# A Combinatorial Note for Harmonic Tensors

Authors: Zhankui Xiao
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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


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## Body

## 1. Introduction and Preliminaries

Letm,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 Bn=Bn(-2m) be the Brauer algebra over K with canonical generators s1,…,sn-1,e1,…,en-1 subject to the following relations:
(1.1)si2=1,ei2=(-2m)ei,eisi=siei=ei,∀1≤i≤n-1,sisj=sjsi,siej=ejsi,eiej=ejei,∀1≤i<j-1≤n-2,sisi+1si=si+1sisi+1,eiei+1ei=ei,ei+1eiei+1=ei+1,∀1≤i≤n-2,siei+1ei=si+1ei,ei+1eisi+1=ei+1si,∀1≤i≤n-2.
Note that Bn 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 Bn on V⊗n which we now recall. Let δij denote the Kronecker delta. For each integer i with 1≤i≤2m, set i′:=2m+1-i. We fix an ordered basis {vi}i=12m of V such that
(1.2)(vi,vj)=0=(vi′,vj′),(vi,vj′)=δij=-(vj′,vi),∀1≤i,j≤m.
For any i,j∈{1,2,…,2m}, let
(1.3)εi,j:={1ifi=j′,i<j,-1ifi=j′,i>j,0otherwise.
For any simple tensor vi1⊗⋯⊗vin∈V⊗n, the right action of Bn on V⊗n is defined on generators by
(1.4)(vi1⊗⋯⊗vin)sj:=-(vi1⊗⋯⊗vij-i⊗vij+1⊗vij⊗vij+2⊗⋯⊗vin),(vi1⊗⋯⊗vin)ej:=εij,ij+1vi1⊗⋯⊗vij-1⊗(∑k=1m‍(vk′⊗vk-vk⊗vk′))⊗vij+2⊗⋯⊗vin.
The sj acts as a signed transposition, and ej acts as a signed contraction. It is well known that the centralizer of the image of the group algebra KSp(V) in EndK(V⊗n) is the image of Bn 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. LetBn(1) be the two-sided ideal of Bn generated by e1. We set
(1.5)W1,n:={v∈V⊗n∣vx=0,∀x∈Bn(1)}.
We call W1,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 Bn/Bn(1)≅K𝔖n, the group algebra of the symmetric group 𝔖n. The right action of Bn on V⊗n gives rise to a right action of K𝔖n on W1,n. We, therefore, have two natural K-algebra homomorphisms
(1.6)φ:(KSn)op⟶EndKSp(V)(W1,n),ψ:KSp(V)⟶EndKSn(W1,n).
In [4], De Concini and Strickland proved that the dimension of W1,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 W1,n in the group algebra K𝔖n. In this paper, we give another combinatorial characterization of Kerφ.For our aim, we need the notation ofdual harmonic tensors. Maliakas in [6] proved that W1,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
(1.7)V⊗nV⊗nBn(1)≅W1,n*,
and, thus, we call V⊗n/V⊗nBn(1) the space of dual harmonic tensors. Therefore, we will only characterize the annihilator of V⊗n/V⊗nBn(1) in the group algebra K𝔖n.


## 2. The Main Results

In this section, we will give an elementary combinatorial characterization of the annihilator ofV⊗n/V⊗nBn(1) in the group algebra K𝔖n. 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 Bn(-2m) on n-tensor space V⊗n.For convenience, we set(2.1)I(2m,n):={(i1,…,in)∣ij∈{1,2,…,2m},∀j}.
For any i_=(i1,⋯,in)∈I(2m,n), we write vi_=vi1⊗⋯⊗vin. For i_∈I(2m,n), an ordered pair (s,t)(1≤s<t≤n) is called a symplectic pair in i_ if is=it′. Two ordered pairs (s,t) and (u,v) are called disjoint if {s,t}∩{u,v}=∅. We define the symplectic lengthℓs(vi_) 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 si of the Brauer algebra in the group algebra K𝔖n. More or less motivated by the work [9] of Härterich, we have the following proposition.Proposition 2.1.
For any simple tensorvi_∈V⊗n there is vi_xm+1∈V⊗nBn(1), where xm+1=∑w∈𝔖m+1‍w.Proof.
If we have proved the proposition over the base fieldℚ of rational numbers, it can be restated as a result in ℤ𝔖n by restriction since xm+1 is a ℤ-linear combination of basis elements of ℤ𝔖n. 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 onn-tensor spaces defined in Section 1, we know that xm+1 only acts on the first m+1 components of vi_. Hence, we can set n=m+1 without loss of the generality. Let vi_=vi1⊗vi2⊗⋯⊗vim+1. If the (m+1)-tuple (i1,i2,…,im+1) has a repeated number, for instance, is=ir with s<r, then obviously vi_xm+1=vi_(s,r)xm+1=-vi_xm+1 and hence vi_xm+1=0, where (s,r) is a transposition.
Then, we assume thati1,i2,…,im+1 are different from each other. Noting that dimℚV=2m, there exists at least one symplectic pair in i_. We assume the symplectic length ℓs(vi_)=s(1≤s≤[(m+1)/2]) and vi_=v1⊗v2m⊗v2⊗v2m-1⊗⋯⊗vs⊗v2m-s+1⊗vs+1⊗⋯⊗vm-s+1 without loss of the generality. Then
(2.2)vi_xm+1=v1⊗v2m⊗v2⊗v2m-1⊗⋯⊗vs⊗v2m-s+1⊗vs+1⊗⋯⊗vm-s+1xm+1=12(v1⊗v1′-v1′⊗v1)⊗v2⊗v2′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1≡12(∑j=2m‍vj′⊗vj-vj⊗vj′)⊗v2⊗v2′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1(mod⁡V⊗(m+1)Bm+1(1))=(∑j=m-s+2m‍vj′⊗vj)⊗v2⊗v2′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1.
In the following, the notation ≡ always means equivalence modV⊗(m+1)Bm+1(1). We abbreviate wj for vj′⊗vj, noting that wj(1,2)=-vj⊗vj′. By the same procedures, we obtain
(2.3)vi_xm+1≡w1′⊗⋯⊗w(k-1)′⊗(∑j=m-s+2m‍wj)⊗w(k+1)′⊗⋯⊗ws′⊗vs+1⊗⋯⊗vm-s+1xm+1,
where 1≤k≤s.
Now we assume for1<l≤s that
(2.4)((l-1)!)vi_xm+1≡w1′⊗⋯⊗(∑j=m-s+2m‍wj)⊗wk1′⊗⋯⊗(∑j=m-s+2m‍wj)⊗wk2′⊗⋯⊗(∑j=m-s+2m‍wj)⊗wk(l-1)′⊗⋯⊗ws′⊗vs+1⊗⋯⊗vm-s+1xm+1,
where the l-1 summands ∑j=m-s+2m‍wj appear at the (k1-1)-th, (k2-1)-th, …, (kl-1-1)-th positions (1≤k1-1<k2-1<⋯<kl-1-1≤s), respectively. We want to prove that
(2.5)(l!)vi_xm+1≡w1′⊗⋯⊗(∑j=m-s+2m‍wj)⊗wk1′⊗⋯⊗(∑j=m-s+2m‍wj)⊗wk2′⊗⋯⊗(∑j=m-s+2m‍wj)⊗wkl′⊗⋯⊗ws′⊗vs+1⊗⋯⊗vm-s+1xm+1,
where the l summands ∑j=m-s+2m‍wj appear at the (k1-1)-th, (k2-1)-th, …, (kl-1)-th positions (1≤k1-1<k2-1<⋯<kl-1≤s), respectively. Without loss of the generality, we only need to prove it for the case 1≤k1-1<k2-1<⋯<kl-1≤l. In fact, we have
(2.6)(2(l-1)!)vi_xm+1≡v1⊗v1′⊗(∑j=m-s+2m‍wj)⊗⋯⊗(∑j=m-s+2m‍wj)⊗v(l+1)⊗v(l+1)′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1+(∑j=m-s+2m‍wj)⊗⋯⊗(∑j=m-s+2m‍wj)⊗vl⊗vl′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1=(v1⊗v1′+vl⊗vl′)⊗(∑j=m-s+2m‍wj)⊗⋯⊗(∑j=m-s+2m‍wj)⊗v(l+1)⊗v(l+1)′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1≡(∑j=2l-1‍wj+∑j=m-s+2m‍wj)⊗(∑j=m-s+2m‍wj)⊗⋯⊗(∑j=m-s+2m‍wj)⊗v(l+1)⊗v(l+1)′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1≡-(l-2)((l-1)!)vi_xm+1+(∑j=m-s+2m‍wj)⊗⋯⊗(∑j=m-s+2m‍wj)⊗v(l+1)⊗v(l+1)′⊗⋯⊗vs⊗vs′⊗vs+1⊗⋯⊗vm-s+1xm+1,
where the last equivalence follows from the induction hypothesis and the fact wj(1,2)=-vj⊗vj′. Hence, we have proved what we desired.
As a consequence, we immediately get that(2.7)(s!)vi_xm+1≡(∑j=m-s+2m‍wj)⊗⋯⊗(∑j=m-s+2m‍wj)⊗vs+1⊗⋯⊗vm-s+1xm+1.
However, m-(m-s+1)=s-1, there must exists a repeated wj in the right hand side of the above equivalence when written as a linear combination of simple tensors. Therefore, vi_xm+1≡0.Theorem 2.2.
The annihilator of the spaceV⊗n/V⊗nBn(1) of dual harmonic tensors in the group algebra K𝔖n is the principal ideal 〈xm+1〉.Proof.
We denoteAnn(V⊗n/V⊗nBn(1)) as the annihilator of the space V⊗n/V⊗nBn(1) of dual harmonic tensors in the group algebra K𝔖n. It follows from Proposition 2.1 that
(2.8)〈xm+1〉⊆Ann(V⊗nV⊗nBn(1)).
On the other hand, by the work of [10], we know that
(2.9)〈xm+1〉=K-Span⁡{ms,tλ∣λ⊢n,l(λ)>m,s,t∈Std(λ)},
where each m𝔰,𝔱λ 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])
(2.10)dim⁡K〈xm+1〉=∑λ⊢n,l(λ)>m‍(dim⁡KSλ)2=dim⁡QQSn-EndQSp(V)(V⊗nV⊗nBn(1))=dim⁡KKSn-EndKSp(V)(V⊗nV⊗nBn(1))=dim⁡KAnn(V⊗nV⊗nBn(1)),
where Sλ denotes the Specht module of K𝔖n associated to λ. This completes the proof of the theorem.LetBn(f) be the two-sided ideal of Bn generated by e1e3⋯e2f-1 with 1≤f≤[n/2]. Let Xm+1∈Bn 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 spaceV⊗n/V⊗nBn(f) of dual partially harmonic tensors of valence f in the algebra Bn/Bn(f) is the principal ideal 〈Xm+1+Bn(f)〉.



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