A Combinatorial Note for Harmonic Tensors

Abstract

We give another characterization of the annihilator of the space of (dual) harmonic tensors in the group algebra of symmetric group.

1. Introduction and Preliminaries

Let 𝑚,𝑛∈ℕ. Let 𝐾 be an infinite field and 𝑉 a 2𝑚-dimensional symplectic vector space over 𝐾 equipped with a skew bilinear form (,). The symplectic group Sp(𝑉) acts naturally on 𝑉 from the left hand side, and hence on the 𝑛-tensor space 𝑉^⊗𝑛. Let 𝐵_𝑛=𝐵_𝑛(−2𝑚) be the Brauer algebra over 𝐾 with canonical generators 𝑠₁,…,𝑠_𝑛−1,𝑒₁,…,𝑒_𝑛−1 subject to the following relations: 𝑠𝑠^2𝑖=1,𝑒^2𝑖=(−2𝑚)𝑒_𝑖,𝑒_𝑖𝑠_𝑖=𝑠_𝑖𝑒_𝑖=𝑒_𝑖,∀1≤𝑖≤𝑛−1,_𝑖𝑠𝑠_𝑗=𝑠_𝑗𝑠_𝑖,𝑠_𝑖𝑒_𝑗=𝑒_𝑗𝑠_𝑖,𝑒_𝑖𝑒_𝑗=𝑒_𝑗𝑒_𝑖,∀1≤𝑖<𝑗−1≤𝑛−2,_𝑖𝑠_𝑖+1𝑠_𝑖=𝑠𝑠_𝑖+1𝑠_𝑖𝑠_𝑖+1,𝑒_𝑖𝑒_𝑖+1𝑒_𝑖=𝑒_𝑖,𝑒_𝑖+1𝑒_𝑖𝑒_𝑖+1=𝑒_𝑖+1,∀1≤𝑖≤𝑛−2,_𝑖𝑒_𝑖+1𝑒_𝑖=𝑠_𝑖+1𝑒_𝑖,𝑒_𝑖+1𝑒_𝑖𝑠_𝑖+1=𝑒_𝑖+1𝑠_𝑖,∀1≤𝑖≤𝑛−2.(1.1) Note that 𝐵_𝑛 is a 𝐾-algebra with dimension (2𝑛−1)!!=(2𝑛−1)⋅(2𝑛−3)⋯3⋅1.

The Brauer algebra was first introduced by Brauer (see [1]) when he studied how the 𝑛-tensor space decomposes into irreducible modules over the orthogonal group or the symplectic group. There is a right action of 𝐵_𝑛 on 𝑉^⊗𝑛 which we now recall. Let 𝛿_𝑖𝑗 denote the Kronecker delta. For each integer 𝑖 with 1≤𝑖≤2𝑚, set 𝑖′∶=2𝑚+1−𝑖. We fix an ordered basis {𝑣_𝑖}^2𝑚𝑖=1 of 𝑉 such that 𝑣_𝑖,𝑣_𝑗=0=𝑣_𝑖^′,𝑣_𝑗′,𝑣_𝑖,𝑣_𝑗′=𝛿_𝑖𝑗=−𝑣_𝑗′,𝑣_𝑖,∀1≤𝑖,𝑗≤𝑚.(1.2) For any 𝑖,𝑗∈{1,2,…,2𝑚}, let 𝜀_𝑖,𝑗∶=⎧⎪⎨⎪⎩1if𝑖=𝑗^−1if𝑖=𝑗_,𝑖<𝑗,0otherwis,e.𝑖>𝑗,(1.3) For any simple tensor 𝑣_𝑖1⊗⋯⊗𝑣_𝑖𝑛∈𝑉^⊗𝑛, the right action of 𝐵_𝑛 on 𝑉^⊗𝑛 is defined on generators by 𝑣_𝑖1𝑣_𝑖1⊗⋯⊗𝑣_𝑖𝑛𝑠_𝑗∶=−𝑣_𝑖1⊗⋯⊗𝑣_{𝑖𝑗−𝑖}⊗𝑣_𝑖𝑗+1⊗𝑣_𝑖𝑗⊗𝑣_𝑖𝑗+2⊗⋯⊗𝑣_𝑖𝑛,⊗⋯⊗𝑣_𝑖𝑛𝑒_𝑗∶=𝜀_{𝑖𝑗,𝑖𝑗+1}𝑣_𝑖1⊗⋯⊗𝑣_𝑖𝑗−1⊗^𝑚_𝑘=1𝑣_𝑘^′⊗𝑣_𝑘−𝑣_𝑘⊗𝑣_𝑘^′⊗𝑣_𝑖𝑗+2⊗⋯⊗𝑣_𝑖𝑛.(1.4) The 𝑠_𝑗 acts as a signed transposition, and 𝑒_𝑗 acts as a signed contraction. It is well known that the centralizer of the image of the group algebra 𝐾Sp(𝑉) in End_𝐾(𝑉^⊗𝑛) is the image of 𝐵_𝑛 and vice versa. This fact is called Schur-Weyl duality (see [1–3]).

There is a variant of the above Schur-Weyl duality as we will describe. Let 𝐵^𝑛(1) be the two-sided ideal of 𝐵_𝑛 generated by 𝑒₁. We set 𝑊_1,𝑛∶=𝑣∈𝑉^⊗𝑛∣𝑣𝑥=0,∀𝑥∈𝐵^𝑛(1).(1.5) We call 𝑊_1,𝑛 the subspace of harmonic tensors or traceless tensors. It should be pointed out that this definition coincides with that given in [4] and [11, Section 2.1] by [5, Corollary 2.6]. Note that 𝐵_𝑛/𝐵^𝑛(1)≅𝐾𝔖_𝑛, the group algebra of the symmetric group 𝔖_𝑛. The right action of 𝐵_𝑛 on 𝑉^⊗𝑛 gives rise to a right action of 𝐾𝔖_𝑛 on 𝑊_1,𝑛. We, therefore, have two natural 𝐾-algebra homomorphisms 𝜑∶𝐾𝔖_𝑛^op⟶End_𝐾Sp(𝑉)𝑊_1,𝑛,𝜓∶𝐾Sp(𝑉)⟶End_𝐾𝔖𝑛𝑊_1,𝑛.(1.6) In [4], De Concini and Strickland proved that the dimension of 𝑊_1,𝑛 is independent of the field 𝐾 and 𝜑 is always surjective. Moreover, they showed that 𝜑 is an isomorphism if 𝑚≥𝑛. When 𝑚<𝑛, in [4, Theorem 3.5] they also described the kernel of 𝜑, that is, the annihilator of 𝑊_1,𝑛 in the group algebra 𝐾𝔖_𝑛. In this paper, we give another combinatorial characterization of Ker𝜑.

For our aim, we need the notation of dual harmonic tensors. Maliakas in [6] proved that 𝑊^∗1,𝑛 has a good filtration when 𝑚≥𝑛 by using the theory of rational representations of symplectic group. He claimed that it is also true for arbitrary 𝑚. This claim was proved by Hu in [5] using representations of algebraic groups and canonical bases of quantized enveloping algebras. Furthermore, [5, Corollary 1.6] shows that 𝑉^⊗𝑛𝑉^⊗𝑛𝐵^𝑛(1)≅𝑊^∗1,𝑛,(1.7) and, thus, we call 𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(1) the space of dual harmonic tensors. Therefore, we will only characterize the annihilator of 𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(1) in the group algebra 𝐾𝔖_𝑛.

2. The Main Results

In this section, we will give an elementary combinatorial characterization of the annihilator of 𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(1) in the group algebra 𝐾𝔖_𝑛. Besides [4, Theorem 3.5], other characterizations of such annihilator can be found in [7, Theorem 4.2] and [8, Theorem 1.3]. We would like to point out that these approaches depend heavily on invariant theory [4] or representation theory [7, 8]. Therefore, the approach of this paper is more elementary and hence is of independent interest for studying the action of the Brauer algebra 𝐵_𝑛(−2𝑚) on 𝑛-tensor space 𝑉^⊗𝑛.

For convenience, we set 𝐼(2𝑚,𝑛)∶=𝑖₁,…,𝑖_𝑛∣𝑖_𝑗∈{1,2,…,2𝑚},∀𝑗.(2.1) For any _{_}𝑖=(𝑖₁,⋯,𝑖_𝑛)∈𝐼(2𝑚,𝑛), we write 𝑣_{𝑖_}=𝑣_𝑖1⊗⋯⊗𝑣_𝑖𝑛. For _{_}𝑖∈𝐼(2𝑚,𝑛), an ordered pair (𝑠,𝑡)(1≤𝑠<𝑡≤𝑛) is called a symplectic pair in _{_}𝑖 if 𝑖_𝑠=𝑖^𝑡. Two ordered pairs (𝑠,𝑡) and (𝑢,𝑣) are called disjoint if {𝑠,𝑡}∩{𝑢,𝑣}=∅. We define the symplectic length ℓ_𝑠(𝑣_{𝑖_}) to be the maximal number of disjoint symplectic pairs (𝑠,𝑡) in _{_}𝑖 (see [3, Page 198]). Without confusion, we will adopt the same symbol for the image of the canonical generator 𝑠_𝑖 of the Brauer algebra in the group algebra 𝐾𝔖_𝑛. More or less motivated by the work [9] of Härterich, we have the following proposition.

Proposition 2.1. For any simple tensor 𝑣_{𝑖_}∈𝑉^⊗𝑛 there is 𝑣_{𝑖_}𝑥_𝑚+1∈𝑉^⊗𝑛𝐵^𝑛(1), where 𝑥_𝑚+1=∑_{𝑤∈𝔖𝑚+1}𝑤.

Proof. If we have proved the proposition over the base field ℚ of rational numbers, it can be restated as a result in ℤ𝔖_𝑛 by restriction since 𝑥_𝑚+1 is a ℤ-linear combination of basis elements of ℤ𝔖_𝑛. Applying the specialization functor 𝐾⊗_ℤ, we obtain the present statement. Therefore, we now assume we work on the base field ℚ.
By the actions of Brauer algebras on 𝑛-tensor spaces defined in Section 1, we know that 𝑥_𝑚+1 only acts on the first 𝑚+1 components of 𝑣_{𝑖_}. Hence, we can set 𝑛=𝑚+1 without loss of the generality. Let 𝑣_{𝑖_}=𝑣_𝑖1⊗𝑣_𝑖2⊗⋯⊗𝑣_𝑖𝑚+1. If the (𝑚+1)-tuple (𝑖₁,𝑖₂,…,𝑖_𝑚+1) has a repeated number, for instance, 𝑖_𝑠=𝑖_𝑟 with 𝑠<𝑟, then obviously 𝑣_{𝑖_}𝑥_𝑚+1=𝑣_{𝑖_}(𝑠,𝑟)𝑥_𝑚+1=−𝑣_{𝑖_}𝑥_𝑚+1 and hence 𝑣_{𝑖_}𝑥_𝑚+1=0, where (𝑠,𝑟) is a transposition.
Then, we assume that 𝑖₁,𝑖₂,…,𝑖_𝑚+1 are different from each other. Noting that dim_ℚ𝑉=2𝑚, there exists at least one symplectic pair in _{_}𝑖. We assume the symplectic length ℓ_𝑠(𝑣_{𝑖_})=𝑠(1≤𝑠≤[(𝑚+1)/2]) and 𝑣_{𝑖_}=𝑣₁⊗𝑣_2𝑚⊗𝑣₂⊗𝑣_2𝑚−1⊗⋯⊗𝑣_𝑠⊗𝑣_{2𝑚−𝑠+1}⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1} without loss of the generality. Then 𝑣_{𝑖_}𝑥_𝑚+1=𝑣₁⊗𝑣_2𝑚⊗𝑣₂⊗𝑣_2𝑚−1⊗⋯⊗𝑣_𝑠⊗𝑣_{2𝑚−𝑠+1}⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1=12𝑣¹⊗𝑣₁^′−𝑣₁^′⊗𝑣₁⊗𝑣₂⊗𝑣₂^′⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠′⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1≡12^𝑚_𝑗=2𝑣_𝑗′⊗𝑣_𝑗−𝑣_𝑗⊗𝑣_𝑗′⊗𝑣²⊗𝑣₂^′⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠′=⊗𝑣_𝑠+1𝑚⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1mod𝑉^⊗(𝑚+1)𝐵^(1)𝑚+1_𝑗=𝑚_−𝑠+2𝑣_𝑗′⊗𝑣_𝑗⊗𝑣²⊗𝑣₂^′⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠′⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1.(2.2) In the following, the notation ≡ always means equivalence mod𝑉^⊗(𝑚+1)𝐵^(1)𝑚+1. We abbreviate 𝑤_𝑗 for 𝑣_𝑗′⊗𝑣_𝑗, noting that 𝑤_𝑗(1,2)=−𝑣_𝑗⊗𝑣_𝑗′. By the same procedures, we obtain 𝑣_{𝑖_}𝑥_𝑚+1≡𝑤₁^′⊗⋯⊗𝑤_(𝑘−1)^′⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^(𝑘+1)′⊗⋯⊗𝑤_𝑠′⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1,(2.3) where 1≤𝑘≤𝑠.
Now we assume for 1<𝑙≤𝑠 that ((𝑙−1)!)𝑣_{𝑖_}𝑥_𝑚+1≡𝑤₁^′⊗⋯⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^𝑘′1⊗⋯⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^𝑘′2⊗⋯⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^{𝑘(𝑙−1)}⊗⋯⊗𝑤_𝑠′⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1,(2.4) where the 𝑙−1 summands ∑^{𝑚𝑗=𝑚−𝑠+2}𝑤_𝑗 appear at the (𝑘₁−1)-th, (𝑘₂−1)-th, …, (𝑘_𝑙−1−1)-th positions (1≤𝑘₁−1<𝑘₂−1<⋯<𝑘_𝑙−1−1≤𝑠), respectively. We want to prove that (𝑙!)𝑣_{𝑖_}𝑥_𝑚+1≡𝑤₁^′⊗⋯⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^𝑘′1⊗⋯⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^𝑘′2⊗⋯⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗𝑤^𝑘′𝑙⊗⋯⊗𝑤_𝑠⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1,(2.5) where the 𝑙 summands ∑^{𝑚𝑗=𝑚−𝑠+2}𝑤_𝑗 appear at the (𝑘₁−1)-th, (𝑘₂−1)-th, …, (𝑘_𝑙−1)-th positions (1≤𝑘₁−1<𝑘₂−1<⋯<𝑘_𝑙−1≤𝑠), respectively. Without loss of the generality, we only need to prove it for the case 1≤𝑘₁−1<𝑘₂−1<⋯<𝑘_𝑙−1≤𝑙. In fact, we have (2(𝑙−1)!)𝑣_{𝑖_}𝑥_𝑚+1≡𝑣₁⊗𝑣₁^′⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗⋯⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗𝑣^(𝑙+1)⊗𝑣_(𝑙+1)⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1+^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗⋯⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗𝑣^𝑙⊗𝑣_𝑙⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1=𝑣₁⊗𝑣₁^′+𝑣_𝑙⊗𝑣_𝑙^′⊗^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗⋯⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗𝑣_(𝑙+1)⊗𝑣_(𝑙+1)⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1≡^𝑙−1_𝑗=2𝑤_𝑗+^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗⋯⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗𝑣_(𝑙+1)⊗𝑣_(𝑙+1)⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1≡−(𝑙−2)((𝑙−1)!)𝑣_{𝑖_}𝑥_𝑚+1+^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗⋯⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗𝑣_(𝑙+1)⊗𝑣_(𝑙+1)⊗⋯⊗𝑣_𝑠⊗𝑣_𝑠⊗𝑣_𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1,(2.6) where the last equivalence follows from the induction hypothesis and the fact 𝑤_𝑗(1,2)=−𝑣_𝑗⊗𝑣_𝑗. Hence, we have proved what we desired.
As a consequence, we immediately get that (𝑠!)𝑣_{𝑖_}𝑥_𝑚+1≡^𝑚_𝑗=𝑚_−𝑠+2𝑤_𝑗⊗⋯⊗^𝑚_{𝑗=𝑚−𝑠+2}𝑤_𝑗⊗𝑣^𝑠+1⊗⋯⊗𝑣_{𝑚−𝑠+1}𝑥_𝑚+1.(2.7) However, 𝑚−(𝑚−𝑠+1)=𝑠−1, there must exists a repeated 𝑤_𝑗 in the right hand side of the above equivalence when written as a linear combination of simple tensors. Therefore, 𝑣_{𝑖_}𝑥_𝑚+1≡0.

Theorem 2.2. The annihilator of the space 𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(1) of dual harmonic tensors in the group algebra 𝐾𝔖_𝑛 is the principal ideal ⟨𝑥_𝑚+1⟩.

Proof. We denote Ann(𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(1)) as the annihilator of the space 𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(1) of dual harmonic tensors in the group algebra 𝐾𝔖_𝑛. It follows from Proposition 2.1 that ⟨𝑥_𝑚+1⟩⊆Ann𝑉^⊗𝑛𝑉^⊗𝑛𝐵^𝑛(1).(2.8) On the other hand, by the work of [10], we know that 𝑥_𝑚+1=𝐾−Span𝑚^{𝜆𝔰,𝔱}∣𝜆⊢𝑛,ℓ(𝜆)>𝑚,𝔰,𝔱∈Std(𝜆),(2.9) where each 𝑚^{𝜆𝔰,𝔱} is the Murphy basis element in [10], and Std(𝜆) denotes the set of standard 𝜆-tableaux with entries in {1,2,…,𝑛}.  In particular, [5, Theorem 1.8] shows that (see also [4]) dim_𝐾𝑥_𝑚+1=_{𝜆⊢𝑛,ℓ(𝜆)>𝑚}dim_𝐾𝑆^𝜆²=dim_ℚℚ𝔖_𝑛−End_ℚSp(𝑉)=dim_𝐾𝑉𝑉^⊗𝑛𝐵^{⊗𝑛𝑛(1)}𝐾𝔖_𝑛−End_𝐾Sp(𝑉)=dim_𝐾𝑉𝑉^⊗𝑛𝐵^{⊗𝑛𝑛(1)}Ann𝑉^⊗𝑛𝑉^⊗𝑛𝐵^𝑛(1),(2.10) where 𝑆^𝜆 denotes the Specht module of 𝐾𝔖_𝑛 associated to 𝜆. This completes the proof of the theorem.

Let 𝐵^𝑛(𝑓) be the two-sided ideal of 𝐵_𝑛 generated by 𝑒₁𝑒₃⋯𝑒_2𝑓−1 with 1≤𝑓≤[𝑛/2]. Let 𝑋_𝑚+1∈𝐵_𝑛 be the element defined in [7, Page 2912]. We end this note by a conjecture which is connected with the invariant theory of classical groups (see [11, 12]).

Conjecture 2.3. The annihilator of the space 𝑉^⊗𝑛/𝑉^⊗𝑛𝐵^𝑛(𝑓) of dual partially harmonic tensors of valence 𝑓 in the algebra 𝐵_𝑛/𝐵^𝑛(𝑓) is the principal ideal ⟨𝑋_𝑚+1+𝐵^𝑛(𝑓)⟩.

Acknowledgments

The author expresses sincerely thankful to Professor Stefaan Caenepeel for his kind considerations and warm help. He would like to thank the anonymous referee for careful reading and for many invaluable comments and suggestions. The work was supported by a research foundation of Huaqiao University (Grant no. 10BS323).