On a Theorem of Lehrer and Zhang

Jun Hu, Zhankui Xiao

Received: September 05, 2011 Revised: November 4, 2011

Communicated by Wolfgang Soergel

Abstract. Let K be an arbitrary field of characteristic not equal to 2. Let m, n ∈ ℕ and V be an m dimensional orthogonal space over K. There is a right action of the Brauer algebra 𝔅_{n}(m) on the n-tensor space V^{⊗n} which centralizes the left action of the orthogonal group O(V). Recently G.I. Lehrer and R.B. Zhang defined certain quasiidempotents E_{i} in 𝔅_{n}(m) (see (1.1)) and proved that the annihilator of V^{⊗n} in 𝔅_{n}(m) is always equal to the two-sided ideal generated by E_{[(m+1)/2]} if char K = 0 or char K > 2(m+1). In this paper we extend this theorem to arbitrary field K with char K ≠ 2 as conjectured by Lehrer and Zhang. As a byproduct, we discover a combinatorial identity which relates to the dimensions of Specht modules over the symmetric groups of different sizes and a new integral basis for the annihilator of V^{⊗m+1} in 𝔅_{m+1}(m).

2010 Mathematics Subject Classification: 20B30, 15A72, 16G99 Keywords and Phrases: Brauer algebras, tensor spaces, symmetric groups, standard tableaux

1 Introduction

Let ℕ be the set of non-negative integers. Let x be an indeterminate over ℤ and 0 < n ∈ ℕ. The Brauer algebra 𝔅_{n}(x) over ℤ[x] was introduced by Richard Brauer (see [1]) when he studied how the n-tensor space V^{⊗n} decomposes into irreducible modules over the orthogonal group O(V) or the symplectic group Sp(V), where V is an orthogonal vector space or a symplectic vector space. It was defined as the free ℤ[x]-module on the basis of the set Bd_{n} of all Brauer n-diagrams, graphs on 2n vertices, and n edges with the property that every vertex is incident to precisely one edge. The multiplication of two Brauer ndiagrams is defined using natural concatenation of diagrams. Precisely, we compose two diagrams D_{1}, D_{2} by identifying the bottom row of vertices in D_{1} with the top row of vertices in D_{2}. The result is a graph, with a certain number, n(D_{1}, D_{2}), of interior loops. After removing the interior loops and the identified vertices, retaining the edges and remaining vertices, we obtain a new Brauer n-diagram D_{1} ∘ D_{2}, the composite diagram. Then we define D_{1} · D_{2} = x^{n(D_{1},D_{2})}D_{1} ∘ D_{2}. For example, let d be the following Brauer 5diagram.

Let d′ be the following Brauer 5-diagram.

Then dd′ is equal to

In general, the multiplication of two elements in 𝔅_{n}(x) is given by the linear extension of a product defined on diagrams. For each integer i with 1 ≤ i ≤ 2n, we define i^{−} := 2n + 1 − i. The Brauer algebra 𝔅_{n}(x) is a free ℤ[x]-module with rank (2n − 1) · (2n − 3) ⋯ 3 · 1. For any ℤ[x]-algebra R with x specialized to δ ∈ R, we define 𝔅_{n}(δ)_{R} := R ⊗_{ℤ[x]} 𝔅_{n}(x).

Now let K be an arbitrary field of characteristic not equal to 2. Let m, n be two positive integers and V an m dimensional orthogonal space over K. Let 𝔅_{n}(m) be the specialized Brauer algebra with parameter m · 1_{K}. There is a right action of 𝔅_{n}(m) on the n-tensor space V^{⊗n} which commutes with the left action of the orthogonal group O(V). If K = ℂ, then by a well-known result of Brauer [1], the canonical homomorphism φ : 𝔅_{n}(m) → End_{O(V)}(V^{⊗n}) is surjective. In general, as long as K is an infinite field of characteristic not equal to 2, the surjection still holds and we actually have a characteristic-free version of the Schur–Weyl duality between 𝔅_{n}(m) and KO(V) on V^{⊗n}. For the proof as well as the symplectic version of these results, we refer the readers to [7], [10] and [11].

The above Schur–Weyl duality is closely related to the second fundamental theorem in invariant theory for O(V). By [12] and [27], 𝔅_{n}(m) is semisimple if and only if m ≥ n − 1. From the representation theoretic point of view, it is desirable to describe the radical of 𝔅_{n}(m) in the non-semisimple case. By [16], the kernel of φ is closely related to the radical of 𝔅_{n}(m). Therefore, it is important to understand the kernel of φ. Note that φ is not injective if and only if n ≥ m + 1. In [16, Theorem 4.8], using the invariant theory for O(V), Gavarini showed that the kernel of φ is spanned by some diagrammatic minors of order m + 1 (which are certain alternating sum of some Brauer n-diagrams). Note that, however, those diagrammatic minors are not necessarily K-linearly independent. In [11, Theorem 1.4, Theorem 6.9], an integral basis for the kernel of φ was obtained. The Brauer algebra 𝔅_{n}(m) can be endowed with a right 𝔖_{2n}-module structure in a way such that Ker φ is an 𝔖_{2n}-submodule (cf. [11] and [15]). So far, to the best of our knowledge, it remains an open question on whether or not there exists a characteristic-free basis for Ker φ which consists of some diagrammatic minors of order m + 1.

In [20, Corollary 5.9], we proved in the symplectic case that Ker φ is always generated by one specific diagrammatic Pfaffian of order 2m + 2. In the quantized type C case, we proved (in [20, Proposition 5.6]) a similar statement under the assumption that the quantum parameter q is generic. Recently, G.I. Lehrer and R.B. Zhang have studied extensively the orthogonal case in [22] by connecting it with the second fundamental theorem of invariant theory for the orthogonal group. For each Brauer n-diagram D ∈ Bd_{n}, the vertices of D are arranged in two rows: the top and bottom rows. The vertices in top row are labeled by the indices 1, 2, ⋯, n from left to right; while the vertices in bottom row are labeled by the indices 1^{−}, ⋯, n^{−} from left to right. The following key definitions are due to them.

Definition 1.1. ([22, Definition 4.2]) Let a, b ∈ ℕ such that 1 ≤ a + b ≤ n. Let Bd(a, b) be the set of all Brauer n-diagrams D such that:

(1) for each integer s with a + b + 1 ≤ s ≤ n, D connects the vertex labeled by s with the vertex labeled by s^{−}; and

(2) for each integer s with s ∈ {1, 2, ⋯, a, (a + 1)^{−}, (a + 2)^{−}, ⋯, (a + b)^{−}}, D connects the vertex labeled by s with the vertex labeled by t for t ∈ {1^{−}, 2^{−}, ⋯, a^{−}, a + 1, a + 2, ⋯, a + b}.

We define

E_{a,b} := ∑_{D∈Bd(a,b)} sign(D)D, E_{i} := E_{i,m+1−i}, ∀ 0 ≤ i ≤ m + 1.

We refer the reader to Definition 2.2 for the definition of sign(D).^{1} Lehrer and Zhang have proved a number of important properties about those E_{i}. In particular, they have proved the following theorem in [22].

Theorem 1.2. ([22, Proposition 4.4, Theorems 4.3, 9.4]) Assume that m < n. Then for each integer i with 0 ≤ i ≤ [(m + 1)/2], E_{i} ∈ Ker φ. Furthermore, if char K = 0 or char K > 2(m + 1) then Ker φ is the two-sided ideal of 𝔅_{n}(m) generated by E_{[(m+1)/2]}.

Furthermore, Lehrer and Zhang have conjectured in [22, Remark 9.5] that the second statement of the above theorem is true for arbitrary field K with char K ≠ 2. The main result in this paper is a proof of this conjecture. In other words, we extend Lehrer and Zhang’s theorem to arbitrary field K with char K ≠ 2. That is,

Theorem 1.3. Let K be an arbitrary field of characteristic other than two. Then Ker φ is always equal to the two-sided ideal generated by E_{[(m+1)/2]}.

As a byproduct, we discover (in Theorem 4.15 and Corollary 4.17) a combinatorial identity which connects to the dimensions of some Specht modules over the symmetric group 𝔖_{2m+2} and the symmetric group 𝔖_{m+1}. We get (in Corollary 4.19) a new integral basis for the annihilator of V^{⊗m+1} in 𝔅_{m+1}(m). The content is organized as follows. In Section 2 we recall some basic knowledge about the structure and representation theory of the Brauer algebras as well as some related combinatorics which are needed later. In Section 3 we prove that the annihilator of V^{⊗n} in 𝔅_{n}(m) is equal to the two-sided ideal generated by E_{0}, E_{1}, …, E_{[(m+1)/2]}. The proof makes essential use of the integral basis of Ker φ obtained in [11]. In Section 4 we prove our main result Theorem 1.3. The proof will proceed in three steps. The main strategy to prove Theorem 1.3 is to transform it into a statement about identification between certain two-sided ideals in the symmetric group algebra K𝔖_{n}. For the latter, we make use of the Young seminormal basis and the Murphy basis theory of the symmetric group algebras as well as the first main result obtained in Section 3.

ACKNOWLEDGMENTS. The first author is supported by the Australian Research Council and the National Natural Science Foundation of China (Grant No. 11171021). The second author is supported by a research foundation of Huaqiao University (Grant No. 10BS323).

1At a first look, the definition of E_{i} which we give here seems to be different with [22, Definition 4.2]. But they are indeed the same. The equivalence between the two definitions follows from a simple counting by the proof given in the paragraph directly below [22, (4.5)].

Documenta Mathematica 17 (2012) 245–270

2 The Brauer algebra

The Brauer algebra 𝔅_{n}(x) can be alternatively defined as the unital associative ℤ[x]-algebra with generators s_{1}, ⋯, s_{n−1}, e_{1}, ⋯, e_{n−1} and relations (see [13]):

s_{i}^{2} = 1, e_{i}^{2} = xe_{i}, e_{i}s_{i} = e_{i} = s_{i}e_{i}, ∀ 1 ≤ i ≤ n − 1,
s_{i}s_{j} = s_{j}s_{i}, s_{i}e_{j} = e_{j}s_{i}, e_{i}e_{j} = e_{j}e_{i}, ∀ 1 ≤ i < j − 1 ≤ n − 2,

s_{i}s_{i+1}s_{i} = s_{i+1}s_{i}s_{i+1}, e_{i}e_{i+1}e_{i} = e_{i}, e_{i+1}e_{i}e_{i+1} = e_{i+1}, ∀ 1 ≤ i ≤ n − 2, s_{i}e_{i+1}e_{i} = s_{i+1}e_{i}, e_{i+1}e_{i}s_{i+1} = e_{i+1}s_{i}, ∀ 1 ≤ i ≤ n − 2.

Note that the subalgebra of 𝔅_{n}(x) generated by s_{1}, ⋯, s_{n−1} is isomorphic to the symmetric group algebra ℤ[x]𝔖_{n} of 𝔖_{n} over ℤ[x] and s_{1}, ⋯, s_{n−1} are the standard Coxeter generators. Let ℓ : 𝔖_{n} → ℕ be the length function on 𝔖_{n} so that ℓ(w) = k if k is minimal such that w = s_{i_{1}} … s_{i_{k}}, for some s_{i_{j}} with 1 ≤ i_{j} < n.

For each integer 1 ≤ j < n, the generator s_{j} corresponds to the Brauer ndiagram with edges connecting the vertices j (respectively, j + 1) on the top row with (j + 1)^{−} (respectively, j^{−}) on the bottom row, and all other edges are vertical, connecting the vertices k and k^{−} on the top and bottom rows for all k ≠ i, i + 1; the generator e_{j} corresponds to the Brauer n-diagram with horizontal edges connecting the vertices j, j + 1 (resp., j^{−}, (j + 1)^{−}) on the top rows (resp., bottom rows), and all other edges are vertical, connecting the vertices k and k^{−} on the top and bottom rows for all k ≠ j, j + 1.

Let R be a commutative integral domain which is an ℤ[x]-algebra such that x is specialized to δ ∈ R. Then both the symmetric group algebra R𝔖_{n} and the Brauer algebra 𝔅_{n}(δ)_{R} are cellular algebras over R (see [26] and [17]). To recall their cellular structures we need some combinatorics. A composition of n is a sequence of nonnegative integer λ = (λ_{1}, λ_{2}, ⋯) with ∑_{i≥1} λ_{i} = n. A composition λ = (λ_{1}, λ_{2}, ⋯) of n is said to be a partition if λ_{1} ≥ λ_{2} ≥ · · ·. In this case, we write λ ⊢ n and |λ| = n. We use P_{n} to denote the set of all the partitions of n. For any composition λ of n, the conjugate of λ is defined to be a partition λ′ = (λ′_{1}, λ′_{2}, ⋯), where λ′_{j} := #{i|λ_{i} ≥ j} for each j ≥ 1. We use 𝔖_{λ} to denote the standard Young subgroup of 𝔖_{n} corresponding to λ. That is

𝔖_{λ} := 𝔖_{{1,2,··· ,λ_{1}}} × 𝔖_{{λ_{1}+1,λ_{1}+2,··· ,λ_{1}+λ_{2}}} × · · · .

Let λ be a composition of n. The Young diagram of λ is defined to be the set

[λ] := {(i, j) | 1 ≤ j ≤ λ_{i}}.

The elements of [λ] are called nodes of λ. A λ-tableau is a bijection 𝔱 : [λ] → {1, 2, ⋯, n}. The symmetric group 𝔖_{n} acts on the set of λ-tableaux from the right hand side by letter permutations. If λ is a partition, then the conjugate of 𝔱 is define to be the λ′-tableau 𝔱′ such that 𝔱′(i, j) := 𝔱(j, i) for any (i, j) ∈ [λ′]. The λ-tableau 𝔱 is row standard if 𝔱(i, j) ≤ 𝔱(i, k) whenever j ≤ k. 𝔱 is standard if both 𝔱 and 𝔱′ are row-standard. Let Std(λ) be the set of standard λ-tableaux.

We denote by 𝔱^{λ} (respectively, 𝔱_{λ}) the standard λ-tableau in which the numbers 1, 2, ⋯, n appear in order along successive rows (respectively, columns). If 𝔱 is a λ-tableau then let d(𝔱) ∈ 𝔖_{n} such that 𝔱^{λ}d(𝔱) = 𝔱 and we shall write Shape(𝔱) = λ. Note that 𝔖_{λ} is the row stabilizer of 𝔱^{λ}. We use D_{λ} to denote the set of distinguished right coset representatives of 𝔖_{λ} in 𝔖_{n}. Then for any d ∈ D_{λ} and w ∈ 𝔖_{λ} we have that ℓ(wd) = ℓ(w) + ℓ(d). Let w_{λ} ∈ 𝔖_{n} such that 𝔱^{λ}w_{λ} = 𝔱_{λ}. Then w_{λ} ∈ D_{λ}.

We define

X_{λ} := ∑_{w∈𝔖_{λ}} w, Y_{λ} := ∑_{w∈𝔖_{λ}}(−1)^{ℓ(w)}w.

Let τ be the R-algebra automorphism of R𝔖_{n} which is defined on generators by τ(s_{i}) = −s_{i} for any 1 ≤ i < n. It is clear that τ^{2} = id and τ(Y_{λ}) = X_{λ}.

Let λ ⊢ n. For any 𝔰, 𝔱 ∈ Std(λ), we define X_{𝔰𝔱} := d(𝔰)^{−1}X_{λ}d(𝔱). By a wellknown result of Murphy [26], the set {X_{𝔰𝔱}|λ ⊢ n, 𝔰, 𝔱 ∈ Std(λ)} is a basis of R𝔖_{n}. We call it the Murphy basis of R𝔖_{n}. It is a cellular basis of R𝔖_{n} in the sense of [17]. Note also that the set {Y_{𝔰𝔱} := d(𝔰)^{−1}Y_{λ}d(𝔱)|λ ⊢ n, 𝔰, 𝔱 ∈ Std(λ)} is a cellular basis of R𝔖_{n} too. We call it the Y Murphy basis of R𝔖_{n}. For both cellular bases the cell modules (i.e., Specht modules) of R𝔖_{n} are labeled by the partitions in P_{n}.

For any λ, μ ∈ P_{n}, we write λ ⊵ μ if ∑_{j=1}^{i} λ_{j} ≥ ∑_{j=1}^{i} μ_{j} for any i ≥ 1. If λ ⊵ μ and λ ≠ μ, then we write λ ⊳ μ. We use (R𝔖_{n})^{⊵λ} (respectively, (R𝔖_{n})^{⊳λ} ) to denote the free R-submodule of R𝔖_{n} spanned by the Murphy basis elements of the form X_{𝔲𝔳} with 𝔲, 𝔳 ∈ Std(μ) and μ ⊵ λ (respectively, μ ⊳ λ). Then both (R𝔖_{n})^{⊵λ} and (R𝔖_{n})^{⊳λ} are two-sided ideals of R𝔖_{n}.

We now recall the cellular structure of the Brauer algebra 𝔅_{n}(m). Let f be an integer with 0 ≤ f ≤ [n/2], where [n/2] is the largest non-negative integer not bigger than n/2. We define

D_{f} := \{d ∈ 𝔖_{n} | \begin{matrix}(2j − 1)d < (2j)d for 1 ≤ j ≤ f \\ (1)d < (3)d < ⋯ < (2f − 1)d \\ (2f + 1)d < (2f + 2)d < ⋯ < (n)d \end{matrix}\}.

For each λ ∈ P_{n−2f}, we denote by Std_{2f}(λ) the set of all the standard λtableaux with entries in {2f + 1, ⋯, n}. The initial tableau 𝔱_{f}^{λ} in this case has the numbers 2f + 1, ⋯, n in order along successive rows. Again, for each 𝔱 ∈ Std_{2f}(λ), let d(𝔱) be the unique element in 𝔖_{{2f+1,··· ,n}} ⊆ 𝔖_{n} with 𝔱_{f}^{λ}d(𝔱) = 𝔱. Let σ ∈ 𝔖_{{2f+1,··· ,n}} and d_{1}, d_{2} ∈ D_{f}. Then d_{1}^{−1}e_{1}e_{3} ⋯ e_{2f−1}σd_{2} corresponds to the Brauer n-diagram where the top horizontal edges connect (2i − 1)d_{1} and (2i)d_{1}, the bottom horizontal edges connect ((2i − 1)d_{2})^{−} and ((2i)d_{2})^{−}, for i = 1, 2, ⋯, f, and the vertical edges connects (j)d_{1} with ((j)d_{2})^{−} for j = 2f + 1, 2f + 2, ⋯, n.

Lemma 2.1. ([10, Corollary 3.3]) With the above notations, the set

{d_{1}^{−1}e_{1}e_{3} ⋯ e_{2f−1}σd_{2} | 0 ≤ f ≤ [n/2], σ ∈ 𝔖_{{2f+1,··· ,n}}, d_{1}, d_{2} ∈ D_{f}}.

is a basis of the Brauer algebra 𝔅_{n}(x)_{R}, which coincides with the natural basis given by Brauer n-diagrams.

Definition 2.2. Let D = d_{1}^{−1}e_{1}e_{3} ⋯ e_{2f−1}σd_{2} ∈ Bd_{n}, where 0 ≤ f ≤ [n/2], σ ∈ 𝔖_{{2f+1,··· ,n}}, d_{1}, d_{2} ∈ D_{f}. Then we define ℓ(D) := ℓ(d_{1})+ℓ(d_{2})+ℓ(σ) and sign(D) := (−1)^{f}(−1)^{ℓ(D)}.

Remark 2.3. 1) We can always draw the Brauer diagram as a “nice diagram” (i.e., in a way such that two edges intersect at most once and there are no self-intersections and no three edges intersect at one point, etc, see [14, 1.1]). If D is represented by a “nice diagram” with f horizontal edges in each row and n(D) is the number of crossings of edges, then sign(D) = (−1)^{f+n(D)}. Moreover, if D ∈ 𝔖_{n} then n(D) coincides with the length function on 𝔖_{n} which we introduced before.

2) For any D_{1}, D_{2} ∈ Bd_{n}, note that in general

sign(D_{1}D_{2}) ≠ sign(D_{1}) sign(D_{2}).

3) Our definition of sign(D) coincides with that of ε(D) in [16, 1.4].

Note that, however, the above basis is not a cellular basis for 𝔅_{n}(x). But if we replace the σ in the above basis by a Murphy basis element of R𝔖_{{2f+1,··· ,n}} then we will get a cellular basis of 𝔅_{n}(x). Precisely, the set

\{d_{1}^{−1}e_{1}e_{3} ⋯ e_{2f−1}(d(𝔰)^{−1}X_{λ}^{(f)}d(𝔱))d_{2} | \begin{matrix}0 ≤ f ≤ [n/2], λ ∈ P_{n−2f} \\ 𝔰, 𝔱 ∈ Std_{2f}(λ),d_{1}, d_{2} ∈ D_{f} \end{matrix}\},

where X_{λ}^{(f)} := ∑_{w∈𝔖_{λ}^{(f)}} w and

𝔖_{λ}^{(f)} := 𝔖_{{2f+1,··· ,2f+λ_{1}}} × 𝔖_{{2f+λ_{1}+1,··· ,2f+λ_{1}+λ_{2}}} × ⋯,

is a cellular basis of the Brauer algebra 𝔅_{n}(x)_{R}. The cell modules of 𝔅_{n}(x)_{R} are labeled by the set of pairs (f, λ), where 0 ≤ f ≤ [n/2] and λ ⊢ n − 2f. For any two pairs (f, λ), (g, μ) with 0 ≤ f, g ≤ [n/2] and λ ∈ P_{n−2f}, μ ∈ P_{n−2g}, we define (f, λ) ⊵ (g, μ) if either f > g or f = g and λ ⊵ μ. If (f, λ) ⊵ (g, μ) and (f, λ) ≠ (g, μ), then we write (f, λ)⊳(g, μ). We use (𝔅_{n}(x))^{⊵(f,λ)} (respectively, (𝔅_{n}(x))^{⊳(f,λ)} ) to denote the free R-submodule of 𝔅_{n}(x) spanned by the cellular basis elements corresponding to those (g, μ, d_{1}, d_{2}, 𝔰, 𝔱) with μ ∈ P_{n−2g}, d_{1}, d_{2} ∈ D_{g}, 𝔰, 𝔱 ∈ Std(μ) and (g, μ) ⊵ (f, λ) (respectively, (g, μ) ⊳ (f, λ)). Then both (𝔅_{n}(x))^{⊵(f,λ)} and (𝔅_{n}(x))^{⊳(f,λ)} are two-sided ideals of 𝔅_{n}(x). In particular, if we denote by 𝔅_{n}(x)^{(f)} the two-sided ideal of 𝔅_{n}(x) generated by e_{1}e_{3} ⋯ e_{2f−1}, then

𝔅_{n}(x)^{(f)} = ∑_{λ⊢n−2f}(𝔅_{n}(x))^{⊵(f,λ)}

is spanned by the cellular basis elements which it contains. Henceforth, we shall write 𝔅_{n}^{(f)} instead of 𝔅_{n}(x)^{(f)} for simplicity.

The Brauer algebra 𝔅_{n}(x) and its specialized version have been studied in a number of references, e.g., [1], [2], [3], [4], [5], [6], [7], [10], [12], [14], [15], [16], [19], [20], [25], [27] and [28]. In the set up of Schur–Weyl duality for orthogonal groups, we only need certain specialized Brauer algebras. Let K be a field of characteristic not equal to 2. Let m ∈ ℕ and V an m-dimensional orthogonal space over K. Let 𝔅_{n}(m)_{ℤ} := ℤ ⊗_{ℤ[x]} 𝔅_{n}(x), where ℤ is regarded as ℤ[x]-algebra by specifying x to m. Let 𝔅_{n}(m) := K ⊗_{ℤ} 𝔅_{n}(m)_{ℤ}, where K is regarded as ℤ-algebra in the natural way. Then there is a right action of 𝔅_{n}(m) on the n-tensor space V^{⊗n} which commutes with the natural left action of O(V). We recall the definition of this action. Let δ_{i,j} denote the value of the usual Kronecker delta. We fix an ordered basis {v_{1}, v_{2}, ⋯, v_{m}} of V such that

(v_{i}, v_{j}) = δ_{i,m+1−j}, ∀ 1 ≤ i, j ≤ m.

The right action of 𝔅_{n}(m) on V^{⊗n} is defined on generators by

(v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{n}})s_{j} := v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{j−1}} ⊗ v_{i_{j+1}} ⊗ v_{i_{j}} ⊗ v_{i_{j+2}} ⊗ ⋯ ⊗ v_{i_{n}},
(v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{n}})e_{j} := δ_{i_{j},m+1−i_{j+1}}v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{j−1}} ⊗ (∑_{k=1}^{m} v_{k} ⊗ v_{m+1−k})
⊗ v_{i_{j+2}} ⊗ ⋯ ⊗ v_{i_{n}}.

Let \overline{K} be the algebraic closure of K. Set V\overline{K} := \overline{K} ⊗_{K} V. Then by the main results in [1], [2], [3], [7], [11] and [29], we know that there is a Schur–Weyl duality between 𝔅_{n}(m)\overline{K} and O(V\overline{K}) on V ⊗n ; \overline{K}. In particular, we have two surjective homomorphisms as follows:

φ\overline{K} : 𝔅_{n}(m)\overline{K} → End_{O(V\overline{K})}(V ⊗n ; \overline{K}), ψ\overline{K} : \overline{K}O(V\overline{K}) → End_{𝔅_{n}(m)\overline{K}} (V ⊗n ; \overline{K}).

Furthermore, φ\overline{K} is injective if and only if m ≥ n. If m < n then dim Ker φ\overline{K} is independent of the characteristic of the field K (as long as char K ≠ 2).

From now on until the end of this section, we assume that m < n. The main results in [11] actually implies that dim\overline{K} Ker φ\overline{K} = dim_{K} Ker φ and Ker φ\overline{K} = \overline{K} ⊗_{K} Ker φ because [11, Theorem 1.4, Theorem 6.9] gave an integral basis for Ker φ\overline{K}. In particular, dim Ker φ is independent of the characteristic of the field K (as long as char K ≠ 2). In the following sections we shall sometimes use Ann_{𝔅_{n}(m)}(V^{⊗n}) to denote the annihilator of V^{⊗n} in 𝔅_{n}(m). By definition, Ann_{𝔅_{n}(m)}(V^{⊗n}) = Ker φ.

3 The annihilator of n-tensor space

In this section, we shall prove that the annihilator of V^{⊗n} in 𝔅_{n}(m) is equal to the two-sided ideal generated by E_{0}, E_{1}, …, E_{[(m+1)/2]}. This generalizes the early result (for the case char K = 0) of Lehrer–Zhang [22, Theorem 6.1].

We first recall a definition and a result given in [22].

Definition 3.1. ([22, Lemma 4.1]) Let S = (i_{1}, ⋯, i_{N}), S′ = (j_{1}, ⋯, j_{N}) be two N-tuples of integers such that {i_{1}, ⋯, i_{N}}, {j_{1}, ⋯, j_{N}} are two disjoint subsets of {1, 2, ⋯, 2n}. Let β be any pairing of the vertices {1, 2, ⋯, 2n} \ {i_{1}, ⋯, i_{N}, j_{1}, ⋯, j_{N}}. For w ∈ 𝔖_{N}, let D_{w}(S, S′, β) be the Brauer diagram with edges {(i_{k}, j_{π(k)}) | k = 1, 2, ⋯, N} ⊔ β. We define

b(S, S′, β) := ∑_{w∈𝔖_{N}} sign(w)D_{w}(S, S′, β) ∈ 𝔅_{n}(m).

Lemma 3.2. ([22, (4.5)]) Let S_{a,b} := (1, 2, ⋯, a, (a+1)^{−}, (a+2)^{−}, ⋯, (a+b)^{−}), S′_{a,b} := (a + 1, a + 2, ⋯, a + b, 1^{−}, 2^{−}, ⋯, a^{−}), β_{a+b} be the pairing (a + b + 1, (a + b + 1)^{−}), (a + b + 2, (a + b + 2)^{−}), ⋯, (n, n^{−}). Then we have that

E_{a,b} = b(S_{a,b}, S′_{a,b}, β_{a+b}).

The advantage of the above alternative description of E_{a,b} lies in that the sign sign(w) before D_{w}(S, S′, β) depends only on w which is more easier to be handled than the sign sign(D_{w}(S, S′, β)). More precisely, up to a sign, b(S, S′, β) depends only on β and the two subsets {i_{1}, ⋯, i_{N}}, {j_{1}, ⋯, j_{N}} but not on the orderings on these two subsets.

For any h ∈ 𝔅_{n}(m), we use ⟨h⟩ to denote the two-sided ideal of 𝔅_{n}(m) generated by h. For any finite set S, we use |S| to denote the cardinality of S.

Lemma 3.3. Let a, b ∈ ℕ such that 1 ≤ a+b ≤ n. Then there exist two elements w_{1}, w_{2} ∈ 𝔖_{a+b} such that E_{b,a} = ±w_{1}E_{a,b}w_{2}. In particular, ⟨E_{a,b}⟩ = ⟨E_{b,a}⟩.

Proof. This is clear by Lemma 3.2. In fact, we can take w_{1} to be the Brauer n-diagram which has

{1, (a + 1)^{−}}, {2, (a + 2)^{−}}, ⋯, {b, (a + b)^{−}},
{b + 1, 1^{−}}, {b + 2, 2^{−}}, ⋯, {b + a, a^{−}},
{r, r^{−}}, for all r ∈ {a + b + 1, a + b + 2, ⋯, n},

as its (vertical) edges; and w_{2} to be the Brauer n-diagram which has

{1, (b + 1)^{−}}, {2, (b + 2)^{−}}, ⋯, {a, (b + a)^{−}},
{a + 1, 1^{−}}, {a + 2, 2^{−}}, ⋯, {a + b, b^{−}},
{r, r^{−}}, for all r ∈ {a + b + 1, a + b + 2, ⋯, n},

as its vertical edges.

The proof of the next lemma uses the original definition of E_{a,b}.

Lemma 3.4. For any positive integers a, b with 1 ≤ a + b ≤ n, we have that E_{a,b} ∈ ⟨E_{a,b−1}⟩ ∩ ⟨E_{a−1,b}⟩.

Proof. For each k ∈ {1, 2, ⋯, a, (a + 1)^{−}, ⋯, (a + b)^{−}}, we use Bd(k; a, b) to denote the subset of the Brauer diagrams in Bd(a, b) which have the edge {k, a + b}. If k = i ∈ {1, 2, ⋯, a} then we use d to denote the Brauer n-diagram which has {k, a + b} and {k^{−}, (a + b)^{−}} as its only horizontal edges and

{r, r^{−}}, for all r ∈ {1, 2, ⋯, n}\{k, a + b},

as its vertical edges. It is clear that sign(d) = −1. By the concatenation rule of Brauer diagrams, it is easy to see that

{D | D ∈ Bd(k; a, b)} = {dD′′ | D′′ ∈ Bd(a, b − 1)}.

We claim that

∑_{D∈Bd(k;a,b)} sign(D)D = −dE_{a,b−1} ∈ ⟨E_{a,b−1}⟩.

To prove this claim, it suffices to show that for each D′′ ∈ Bd(a, b − 1),

sign(dD′′) = sign(d) sign(D′′) = − sign(D′′).  (3.5)

Note that when concatenating a “nice diagram” for d with a “nice diagram” for D′′ and transforming it into a “nice diagram” for dD′′, the only transformation which can possibly change the parity of ℓ(d) + 1 + ℓ(D′′) + f (where 2f is the number of horizontal edges of D′′) is the following type of edge which was drawn in red color (where D′′ ∈ Bd(a, b − 1)):

That is, we need to eliminate the self-intersection in the following picture.

However, by eliminating the above self-intersection and making it into an edge in a “nice diagram” for dD′′ has the effect of removing one horizontal edge on the top rows of D′′ together with eliminating 2k − 1 crossing on this concatenation diagram for some k ∈ ℕ. To be more precise, when we eliminate the self-intersection in Figure 1.5, the immediate effect is that we will remove one top horizontal edge of D′′ as well as one crossing from the concatenation diagram. However, there are possibly some more crossings which will be removed. These are the crossings arising from the vertices inside the area circled by the edge in Figure 1.5. If a vertex γ inside the area connects with another vertex which is also inside the circled area then these two interior vertices will contribute two crossings with the red line which will finally be eliminated; otherwise γ itself will connect with two different vertices outside the circled area and hence will produce two crossings with the red line which will finally be eliminated. To sum all, the sign finally remains unchanged. This proves (3.5).

If k = i^{−} ∈ {(a + 1)^{−}, (a + 2)^{−}, ⋯, (a + b)^{−}} then we use d′ to denote the Brauer n-diagram which has

\begin{matrix}\{i, (i + 1)^{−}\}, \{i + 1, (i + 2)^{−}\}, ⋯, \{a + b − 1, (a + b)^{−}\}, \{a + b, i^{−}\}, \\ \{r, r^{−}\}, for all r ∈ \{1, 2, ⋯, i − 1\} ⊔ \{a + b + 1, a + b + 2, ⋯, n\}, \end{matrix}

as its (vertical) edges. It is clear that sign(D_{2}) = (−1)^{a+b−i}. Then by a similar argument as in the case k = i, we can deduce that

∑_{D∈Bd(k;a,b)} sign(D)D = (−1)^{a+b−i}E_{a,b−1}d′ ∈ ⟨E_{a,b−1}⟩.

Therefore, we have that

E_{a,b} = ∑_{k∈{1,2,··· ,a,(a+1)^{−},··· ,(a+b)^{−}}} ∑_{D∈Bd(k;a,b)} sign(D)D ∈ ⟨E_{a,b−1}⟩.

This proves E_{a,b} ∈ ⟨E_{a,b−1}⟩. It remains to prove that E_{a,b} ∈ ⟨E_{a−1,b}⟩. Exchanging the roles of a and b and using Lemma 3.3, we see that

⟨E_{a,b}⟩ = ⟨E_{b,a}⟩ ⊆ ⟨E_{b,a−1}⟩ = ⟨E_{a−1,b}⟩.

as required. This completes the proof of the lemma.

Note that if a ≥ 1 and b = 0 (respectively, if a = 0 and b ≥ 1) then, by the theory of symmetric group, we have that E_{a,0} ∈ ⟨E_{a−1,0}⟩ (respectively, E_{0,b} ∈ ⟨E_{0,b−1}⟩).

Let A^{(1)}, A^{(2)}, A^{(3)}, A^{(4)} be four subsets of indices such that

(a) A^{(i)} ∩ A^{(j)} = ∅ for any 1 ≤ i ≠ j ≤ 4; and

(b) A^{(1)} ⊔ A^{(3)} ⊆ {1, 2, ⋯, n}, A^{(2)} ⊔ A^{(4)} ⊆ {1^{−}, 2^{−}, ⋯, n^{−}}; and

(c) |A^{(1)}| + |A^{(2)}| = |A^{(3)}| + |A^{(4)}|.

Recall that i^{−} = 2n + 1 − i for each 1 ≤ i ≤ 2n. We set n_{0} := |A^{(1)}| + |A^{(2)}|, and

{a_{1}, a_{2}, ⋯, a_{2n−2n_{0}}} := {1, 2, ⋯, 2n} \ ⨆_{k=1}^{4} A^{(k)}.

Let (i_{1}, j_{1}, i_{2}, j_{2}, ⋯, i_{n−n_{0}}, j_{n−n_{0}}) be a fixed permutation of {a_{1}, ⋯, a_{2n−2n_{0}}}. Set

\begin{matrix}i := (i_{1}, i_{2}, ⋯, i_{n−n_{0}}), j := (j_{1}, j_{2}, ⋯, j_{n−n_{0}}). \\ a_{11} := |A^{(1)}|, a_{12} := |A^{(2)}|, a_{21} := |A^{(3)}|, a_{22} := |A^{(4)}|. \end{matrix}

Let β_{i,j} be the pairing (i_{1}, j_{1}), (i_{2}, j_{2}), ⋯, (i_{n−n_{0}}, j_{n−n_{0}}). We fix an ordering on A^{(1)} ⊔ A^{(2)} and an ordering on A^{(3)} ⊔ A^{(4)} respectively. We define S_{A}, S′_{A} to be the corresponding n_{0}-tuples with respect to the two orderings. As we said before, for different choices of orderings, b(S_{A}, S′_{A}, β_{i,j}) differs only by a sign.

Lemma 3.6. With notations as above, we have that

b(S_{A}, S′_{A}, β_{i,j}) ∈ ⟨E_{a_{11}+k,a_{12}−k}⟩,

for some integers 0 ≤ k ≤ min{a_{12}, n − a_{11}}.

Proof. Assume that

S_{A} = (q_{1}, ⋯, q_{a_{11}}, p_{a_{22}+1}^{−}, ⋯, p_{a_{22}+a_{12}}^{−}),

S′_{A} = (q_{a_{11}+1}, ⋯, q_{a_{11}+a_{21}}, p_{1}^{−}, ⋯, p_{a_{22}}^{−}),

{q_{a_{11}+a_{21}+1}, ⋯, q_{n}} = {i_{1}, ⋯, i_{n−n_{0}}, j_{1}, ⋯, j_{n−n_{0}}} ∩ {1, 2, ⋯, n},

{p_{a_{12}+a_{22}+1}^{−}, ⋯, p_{n}^{−}} = {i_{1}, ⋯, i_{n−n_{0}}, j_{1}, ⋯, j_{n−n_{0}}} ∩ {1^{−}, 2^{−}, ⋯, n^{−}},

where a_{11} + a_{12} = a_{21} + a_{22} = n_{0}. In other words,

\begin{matrix}A^{(1)} = \{q_{1}, ⋯, q_{a_{11}}\}, A^{(2)} = \{p_{a_{22}+1}^{−}, ⋯, p_{a_{22}+a_{12}}^{−}\}, \\ A^{(3)} = \{q_{a_{11}+1}, ⋯, q_{a_{11}+a_{21}}\}, A^{(4)} = \{p_{1}^{−}, ⋯, p_{a_{22}}^{−}\}. \end{matrix}

We divide the proof into three cases:

Case 1. a_{11} = a_{22} and a_{12} = a_{21}. In this case, We use σ_{1} to denote the Brauer diagram which has the following edges

\begin{matrix}\{k, q_{k}^{−}\}, for 1 ≤ k ≤ a_{11} + a_{21}, \\ \{a_{11} + a_{21} + l, q_{a_{11}+a_{21}+l}^{−}\}, for 1 ≤ l ≤ n − a_{11} − a_{21}, \end{matrix}

and use σ_{2} to denote the Brauer diagram which has the following edges

\begin{matrix}\{p_{k}, k^{−}\}, for 1 ≤ k ≤ a_{12} + a_{22}, \\ \{p_{a_{12}+a_{22}+l}, (a_{12} + a_{22} + l)^{−}\}, for 1 ≤ l ≤ n − a_{12} − a_{22}. \end{matrix}

Then σ_{1}, σ_{2} are both elements in the symmetric group 𝔖_{n}. The pairing β_{i,j} and the elements σ_{1}, σ_{2} determine a pairing β on the set of vertices {n_{0}+1, n_{0}+2, ⋯, n, (n_{0}+1)^{−}, (n_{0}+2)^{−}, ⋯, n^{−}}, and hence a Brauer (n − n_{0})-diagram D. Since the number of top horizontal edges of D is the same as the number of the bottom horizontal edges of D, we can clearly write D = D_{1}D_{0}D_{2} such that D_{1}, D_{0}, D_{2} ∈ Bd_{n−n_{0}} and D_{0} contains only the vertical edges of the form (k, k^{−}) with n_{0} + 1 ≤ k ≤ n. We extend the Brauer diagrams D_{1}, D_{2} ∈ Bd_{n−n_{0}} to be Brauer diagrams D′_{1}, D′_{2} ∈ Bd_{n} by adding the vertical edges (k, k^{−}) with 1 ≤ k ≤ n_{0} to their left-hand sides. Then it follows directly from the alternative description of E_{a,b} given in Lemma 3.2 that

σ_{1}b(S_{A}, S′_{A}, β_{i,j})σ_{2} = D′_{1}E_{a_{11},a_{12}}D′_{2}.

It follows (by Lemma 3.3) that

b(S_{A}, S′_{A}, β_{i,j}) = σ_{1}^{−1}D′_{1}E_{a_{11},a_{12}}D′_{2}σ_{2}^{−1} ∈ ⟨E_{a_{11},a_{12}}⟩ = ⟨E_{a_{12},a_{11}}⟩,

as required. Case 2. a_{11} > a_{22} and a_{12} < a_{21}. We set d := a_{11} − a_{22}. Then d ∈ ℕ and d ≥ 1. We use induction on d. It is clear that a_{11} ≥ 1. In this case, for each Brauer diagram D′′ involved in b(S_{A}, S′_{A}, β_{i,j}), since a_{11} + a_{21} > a_{12} + a_{22}, there must be some bottom horizontal edges in D′′ of the form {p_{a}^{−}, p_{b}^{−}} with a, b > a_{12} + a_{22}. In other words, there must exist a pairing (i_{s_{0}}, j_{s_{0}}) ∈ β_{i,j} such that i_{s_{0}}, j_{s_{0}} ∈ {1^{−}, 2^{−}, ⋯, n^{−}}. Let D_{3} be the Brauer diagram which has the horizontal edges {i_{s_{0}}^{−}, j_{s_{0}}^{−}}, {i_{s_{0}}, j_{s_{0}}} and vertical edges {j, j^{−}} for any j ∈ {1, 2, ⋯, n} \ {i_{s_{0}}^{−}, j_{s_{0}}^{−}}. Define

\begin{matrix}S_{B} := S_{A}, S′_{B} := (S′_{A} \ \{q_{a_{11}+a_{21}}\}) ∪ \{i_{s_{0}}\}, \\ β′ := \{(i_{s}, j_{s})|1 ≤ s ≤ n − n_{0}, (i_{s}, j_{s}) ≠ (i_{s_{0}}, j_{s_{0}})\} ⊔ \{(q_{a_{11}+a_{21}}, j_{s_{0}})\}. \end{matrix}

Then it is clear that

b(S_{A}, S′_{A}, β_{i,j}) = b(S_{B}, S′_{B}, β′)D_{3} ∈ ⟨b(S_{B}, S′_{B}, β′)⟩.

Now we are in a position to apply induction hypothesis to b(S_{B}, S′_{B}, β′). This proves the lemma in Case 2.

Case 3. a_{11} < a_{22} and a_{12} > a_{21}. We set d := a_{22} − a_{11}. Then d ∈ ℕ and d ≥ 1. We use induction on d. It is clear that a_{12} ≥ 1. In this case, for each Brauer diagram D̂ involved in b(S_{A}, S′_{A}, β_{i,j}), since a_{11} + a_{21} < a_{12} + a_{22}, there must be some top horizontal edges in D̂ of the form {q_{a}, q_{b}} with a, b > a_{11} + a_{21}. In other words, there must exist a pairing (i_{s_{0}}, j_{s_{0}}) ∈ β_{i,j} such that i_{s_{0}}, j_{s_{0}} ∈ {1, 2, ⋯, n}. Let D′_{3} be the Brauer diagram which has the horizontal edges {i_{s_{0}}^{−}, j_{s_{0}}^{−}}, {i_{s_{0}}, j_{s_{0}}} and vertical edges {j, j^{−}} for any j ∈ {1, 2, ⋯, n} \ {i_{s_{0}}, j_{s_{0}}}. Define

\begin{matrix}S_{B} := (S_{A} \ \{p_{a_{12}+a_{22}}^{−}\}) ∪ \{i_{s_{0}}\}, S′_{B} := S′_{A}, \\ β′ := \{(i_{s}, j_{s})|1 ≤ s ≤ n − n_{0}, (i_{s}, j_{s}) ≠ (i_{s_{0}}, j_{s_{0}})\} ⊔ \{(j_{s_{0}}, p_{a_{12}+a_{22}}^{−})\}. \end{matrix}

Then it is clear that

b(S_{A}, S′_{A}, β_{i,j}) = D′_{3}b(S_{B}, S′_{B}, β′) ∈ ⟨b(S_{B}, S′_{B}, β′)⟩.

Now we are in a position to apply induction hypothesis to b(S_{B}, S′_{B}, β′). This proves the lemma in Case 3. Hence we complete the proof of the lemma.

For the sake of simplicity, we shall abbreviate the partition (\underbrace{a, ⋯, a}_{k copies}) as (a^{k}).

Definition 3.7. ([11, Theorem 1.4]) We set

\begin{aligned}(2P_{n})′ & := \{λ̃ := (λ_{1}, λ_{1}, λ_{2}, λ_{2}, ⋯) ⊢ 2n | λ = (λ_{1}, λ_{2}, ⋯) ∈ P_{n}\}, \\ T_{m} & := \{(ν, 𝔱) | 𝔱 ∈ Std(ν̃), (m + 1, 1^{n−m−1}) ⊴ ν ∈ P_{n}\}. \end{aligned}

Now we are in a position to state the main result of this section.

Theorem 3.8. Let K be an arbitrary field of characteristic other than two. If n > m, then

Ann_{𝔅_{n}(m)} (V^{⊗n}) = ⟨E_{0}, E_{1}, ⋯, E_{[ \frac{m+1}{2} ]}⟩ .

Proof. By [11, Theorem 1.4 and Theorem 6.9], we know that Ann_{𝔅_{n}(m)} (V^{⊗n}) has a basis consisting of elements of the form Y_{ν,𝔱}, where (ν, 𝔱) ∈ T_{m}. It remains to show (by the first statement of Theorem 1.2) that each Y_{ν,𝔱} belongs the twosided ideal generated by E_{0}, E_{1}, ⋯, E_{[ \frac{m+1}{2} ]}.

Let (ν, 𝔱) ∈ T_{m}. By the definition (see [11, §6])

Y_{ν,𝔱} = (−1)^{ℓ(d(𝔱))}Y_{ν} ∗ d(𝔱) = (−1)^{ℓ(d(𝔱))} ∑_{i,j}(±b(S_{A}, S′_{A}, β_{i,j})),

where “ ∗ ” denotes the permutation action of 𝔖_{2n} on 𝔅_{n}(m) (see [11, Section 6]), and

\begin{aligned}A^{(1)} : & = \{(i)d(𝔱) | i = 1, 2, 3, ⋯, ν_{1}\} ∩ \{1, 2, ⋯, n\}, \\ A^{(2)} : & = \{(i)d(𝔱) | i = 1, 2, 3, ⋯, ν_{1}\} ∩ \{1^{−}, 2^{−}, ⋯, n^{−}\}, \\ A^{(3)} : & = \{(i^{−})d(𝔱) | i = 1, 2, 3, ⋯, ν_{1}\} ∩ \{1, 2, ⋯, n\}, \\ A^{(4)} : & = \{(i^{−})d(𝔱) | i = 1, 2, 3, ⋯, ν_{1}\} ∩ \{1^{−}, 2^{−}, ⋯, n^{−}\}, \end{aligned}

with |A^{(1)}| + |A^{(2)}| = |A^{(3)}| + |A^{(4)}| = ν_{1}, S_{A}, S′_{A} are defined by using certain prefixed ordering on the sets A^{(1)} ⊔ A^{(2)}, A^{(3)} ⊔ A^{(4)}respectively, and

i := (i_{1}, i_{2}, ⋯, i_{n−ν_{1}}), j := (j_{1}, j_{2}, ⋯, j_{n−ν_{1}})

such that (i_{1}, j_{1}, i_{2}, j_{2}, ⋯, i_{n−ν_{1}}, j_{n−ν_{1}}) runs over a subset of permutations of the integers in {1, 2, ⋯, 2n} \ ⨆_{k=1}^{4} A^{(k)}.

By Lemma 3.6, we obtain

b(S_{A}, S′_{A}, β_{i,j}) ∈ ⟨E_{|A^{(1)}|+k,|A^{(2)}|−k}⟩ ,

for some integers k with 0 ≤ k ≤ min{|A^{(2)}|, n − |A^{(1)}|}. Note that the condition (ν, 𝔱) ∈ T_{m} implies that |A^{(1)}| + |A^{(2)}| = ν_{1} ≥ m + 1. It follows from Lemma 3.4 that b(S_{A}, S′_{A}, β_{i,j}) belongs to the two-sided ideal generated by E_{0}, E_{1}, ⋯, E_{m+1}.

On the other hand, it is clear that for any integer [\frac{m+1}{2}] + 1 ≤ i ≤ m + 1, there exists σ′_{1}, σ′_{2} ∈ 𝔖_{n}, such that E_{i} = ±σ′_{1}E_{m+1−i}σ′_{2} ∈ ⟨E_{m+1−i}⟩. As a consequence, we get that

b(S_{A}, S′_{A}, β_{i,j}) ∈ ⟨E_{0}, E_{1}, ⋯, E_{[(m+1)/2]}⟩,

as required. This completes the proof of the theorem.

4 Proof of Theorem 1.3

In this section we shall give the main result of this paper. That is, the proof of Theorem 1.3.

We shall proceed the proof in three steps. The first step is to prove a statement (Theorem 4.10) about identification between certain two-sided ideals in the symmetric group algebra K𝔖_{n}. To this end, we need to recall the seminormal basis ([18], [9], [24]) of the symmetric group algebra. We shall follow the approach in [24]. Note that [24] only consider the seminormal basis of the (cyclotomic) Hecke algebra with q ≠ 1. The symmetric group case (i.e., q = 1) is similar and may be proved using the same arguments. The only real difference between the cases q ≠ 1 and q = 1 is the choice of content function: if q ≠ 1 then cont_{𝔱}(k) = ξ^{c−r}, when 𝔱(r, c) = k, and if q = 1 then, instead, cont_{𝔱}(k) = c − r. Analogous minor ‘logarithmic’ adjustments are required in the argument below when q = 1.

Set L_{1} := 0 and define L_{i+1} := s_{i}L_{i}s_{i} + s_{i} for i = 1, ⋯, n − 1. The elements L_{1}, ⋯, L_{n} are called the Jucys–Murphy operators of the symmetric group 𝔖_{n}. Let λ ⊢ n and 𝔱 ∈ Std(λ). For any integer 1 ≤ k ≤ n, we define cont_{𝔱}(k) = j − i if k appears in row i and column j in 𝔱. Let

R(k) := {d ∈ ℤ | |d| < k and d ≠ 0 if k = 2, 3}, which is the complete set of possible contents cont_{𝔱}(k) as 𝔱 runs over the set of standard tableaux.

Definition 4.1. ([24, Definition 2.4]) Let λ ⊢ n and 𝔰, 𝔱 ∈ Std(λ).

(i) Let F_{𝔱} := ∏_{k=1}^{n} ∏_{\begin{matrix}c∈R(k) \\ cont_{𝔱}(k)≠c \end{matrix}} \frac{L_{k} − c}{cont_{𝔱}(k) − c}.

(ii) Let f_{𝔰𝔱} := F_{𝔰}X_{𝔰𝔱}F_{𝔱}.

Let λ ⊢ n and 𝔱 ∈ Std(λ). For each integer 1 ≤ k ≤ n we use 𝔱_{k} to denote the subtableau of 𝔱 which contains {1, 2, ⋯, k}. If γ = (i, j) ∈ [λ] such that [λ]\{γ} is again the Young diagram of a partition μ. Then we call γ a removable node of λ and an addable node of μ. For any two nodes γ = (i, j), γ′ = (i′, j′) we say that γ is below γ′, or γ′ is above γ if i > i′.

Definition 4.2. ([24, (2.8)]^{2}, [21, 3.15]) Let λ ⊢ n and 𝔱 ∈ Std(λ). For k = 1, ⋯, n, let 𝒜_{𝔱}(k) be the set of addable nodes of the partition Shape(𝔱_{k}) which are below 𝔱^{−1}(k). Similarly, let ℛ_{𝔱}(k) be the set of removable nodes of Shape(𝔱_{k}) which are below 𝔱^{−1}(k). Now define

γ_{𝔱} = ∏_{k=1}^{n} \frac{∏_{α∈𝒜_{𝔱}(k)} (cont_{𝔱}(k) − cont(α))}{∏_{ρ∈ℛ_{𝔱}(k)} (cont_{𝔱}(k) − cont(ρ))} ∈ ℚ,

and f̃_{𝔰𝔱} := γ_{𝔱}^{−1}f_{𝔰𝔱} for any 𝔰 ∈ Std(λ).

Lemma 4.3. ([24, (2.9)]) Let λ ⊢ n. Then

γ_{𝔱^{λ}} = [λ]! := ∏_{i≥1} λ_{i}! .

Theorem 4.4. ([24, (2.14), (2.15)])

(1) {f̃_{𝔰𝔱} | 𝔰, 𝔱 ∈ Std(λ), λ ⊢ n} is a basis of matrix units in ℚ𝔖_{n}.

(2) Let λ ⊢ n and 𝔱 ∈ Std(λ), then F_{𝔱} = f_{𝔱𝔱}/γ_{𝔱} and F_{𝔱} is a primitive idempotent in ℚ𝔖_{n} with S^{λ} ≅ F_{𝔱}ℚ𝔖_{n}.

(3) For any λ ⊢ n let F_{λ} := ∑_{𝔱∈Std(λ)} F_{𝔱}. Then F_{λ} is a primitive central idempotent in ℚ𝔖_{n}.

(4) {F_{λ} | λ ⊢ n} is a complete set of primitive central idempotent in ℚ𝔖_{n}

and

1 = ∑_{λ⊢n} F_{λ} = ∑_{𝔱 standard} F_{𝔱}.

2We remark that there is a typos in [24, Page 704, Line 9], y ≺ x there should be replaced by y ≻ x, cf. [21, 3.15].

Documenta Mathematica 17 (2012) 245–270

Lemma 4.5. ([24, Proposition 2.6]) Let λ ⊢ n and 𝔰, 𝔱 ∈ Std(λ). Then

f_{𝔰𝔱} ≡ X_{𝔰𝔱} + ∑_{\begin{matrix}𝔲,𝔳∈Std(λ) \\ 𝔲⊳𝔰,𝔳⊳𝔱 \end{matrix}}a_{𝔲𝔳}X_{𝔲𝔳} (mod (ℚ𝔖_{n})^{⊳λ} ),

where a_{𝔲,𝔳} ∈ ℚ for each 𝔲, 𝔳 ∈ Std(λ).

Definition 4.6. ([8, Section 4]) Let λ ⊢ n. We define

z_{λ} := X_{λ}w_{λ}Y_{λ′}.

Lemma 4.7. ([8], [26]) Let λ ⊢ n and w ∈ 𝔖_{n}. Then

(1) (ℤ𝔖_{n})^{⊳λ} Y_{λ′} = 0 = Y_{λ′}(ℤ𝔖_{n})^{⊳λ} ;

(2) If w ≠ w_{λ} and ℓ(w) ≤ ℓ(w_{λ}), then X_{λ}wY_{λ′} = 0 in ℤ𝔖_{n}.

Proof. (1) follows from [26, Lemma 4.12]. It remains to prove (2). Assume that w ≠ w_{λ} and ℓ(w) ≤ ℓ(w_{λ}). If ℓ(w) < ℓ(w_{λ}), then by [26, Corollary 4.13] we see that X_{λ}wY_{λ′} = 0. Now assume ℓ(w) = ℓ(w_{λ}). Then by [8, Lemma 1.5] 𝔱^{λ}w ∉ Std(λ) because w ≠ w_{λ}. For any 𝔱 ∈ Std(λ) and integer 1 ≤ k < n, it is well-known that

X_{λ}d(𝔱)s_{k} = \begin{cases}X_{λ}d(𝔱), & if k, k + 1 are in the same row of 𝔱; \\ X_{λ}d(𝔱s_{k}), & if 𝔱s_{k} ∈ Std(λ), \end{cases}

and if k, k + 1 are in the same column of 𝔱, then (by [23, Corollary 3.21])

X_{λ}d(𝔱)s_{k} ≡ −X_{λ}d(𝔱) + ∑_{𝔱⊲𝔳∈Std(λ)} r_{𝔳}X_{λ}d(𝔳) (mod (ℤ𝔖_{n})^{⊳λ} ),

where r_{𝔳} ∈ ℤ for each 𝔳. By [23, Theorem 3.8], 𝔱 ⊲ 𝔳 ∈ Std(λ) only if ℓ(d(𝔱)) > ℓ(d(𝔳)). As a result (of the fact 𝔱^{λ}w ∉ Std(λ)), we see that

X_{λ}w ≡ ∑_{\begin{matrix}𝔱∈Std(λ) \\ ℓ(d(𝔱))<ℓ(w)=ℓ(w_{λ}) \end{matrix}} b_{𝔱}X_{λ}d(𝔱) (mod (ℤ𝔖_{n})^{⊳λ} ),

where b_{𝔱} ∈ ℤ for each 𝔱. Now, applying the result (1) (which we have just proved) and [26, Corollary 4.13] again,

X_{λ}wY_{λ′} = ∑_{\begin{matrix}𝔱∈Std(λ) \\ ℓ(d(𝔱))<ℓ(w)=ℓ(w_{λ}) \end{matrix}} b_{𝔱}X_{λ}d(𝔱)Y_{λ′} = 0,

as required. This completes the proof of (2).

Lemma 4.8. ([24, (3.13)]) Let λ ⊢ n. Then

z_{λ} = γ_{𝔱^{λ′}} f_{𝔱^{λ}𝔱_{λ}}.

Note that in the right hand side of the above lemma the coefficient is γ_{𝔱^{λ′}} instead of γ′_{𝔱^{λ′}} because we have specialized q to 1.

The next lemma is a key observation to the proof of Theorem 4.10.

Lemma 4.9. Let λ = (n − k, k) ⊢ n, where k ∈ ℤ such that 0 ≤ k ≤ n/2. Then

(1) z_{λ} = 2^{k}f_{𝔱^{λ}𝔱_{λ}} ≡ 2^{k}X_{𝔱^{λ}𝔱_{λ}} (mod (ℤ𝔖_{n})^{⊳λ} ).

(2) Y_{λ′}w_{λ′}X_{λ}w_{λ}Y_{λ′} = 2^{k}Y_{λ′}w_{λ′}X_{λ}w_{λ}.

Proof. By Lemma 4.8 and Lemma 4.3, we get that z_{λ} = 2^{k}f_{𝔱^{λ}𝔱_{λ}}. Applying Lemma 4.5, we get that

z_{λ} = 2^{k}f_{𝔱^{λ}𝔱_{λ}} ≡ 2^{k}X_{𝔱^{λ}𝔱_{λ}} (mod (ℚ𝔖_{n})^{⊳λ} ).

Since z_{λ}, 2^{k}X_{𝔱^{λ}𝔱_{λ}} ∈ ℤ𝔖_{n}, we deduce that

z_{λ} = 2^{k}f_{𝔱^{λ}𝔱_{λ}} ≡ 2^{k}X_{𝔱^{λ}𝔱_{λ}} (mod (ℤ𝔖_{n})^{⊳λ} ).

This proves (1). Note that w_{λ}^{−1} = w_{λ′}. By [8, (4.1)] and applying the antiautomorphism ∗ we know that Y_{λ′}w_{λ′}(ℤ𝔖_{n})^{⊳λ} = 0. Now (2) follows from this equality and (1).

Let λ, μ be two compositions of n. A λ-tableau of type μ is a map S : [λ] → {1, 2, ⋯, d} such that μ_{i} = #{γ ∈ [λ]|S(γ) = i} for i ≥ 1. A λ-tableau S of type μ is row semistandard if the entries in each row of S are non-decreasing from left to right; S is semistandard if (i) λ is a partition; and (ii) S is row semistandard and the entries in each column of S are strictly increasing from top to bottom. If 𝔱 ∈ Std(λ) then we define μ(𝔱) to be the λ-tableau obtained from 𝔱 by replacing each entry i in 𝔱 by r if i appears in row r of 𝔱^{μ}. It is clear that μ(𝔱) is a row semistandard λ-tableau of type μ. Recall our definition (see Section 2) of X_{λ}, Y_{λ} for each composition λ. For each integer i with 0 ≤ i ≤ n, we set

X_{i} := X_{(i,n−i)}, Y_{i} := Y_{(i,n−i)}.

For any h ∈ K𝔖_{n}, we use ⟨h⟩_{0} to denote the two-sided ideal of K𝔖_{n} generated by h. The next theorem is the first step in the direction towards the proof of Theorem 1.3.

Theorem 4.10. Let K be a field of characteristic other than two. Let a be an integer with 0 ≤ a ≤ n/2. Then we have that

⟨X_{n−a}⟩_{0} = (K𝔖_{n})^{⊵(n−a,a)} .

Proof. By the cellular structure of K𝔖_{n}, we see that ⟨X_{n−a}⟩_{0} ⊆ (K𝔖_{n})^{⊵(n−a,a)} and

dim(K𝔖_{n})^{⊵(n−a,a)} = n_{a} := ∑_{(n−a,a)⊴λ⊢n} (# Std(λ))^{2}.  (4.11)

To prove the theorem, it suffices to find at least n_{a} K-linear independent elements in the two-sided ideal ⟨X_{n−a}⟩_{0}.

Let λ ⊢ n be a partition such that λ⊵μ := (n−a, a). Then λ := (n−k, k), where k ∈ ℤ with 0 ≤ k ≤ a ≤ n/2. Let S_{k} be the following (unique) semistandard λ-tableau of type μ:

S_{k} := \overbrace{1, 1, …, 1,}^{n − a copies} \overbrace{2, …, 2,}^{a − k copies} \underbrace{2, …, 2}_{k copies}

We define S_{0}(λ) := {𝔱 ∈ Std(λ) | μ(𝔱) = S_{k}}. By [26, Section 7],

∑_{𝔱∈S_{0}(λ)} X_{λ}d(𝔱) ∈ (K𝔖_{n})X_{n−a} ⊆ ⟨X_{n−a}⟩_{0}.  (4.12)

It is clear that S_{0}(λ) ≠ ∅. We divide the remaining proof into three steps:

Step 1. We claim that Y_{λ′}w_{λ′}X_{λ}(∑_{𝔱∈S_{0}(λ)} d(𝔱))K𝔖_{n} = Y_{λ′}w_{λ′}X_{λ}K𝔖_{n}. Let 𝔱_{0} ∈ S_{0}(λ) such that ℓ(d(𝔱_{0})) is maximal. Then ℓ(d(𝔰)) ≤ ℓ(d(𝔱_{0})) for any 𝔰 ∈ S_{0}(λ). Furthermore, by [8, Lemma 1.5],

ℓ(w_{λ}) = ℓ(d(𝔱_{0})) + ℓ(d(𝔱_{0})^{−1}w_{λ}).

We set w := d(𝔱_{0})^{−1}w_{λ}. By Lemma 4.7, we deduce that X_{λ}d(𝔰)wY_{λ′} = 0 for any 𝔱_{0} ≠ 𝔰 ∈ S_{0}(λ). Now multiplying wY_{λ′} and applying Lemma 4.7 and Lemma 4.9, we get that

\begin{matrix}Y_{λ′}w_{λ′}X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))wY_{λ′} \\ = Y_{λ′}w_{λ′}X_{λ}w_{λ}Y_{λ′} & = 2^{k}Y_{λ′}w_{λ′}X_{λ}w_{λ}. \end{matrix}

Since char K ≠ 2, 2^{k} is invertible in K. The above equality implies that

Y_{λ′}w_{λ′}X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))K𝔖_{n} = Y_{λ′}w_{λ′}X_{λ}w_{λ}K𝔖_{n} = Y_{λ′}w_{λ′}X_{λ}K𝔖_{n},

as required. This proves our claim. Furthermore, it is well-known that the elements in {Y_{λ′}w_{λ′}X_{λ}d(𝔱)|𝔱 ∈ Std(λ)} form a K-basis of Y_{λ′}w_{λ′}X_{λ}K𝔖_{n}. In particular, we have that dim Y_{λ′}w_{λ′}X_{λ}K𝔖_{n} = # Std(λ).

Step 2. We define

N_{λ} := \frac{X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))K𝔖_{n}}{X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))K𝔖_{n} ⋂(K𝔖_{n})^{⊳λ}} .

Documenta Mathematica 17 (2012) 245–270

We claim that n_{λ} := dim N_{λ} ≥ # Std(λ).

In fact, by Lemma 4.7, the left multiplication by Y_{λ′}w_{λ′} induces a surjective homomorphism from N_{λ} onto Y_{λ′}w_{λ′}X_{λ}(∑_{𝔱∈S_{0}(λ)} d(𝔱))K𝔖_{n}. By the main

result of Step 1, we know that Y_{λ′}w_{λ′}X_{λ}(∑_{𝔱∈S_{0}(λ)} d(𝔱))K𝔖_{n} = Y_{λ′}w_{λ′}X_{λ}K𝔖_{n} and has dimension # Std(λ). It follows that n_{λ} := dim N_{λ} ≥ # Std(λ), as required. This proves our claim.

As a consequence, we can find u_{1}, ⋯, u_{n_{λ}} ∈ 𝔖_{n} such that the natural image of the following elements

X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{1}, ⋯, X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{n_{λ}}

in N_{λ} form a K-basis of N_{λ}. For each 𝔰 ∈ Std(λ) and each integer 1 ≤ j ≤ n_{λ}, we define

v_{𝔰,j} := d(𝔰)^{−1}X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{j}.

By construction, it is clear that v_{𝔰,j} ∈ ⟨X_{n−a}⟩_{0}.

Step 3. We claim that the elements in the following set

{v_{𝔰,j} | 𝔰 ∈ Std(λ), λ ⊢ n, 1 ≤ j ≤ n_{λ}}  (4.13)

are K-linearly independent.

In fact, assume that

∑_{\begin{matrix}λ⊢n,𝔰∈Std(λ) \\ 1≤j≤n_{λ} \end{matrix}} c_{𝔰,j}v_{𝔰,j} = ∑_{\begin{matrix}λ⊢n,𝔰∈Std(λ) \\ 1≤j≤n_{λ} \end{matrix}} c_{𝔰,j}d(𝔰)^{−1}X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{j} = 0, (4.14)

where c_{𝔰,j} ∈ K for each 𝔰, j. Set

Σ_{0} := {(𝔰, j) | 𝔰 ∈ Std(λ), λ ⊢ n, 1 ≤ j ≤ n_{λ}, c_{𝔰,j} ≠ 0}.

Suppose that Σ_{0} ≠ ∅. We choose an 𝔰 such that (𝔰, j) ∈ Σ_{0} for some j and Shape(𝔰) = λ is minimal under the dominance order “⊲”. By the property of the cellular basis {X_{𝔰,𝔱}} we know that

X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{j} ≡ ∑_{𝔱∈Std(λ)} r_{j,𝔱}X_{𝔱^{λ},𝔱} (mod (K𝔖_{n})^{⊳λ} ),

and hence

v_{𝔰,j} ≡ ∑_{𝔱∈Std(λ)} r_{j,𝔱}X_{𝔰,𝔱} (mod (K𝔖_{n})^{⊳λ} ).

By the main result in Step 2, we know that X_{λ}(∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{j} ∉ (K𝔖_{n})^{⊳λ} . It follows that at least one of those coefficients r_{j,𝔱} is nonzero. Combing this fact and the minimality of λ and (4.14) we can deduce that for each 𝔰 ∈ Std(λ) and each 1 ≤ j ≤ n_{λ},

∑_{1≤j≤n_{λ}} c_{𝔰,j}d(𝔰)^{−1}X_{λ}( ∑_{𝔱∈S_{0}(λ)} d(𝔱))u_{j} ∈ (K𝔖_{n})^{⊳λ} .

Now applying the main result of Step 2 again, we deduce that c_{𝔰,j} = 0 for each 1 ≤ j ≤ n_{λ}, a contradiction to the definition of Σ_{0}. This completes the proof of our claim.

As a consequence, we see that the two-sided ideal ⟨X_{n−a}⟩_{0} contains at least

∑_{(n−a,a)⊴λ⊢n} n_{λ}(# Std(λ)) ≥ ∑_{(n−a,a)⊴λ⊢n} (# Std(λ))^{2} = n_{a}

K-linearly independent elements. This implies that we must have that

dim⟨X_{n−a}⟩_{0} = n_{a} = dim(K𝔖_{n})^{⊵(n−a,a)}

and hence ⟨X_{n−a}⟩_{0} = (K𝔖_{n})^{⊵(n−a,a)} . This completes the proof of the theorem.

Our second step in the direction towards Theorem 1.3 is the proof of the following purely combinatorial identity.

Theorem 4.15. Let K be a field of arbitrary characteristic. Let S^{(m+1,m+1)} (respectively, S^{(m+1−k,k)}) be the Specht module of the symmetric group algebra K𝔖_{2m+2} (respectively, K𝔖_{m+1}) corresponding to (m + 1, m + 1) (respectively, (m + 1 − k, k), where 0 ≤ k ≤ [(m + 1)/2]. Then we have that

∑_{k=0}^{[(m+1)/2]}(dim_{K} S^{(m+1−k,k)})^{2} = dim_{K} S^{(m+1,m+1)}.

Proof. We shall give a representation theoretic argument to prove the above combinatorial identity. Note that the dimension of each Specht module is independent of the field. We can assume without loss of generality that K = ℂ.

We use 𝔅_{m+1}(m)_{ℂ} to denote the specialized Brauer algebra over ℂ with parameter m. In other words, we assume that n = m + 1 for the moment. Then by [12], 𝔅_{m+1}(m)_{ℂ} is semisimple. We have a natural surjective homomorphism φ_{ℂ} : 𝔅_{m+1}(m)_{ℂ} ↠ End_{ℂO(V_{ℂ})}(V_{ℂ}^{⊗m+1}), where V_{ℂ} is an orthogonal ℂ-vector space with dimension m. Note that ℂ𝔖_{2m+2} is semisimple and each Specht module ℂ𝔖_{2m+2} over is irreducible. By [11] we see that

dim Ker φ_{ℂ} = dim_{ℂ} S_{ℂ}^{(m+1,m+1)},  (4.16)

where we use S_{ℂ}^{(m+1,m+1)} to denote the simple (Specht) module of ℂ𝔖_{2m+2} corresponding to the partition (m + 1, m + 1). As before, we use 𝔅_{m+1}^{(1)} to denote the two-sided ideal of 𝔅_{m+1}(m)_{ℂ} generated by e_{1}. Equivalently, 𝔅_{m+1}^{(1)} is the subspace of 𝔅_{m+1}(m)_{ℂ} spanned by those Brauer (m+1)-diagrams which contain horizontal edges. We claim that Ker φ_{ℂ} ∩ 𝔅_{m+1}^{(1)} = {0}.

In fact, by [22, Theorem 4.3], we know that Ker φ_{ℂ} is equal to the two-sided ideal ⟨E_{[(m+1)/2]}⟩. It suffices to show that ⟨E_{[(m+1)/2]}⟩ ∩ 𝔅_{m+1}^{(1)} = {0}. Since 𝔅_{m+1}^{(1)} is a two-sided ideal of the semisimple ℂ-algebra 𝔅_{m+1}(m)_{ℂ}, we have that 𝔅_{m+1}(m)_{ℂ} = 𝔅_{m+1}^{(1)} ⊕ I_{0} for some two-sided ideal I_{0} of 𝔅_{m+1}(m)_{ℂ}. In particular, xI_{0} = 0 = I_{0}x for any x ∈ 𝔅_{m+1}^{(1)}. We define

J_{0} := {y ∈ 𝔅_{m+1}(m)_{ℂ} | yx = 0 = xy for any x ∈ 𝔅_{m+1}^{(1)}}.

Then J_{0} is a two-sided ideal of 𝔅_{m+1}(m)_{ℂ} and I_{0} ⊆ J_{0}. Thus 𝔅_{m+1}(m)_{ℂ} = J_{0} + 𝔅_{m+1}^{(1)}. Since J_{0}x = 0 for any x ∈ 𝔅_{m+1}^{(1)}, it follows that every right simple submodule of J_{0} is a simple module over 𝔅_{m+1}(m)_{ℂ}/𝔅_{m+1}^{(1)} ≅ ℂ𝔖_{m+1}. Using the Wedderburn theorem for semisimple algebras we get that

J_{0} = ⨁_{λ⊢m+1}(S_{ℂ}^{λ})^{⊕a_{λ}},

where a_{λ} ∈ ℕ such that 0 ≤ a_{λ} ≤ dim S_{ℂ}^{λ} for each λ ⊢ m + 1. As a result, we deduce that

dim J_{0} ≤ ∑_{λ⊢m+1}(dim S_{ℂ}^{λ})^{2} = dim_{ℂ} ℂ𝔖_{m+1} = dim_{ℂ} 𝔅_{m+1}(m)_{ℂ} − dim_{ℂ} 𝔅_{m+1}^{(1)},

because 𝔅_{m+1}(m)_{ℂ}/𝔅_{m+1}^{(1)} ≅ ℂ𝔖_{m+1}. This in turn forces

J_{0} ⨁ 𝔅_{m+1}^{(1)} = J_{0} + 𝔅_{m+1}^{(1)} = 𝔅_{m+1}(m)_{ℂ}.

In particular, J_{0} ∩ 𝔅_{m+1}^{(1)} = {0}. On the other hand, by [22, Corollary 5.13] we know that ⟨E_{[(m+1)/2]}⟩ ⊆ J_{0}, from which our claim follows at once.

We write a_{0} := m + 1 − [(m + 1)/2]. Then a_{0} ≥ m + 1 − a_{0} and hence (a_{0}, m + 1 − a_{0}) is a partition of m + 1. Note that (by Lemma 3.3)

\begin{matrix}⟨E_{[(m+1)/2]}⟩ = ⟨E_{[(m+1)/2],a_{0}}⟩ = ⟨E_{a_{0},[(m+1)/2]}⟩ = ⟨E_{a_{0}}⟩, \\ E_{a_{0}} = E_{a_{0},m+1−a_{0}} ≡ Y_{a_{0},m+1−a_{0}} ≡ Y_{a_{0}} (mod 𝔅_{m+1}^{(1)}). \end{matrix}

By Theorem 4.10, we know that ⟨Y_{a_{0}}⟩_{0} = (ℂ𝔖_{m+1})^{⊵(a_{0},m+1−a_{0})} . Combining these fact together with the equality

⟨E_{[(m+1)/2]}⟩ ∩ 𝔅_{m+1}^{(1)} = Ker φ_{ℂ} ∩ 𝔅_{m+1}^{(1)} = {0} which we have just proved, we can deduce that

dim Ker φ_{ℂ} = dim⟨E_{[(m+1)/2]}⟩ = dim(ℂ𝔖_{m+1})^{⊵(a_{0},m+1−a_{0})}

= ∑_{k=0}^{[(m+1)/2]}(dim_{ℂ} S_{ℂ}^{(m+1−k,k)})^{2}.

Finally, comparing the above equality with (4.16), we complete the proof of the theorem.

By the well-known hook formula (cf. [23]) for the dimension of Specht modules, we get that

dim S^{(m+1)} = 1, dim S^{(m,1)} = m,

and for each 2 ≤ k ≤ [(m + 1)/2],

\begin{aligned}dim S^{(m+1−k,k)} & = \frac{(m + 1)!}{∏_{(i,j)∈[(m+1−k,k)]} h_{i,j}^{(m+1−k,k)}} \\ & = \frac{(m + 1)m ⋯ (m − k + 4)(m − k + 3)(m − 2k + 2)}{k!}, \end{aligned}

where h_{i,j}^{λ} := λ_{i} + λ′_{j} − i − j + 1 is the (i, j)-hook length of the partition λ. Similarly,

\begin{aligned}dim S^{(m+1,m+1)} & = \frac{(2m + 2)!}{∏_{(i,j)∈[(m+1,m+1)]} h_{i,j}^{(m+1,m+1)}} \\ & = \frac{(2m + 2)(2m + 1) ⋯ (m + 3)}{(m + 1)!}. \end{aligned}

As a result, we have the following corollary.

Corollary 4.17. We have the following identity:

\begin{matrix}1 + m^{2} + ∑_{k=2}^{[(m+1)/2]}(\frac{(m + 1)m ⋯ (m − k + 3)(m + 2 − 2k)}{k!})^{2} \\ = \frac{(2m + 2)(2m + 1) ⋯ (m + 3)}{(m + 1)!}. \end{matrix}

Proof. This follows directly from Theorem 4.15 and the hook length formulae for the dimensions of Specht modules over symmetric groups.

Now we are in the final step to prove Theorem 1.3.

Theorem 4.18. Let K be an arbitrary field of characteristic other than two. Then the annihilator of V^{⊗n} in 𝔅_{n}(m) is the two-sided ideal generated by E_{[(m+1)/2]}.

Proof. By Theorem 3.8, it suffices to show that for each integer i with 0 ≤ i < [(m + 1)/2], E_{i} lies in the two-sided ideal of 𝔅_{n}(m) generated by E_{[(m+1)/2]}. For simplicity, we write a := [(m + 1)/2]. By Lemma 3.3, we know that ⟨E_{j}⟩ = ⟨E_{m+1−j}⟩ for each 0 ≤ j ≤ m + 1. Therefore, it suffices to show that for each integer i with 0 ≤ i < a, E_{(m+1−i,i)} lies in the two-sided ideal of 𝔅_{n}(m) generated by E_{m+1−a} := E_{(m+1−a,a)}. Note that for each integer 0 ≤ i ≤ m + 1, E_{i} ∈ 𝔅_{m+1}(m) ⊆ 𝔅_{n}(m). Without loss of generality, we can assume that n = m + 1 henceforth. We consider the natural homomorphism φ_{K} : 𝔅_{m+1}(m) → End_{K}(V^{⊗m+1}). By Corollary 4.17, we have that

\begin{aligned}(m + 1)_{a} & := 1 + m^{2} + ∑_{k=2}^{[(m+1)/2]}(\frac{(m + 1)m ⋯ (m − k + 3)(m + 2 − 2k)}{k!})^{2} \\ & = \frac{(2m + 2)(2m + 1) ⋯ (m + 3)}{(m + 1)!}. \end{aligned}

By [11, Lemma 7.1], Theorem 4.15 and Corollary 4.17, we have that

dim_{K} Ann_{𝔅_{m+1}(m)}(V^{⊗m+1}) = dim_{K} Ker φ_{K} = (m + 1)_{a}.

On the other hand, since E_{m+1−a} ≡ Y_{m+1−a} (mod 𝔅_{n}^{(1)}), we deduce (from Theorem 4.15, Corollary 4.17 and the Y -Murphy basis for K𝔖_{m+1}) that ⟨E_{m+1−a}⟩ contains at least (m + 1)_{a} K-linearly independent elements. Since ⟨E_{m+1−a}⟩ ⊆ Ker φ_{K}, it follows that ⟨E_{m+1−a}⟩ = Ker φ_{K} and hence for each integer 0 ≤ i ≤ [(m+1)/2], E_{i} ∈ ⟨E_{m+1−a}⟩ = ⟨E_{a}⟩ = ⟨E_{[(m+1)/2]}⟩, as required. This completes the proof of the Theorem.

The following corollary give a new integral basis for the annihilator of V^{⊗m+1} in 𝔅_{m+1}(m).

Corollary 4.19. Let K be a field of characteristic other than two. Then the elements in the following set

{d(𝔰)^{−1}E_{[(m+1)/2]}d(𝔱) | 𝔰, 𝔱 ∈ Std(m + 1 − k, k), 0 ≤ k ≤ [(m + 1)/2]}

form a K-basis of Ann_{𝔅_{m+1}(m)}(V^{⊗m+1})

Proof. This follows directly from the proof of Theorem 4.18.

References

[1] R. Brauer, On algebras which are connected with semisimple continuous groups, Ann. Math. 38 (1937), 857–872.

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Jun Hu School of Mathematics Beijing Institute of Technology Beijing, 100081 P.R. China

&

School of Mathematics and Statistics University of Sydney NSW, 2006, Australia junhu303@yahoo.com.cn Zhankui Xiao School of Mathematical Sciences Huaqiao University Quanzhou, Fujian, 362021 P. R. China zhkxiao@gmail.com