m ε_{max(i,j)}, χ(\binom{H_{k}}{m}) := 0

where α = ε_{i} − ε_{j} (i ≠ j), and by declaring that χ(XY) = χ(X) + χ(Y) whenever X, Y are Kostant monomials.

We write any A ∈ ℕ^{Φ^{+}} in “multi-index” form A = (a_{α})_{α∈Φ^{+}} and set |A| := ∑_{α∈Φ^{+}} a_{α}. For A, C ∈ ℕ^{Φ^{+}} we write

e_{A} = ∏_{α∈Φ^{+}} x_{α}^{(a_{α})}, f_{C} = ∏_{α∈Φ^{−}} x_{α}^{(c_{α})}

where the products in e_{A} and f_{C} are taken relative to any fixed orders on Φ^{+} and Φ^{−}.

Theorem 3. The integral Schur algebra S_{ℤ}(n, d) is the subring of S(n, d) generated by all divided powers e_{j}^{(m)}, f_{j}^{(m)}(j ∈ {1, …, n − 1}, m ∈ ℕ). Moreover, this algebra has a ℤ-basis consisting of all

e_{A}1_{λ}f_{C} (χ(e_{A}f_{C}) ⪯ λ) (B1) and another such basis consisting of all

f_{A}1_{λ}e_{C} (χ(f_{A}e_{C}) ⪯ λ) (B2)

where A, C ∈ ℕ^{Φ^{+}}, λ ∈ Λ(n, d), and where ⪯ denotes the componentwise partial ordering on ℕ^{n}.

Finally, we have another presentation of the Schur algebra by generators and relations. This presentation has the advantage that it possesses a quantization of the same form, in which we can set v = 1 to recover the classical version.

Theorem 4. For each i ∈ {1, …, n − 1} write α_{i} := ε_{i} − ε_{i+1}. The algebra S(n, d) is the associative algebra (with 1) given by generators 1_{λ} (λ ∈ Λ(n, d)), e_{i}, f_{i} (i ∈ {1, …, n − 1}) subject to the relations

\begin{matrix}1_{λ}1_{μ} = δ_{λ,μ}1_{λ} (λ, μ ∈ Λ(n, d)) \\ ∑_{λ∈Λ(n,d)} 1_{λ} = 1 \end{matrix} (S1)

e_{i}1_{λ} = \begin{cases}1_{λ+α_{i}}e_{i} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

f_{i}1_{λ} = \begin{cases}1_{λ−α_{i}}f_{i} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}e_{i} = \begin{cases}e_{i}1_{λ−α_{i}} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}f_{i} = \begin{cases}f_{i}1_{λ+α_{i}} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

(S2)

e_{i}f_{j} − f_{j}e_{i} = δ_{ij} ∑_{λ∈Λ(n,d)}(λ_{j} − λ_{j+1})1_{λ} (S3)

along with the Serre relations (R4), (R5), for i, j ∈ {1, …, n − 1}.

5. Main results: quantized case

Our main results in the quantized case are similar in form to those in the classical case. The first result describes a presentation by generators and relations of the quantized Schur algebra over the rational function field ℚ(v). This presentation is compatible with the usual presentation (see section 3) of U = U_{v}(𝔤𝔩_{n}).

Theorem 1′. Over ℚ(v), the q-Schur algebra S(n, d) is isomorphic with the associative algebra (with 1) on the same generators as for U subject to the same relations (Q1)–(Q5) as for U, together with the additional relations:

K_{1}K_{2} ⋯ K_{n} = v^{d} (Q6)

(K_{j} − 1)(K_{j} − v)(K_{j} − v^{2}) ⋯ (K_{j} − v^{d}) = 0, (j = 1, ⋯, n). (Q7)

The next result gives a basis for the q-Schur algebra which is the analogue of the Poincare-Birkhoff-Witt (PBW) type basis of U, given in Lusztig [9, Prop. 1.13].

Theorem 2′. The algebra S(n, d) has a “truncated PBW type” basis which can be described as follows. Fix an integer k_{0} with 1 ≤ k_{0} ≤ n and set

G′ = {E_{α}, F_{α} | α ∈ Φ^{+}} ∪ {K_{k} | k ∈ {1, …, n} − {k_{0}} }

and fix some ordering for this set. Then the set of all monomials in G′ (with specified order) of total degree at most d is a basis for S(n, d).

Note that setting v = 1 in the basis of Theorem 2′ does not yield the basis of Theorem 2 since K_{i} acts as the identity when v = 1.

Our next result constructs the integral q-Schur algebra S_{A}(n, d) in terms of the generators given above. For B in ℕ^{n}, we write

K_{B} = ∏_{j=1}^{n} [\begin{matrix}K_{j} \\ b_{j} \end{matrix}] .

where for indeterminates X, X^{−1} satisfying XX^{−1} = X^{−1}X = 1 and any t ∈ ℕ we formally set

[\begin{matrix}X \\ t \end{matrix}] := ∏_{s=1}^{t} \frac{Xv^{−s+1} − X^{−1}v^{s−1}}{v^{s} − v^{−s}} ,

an expression that makes sense if X is replaced by any invertible element of a ℚ(v)-algebra.

Given λ ∈ Λ(n, d) we set (when we are in the quantum case) 1_{λ} := K_{λ}. It turns out that, just as in the classical case, the collection {1_{λ}} as λ varies over Λ(n, d) forms a set of pairwise orthogonal idempotents in S_{A}(n, d) which sum to the identity element.

Let [m] denote the quantum integer [m] := (v^{m} −v^{−m})/(v −v^{−1}) and set [m]! := [m][m − 1] ⋯ [1]. Then the q-analogues of the divided powers of root vectors are defined to be E_{α}^{(m)} := E_{α}/[m]! and F_{α}^{(m)} := F_{α}/[m]!. Any product in U of elements of the form

E_{α}^{(m)}, F_{α}^{(m)}, [\begin{matrix}K_{j} \\ m \end{matrix}] (m ∈ ℕ, α ∈ Φ, j ∈ \{1, …, n\}),

taken in any order, will be called a Kostant monomial. As before, the set of Kostant monomials is multiplicatively closed, and spans U_{A}. The definition of content χ of a Kostant monomial is obtained similarly, by setting

χ(E_{α}^{(m)}) = χ(F_{α}^{(m)}) := m ε_{max(i,j)}, χ([\begin{matrix}K_{l} \\ m \end{matrix}]) := 0

where α = ε_{i} − ε_{j} ∈ Φ^{+}, and by declaring that χ(XY) = χ(X) + χ(Y) whenever X, Y are Kostant monomials. For A, C ∈ ℕ^{Φ^{+}} we write

E_{A} = ∏_{α∈Φ^{+}} E_{α}^{(a_{α})}, F_{C} = ∏_{α∈Φ^{+}} F_{α}^{(c_{α})}

where the products in E_{A} and F_{C} are taken relative to any two specified orderings on Φ^{+}.

Theorem 3′. The integral q-Schur algebra S_{A}(n, d) is the subring of S(n, d) generated by all quantum divided powers E_{j}^{(m)}, F_{j}^{(m)} (j ∈ {1, …, n − 1}, m ∈ ℕ), along with the elements [\begin{matrix}K_{j} \\ m \end{matrix}] (j ∈ \{1, …, n\}, m ∈ ℕ). Moreover, this algebra has a basis

over A consisting of all

(B1′)  E_{A}1_{λ}F_{C}  (χ(e_{A}f_{C}) ⪯ λ)

and another such basis consisting of all

(B2′)  F_{A}1_{λ}E_{C}  (χ(f_{A}e_{C}) ⪯ λ)

where A, C ∈ ℕ^{Φ^{+}}, λ ∈ Λ(n, d), and where ⪯ denotes the componentwise partial ordering on ℕ^{n}.

Unlike the truncated PBW basis, the bases of Theorem 3′ do specialize when v = 1 to their classical analogues given in Theorem 3.

Finally, we have another presentation of the q-Schur algebra by generators and relations. This presentation has the advantage that by setting v = 1, we recover the classical version given in Theorem 4. The relations of the presentation are similar to relations that hold for Lusztig’s modified form U̇ of U. (See [9, Chap. 23].)

Theorem 4′. For each i ∈ {1, …, n−1} write α_{i} := ε_{i} −ε_{i+1}. The algebra S(n, d) is the associative algebra (with 1) given by generators 1_{λ} (λ ∈ Λ(n, d)), E_{i}, F_{i} (i ∈ {1, …, n − 1}) subject to the relations

\begin{matrix}1_{λ}1_{μ} = δ_{λ,μ}1_{λ} (λ, μ ∈ Λ(n, d)) \\ ∑_{λ∈Λ(n,d)} 1_{λ} = 1 \end{matrix} (S1′)

E_{i}1_{λ} = \begin{cases}1_{λ+α_{i}}E_{i} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

F_{i}1_{λ} = \begin{cases}1_{λ−α_{i}}F_{i} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}E_{i} = \begin{cases}E_{i}1_{λ−α_{i}} & if λ − α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

1_{λ}F_{i} = \begin{cases}F_{i}1_{λ+α_{i}} & if λ + α_{i} ∈ Λ(n, d) \\ 0 & otherwise \end{cases}

(S2′)

E_{i}F_{j} − F_{j}E_{i} = δ_{ij} ∑_{λ∈Λ(n,d)}[λ_{j} − λ_{j+1}]1_{λ} (S3′)

along with the q-Serre relations (Q4), (Q5), for i, j ∈ {1, …, n − 1}.

6. Other results

6.1. Triangular decomposition. One can define the plus, minus, and zero parts of Schur algebras in terms of the generators, as follows. (These subalgebras have been studied before.) The plus part S^{+}(n, d) (resp., the minus part S^{−}(n, d)) is the subalgebra of S(n, d) generated by all x_{α}, α ∈ Φ^{+} (resp., α ∈ Φ^{−}). The zero part S^{0}(n, d) is the subalgebra generated by all H_{j}, j = 1, …, n. We also have the Borel Schur algebras S^{⩾0}(n, d) (resp., S^{⩽0}(n, d)), the subalgebra generated by S^{+}(n, d) (resp., S^{−}(n, d)) together with S^{0}(n, d).

The appellations S_{ℤ}^{+}(n, d), S_{ℤ}^{−}(n, d), S_{ℤ}^{0}(n, d), S_{ℤ}^{⩾0}(n, d), S_{ℤ}^{⩽0}(n, d) will denote the intersection of the appropriate algebra from above with the integral form S_{ℤ}(n, d).

The algebra S = S(n, d) has a triangular decomposition S = S^{+}S^{0}S^{−}. We show that in this decomposition one can permute the three factors in any order. Moreover, the same result holds over ℤ.

The zero part S_{ℤ}^{0}(n, d) is the algebra generated by all \binom{H_{j}}{m} for j = 1, …, n and

m ∈ ℕ. We give in [4] a presentation of S^{0}(n, d) by generators and relations. In particular, we prove that H_{B} = 0 whenever |B| > d (B ∈ ℕ^{n}) and that the set of (pairwise orthogonal) idempotents 1_{λ}, λ ∈ Λ(n, d), is a ℤ-basis of S_{ℤ}^{0}(n, d).

S_{ℤ}^{+}(n, d) (resp., S_{ℤ}^{−}(n, d)) is the algebra generated by all divided powers x_{α}^{(m)} for α ∈ Φ^{+} (resp., α ∈ Φ^{−}) and m ∈ ℕ. It is an easy consequence of the commutation formulas (S2) that each generator x_{α} is nilpotent of index d + 1; see [4] for details. Moreover, from our main results we see easily that the set of all e_{A} (resp., f_{A}) such that |A| ≤ d is a ℤ-basis for the algebra S_{ℤ}^{+}(n, d) (resp., S_{ℤ}^{−}(n, d)).

It also follows immediately from our results that the set of all e_{A}1_{λ} (resp., 1_{λ}f_{A}) satisfying χ(e_{A}) ⪯ λ is an integral basis for the Borel Schur algebra S_{ℤ}^{⩾0}(n, d) (resp., S_{ℤ}^{⩽0}(n, d)).

Similar statements to the above hold in the quantum case. In particular, as a corollary of the commutation formulas (S2′) one can give a simple proof of [7, Prop. 2.3].

6.2. Explicit reduction formulas. Fix a positive root α and write α = ε_{i} − ε_{j} for i < j.

Then from [2] we have the following reduction formulas in S_{ℤ}(n, d), for any a, b, c ∈ ℕ:

(1) f_{α}^{(a)}\binom{H_{j}}{b}e_{α}^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}f_{α}^{(a−k)}\binom{H_{j}}{b + k}e_{α}^{(c−k)}

(2) e_{α}^{(a)}\binom{H_{i}}{b}f_{α}^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s}\binom{k − 1}{s − 1}\binom{b + k}{k}e_{α}^{(a−k)}\binom{H_{i}}{b + k}f_{α}^{(c−k)}

where s = a + b + c − d and s ≥ 1.

We do not have a q-analogue of these formulas.

The results of [3] give another type of reduction formula for S_{A}(n, d). If b_{1}, b_{2} ∈ ℕ satisfy b_{1} + b_{2} = d, set λ := b_{1}ε_{i} + b_{2}ε_{j} ∈ Λ(n, d). Then for all s ≥ 1 we have:

(3)  E_{α}^{(a)}1_{λ}F_{α}^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{1} + k \\ k \end{matrix}] E_{α}^{(a−k)}1_{λ+kα}F_{α}^{(c−k)}

(4)  F_{α}^{(a)}1_{λ}E_{α}^{(c)} = ∑_{k=s}^{min(a,c)}(−1)^{k−s} [\begin{matrix}k − 1 \\ s − 1 \end{matrix}] [\begin{matrix}b_{2} + k \\ k \end{matrix}] F_{α}^{(a−k)}1_{λ−kα}E_{α}^{(c−k)}

where s = a + b_{1} + c − d in (3) and s = a + b_{2} + c − d in (4).

The classical analogues of formulas (3) and (4) hold in S(n, d).

6.3. Another presentation (for n = 2). We have the following result from [2], which presents S(2, d) as a quotient of U(𝔰𝔩_{2}).

Theorem 5. Over ℚ, the Schur algebra S(2, d) is isomorphic with the associative algebra (with 1) generated by e, f, h subject to the relations:

he − eh = 2e;  ef − fe = h;  hf − fh = −2f
(h + d)(h + d − 2) ⋯ (h − d + 2)(h − d) = 0.

Moreover, this algebra has a “truncated PBW” basis over ℚ consisting of all f^{a}h^{b}e^{c} such that a + b + c ≤ d.

Note that if we eliminate the last relation we have the usual presentation of U(𝔰𝔩_{2}) over ℚ. The last relation is the minimal polynomial of h in the representation on tensor space. The problem of presenting S(n, d) as a quotient of U(𝔰𝔩_{n}) seems to be more difficult for n > 2.

We also have from [3] the following q-version of the above, which presents the q-Schur algebra S(2, d) as a quotient of the quantized enveloping algebra U(𝔰𝔩_{2}).

Theorem 5′. Over ℚ(v), the quantum Schur algebra S(2, d) is isomorphic to the algebra generated by E, F, K^{±1} subject to the relations:

KK^{−1} = K^{−1}K = 1

KEK^{−1} = v^{2}E KF K^{−1} = v^{−2}F

EF − F E = \frac{K − K^{−1}}{v − v^{−1}}

(K − v^{d})(K − v^{d−2}) ⋯ (K − v^{−d+2})(K − v^{−d}) = 0.

6.4. Hecke algebras. Suppose that n ≥ d. Let ω = (1^{d}). Then the subalgebra 1_{ω}S(n, d)1_{ω} is isomorphic with the Hecke algebra H(Σ_{d}). The nonzero elements of the basis (B1′) of the form 1_{ω}E_{A}F_{C}1_{ω} form a basis of the Hecke algebra; similarly for elements of the basis (B2′) of the form 1_{ω}F_{A}E_{C}1_{ω}.

Moreover, taking d = n, we can see that H = H(Σ_{n}) is generated by the elements 1_{ω}E_{i}F_{i}1_{ω} (1 ≤ i ≤ n − 1). Alternatively, H is generated by the 1_{ω}F_{i}E_{i}1_{ω} (1 ≤ i ≤ n − 1). One can easily describe the relations on these generators, thus obtaining a presentation of H which is closely related to one in [10].

References

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Loyola University Chicago, Chicago, Illinois 60626 U.S.A.

E-mail address: doty@math.luc.edu

Loyola University Chicago, Chicago, Illinois 60626 U.S.A.

E-mail address: tonyg@math.luc.edu