A Combinatorial Note for Harmonic Tensors

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


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 s 1 , . . ., s n−1 , e 1 , . . ., e n−1 subject to the following relations: −2m e i , e i s i s i e 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.

1.1
Note that B n is a K-algebra with dimension 2n − 1 !! 2n − 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 International Journal of Mathematics and Mathematical Sciences Kronecker delta.For each integer i with 1 ≤ i ≤ 2m, set i : 2m 1 − i.We fix an ordered basis For any i, j ∈ {1, 2, . . ., 2m}, let , the right action of B n on V ⊗n is defined on generators by 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 1 n be the two-sided ideal of B n generated by e 1 .We set 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 1 n ∼ 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 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 Ker ϕ.
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 and, thus, we call V ⊗n /V ⊗n B 1 n the space of dual harmonic tensors.Therefore, we will only characterize the annihilator of V ⊗n /V ⊗n B 1 n in the group algebra KS n .

The Main Results
In this section, we will give an elementary combinatorial characterization of the annihilator of V ⊗n /V ⊗n B 1 n 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 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 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
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 If the m 1 -tuple i 1 , i 2 , . . ., 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 i 1 , i 2 , . . ., i m 1 are different from each other.Noting that dim Q V 2m, there exists at least one symplectic pair in i.We assume the symplectic length International Journal of Mathematics and Mathematical Sciences

2.2
In the following, the notation ≡ always means equivalence mod V ⊗ 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 where the l − 1 summands m j m−s 2 w j appear at the where the l summands m m−s 2 w j appear at the respectively.Without loss of the generality, we only need to prove it for the case 1 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.

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As a consequence, we immediately get that 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 ⊗n B 1 n of dual harmonic tensors in the group algebra KS n is the principal ideal x m 1 .
Proof.We denote Ann V ⊗n /V ⊗n B 1 n as the annihilator of the space V ⊗n /V ⊗n B 1 n of dual harmonic tensors in the group algebra KS n .It follows from Proposition 2.1 that On the other hand, by the work of 10 , we know that where each m λ s,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 where S λ denotes the Specht module of KS n associated to λ.This completes the proof of the theorem.
Let B f n be the two-sided ideal of B n generated by e 1 e 3 • • • 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 ⊗n B f n of dual partially harmonic tensors of valence f in the algebra B n /B f n is the principal ideal X m 1 B f n .