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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 316389, 7 pages
http://dx.doi.org/10.1155/2012/316389
Research Article

A Combinatorial Note for Harmonic Tensors

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

Received 25 March 2012; Accepted 16 May 2012

Academic Editor: Stefaan Caenepeel

Copyright © 2012 Zhankui Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. 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 𝑉𝑛. Let 𝐵𝑛=𝐵𝑛(2𝑚) be the Brauer algebra over 𝐾 with canonical generators 𝑠1,,𝑠𝑛1,𝑒1,,𝑒𝑛1 subject to the following relations: 𝑠2𝑖=1,𝑒2𝑖=(2𝑚)𝑒𝑖,𝑒𝑖𝑠𝑖=𝑠𝑖𝑒𝑖=𝑒𝑖𝑠,1𝑖𝑛1,𝑖𝑠𝑗=𝑠𝑗𝑠𝑖,𝑠𝑖𝑒𝑗=𝑒𝑗𝑠𝑖,𝑒𝑖𝑒𝑗=𝑒𝑗𝑒𝑖𝑠,1𝑖<𝑗1𝑛2,𝑖𝑠𝑖+1𝑠𝑖=𝑠𝑖+1𝑠𝑖𝑠𝑖+1,𝑒𝑖𝑒𝑖+1𝑒𝑖=𝑒𝑖,𝑒𝑖+1𝑒𝑖𝑒𝑖+1=𝑒𝑖+1𝑠,1𝑖𝑛2,𝑖𝑒𝑖+1𝑒𝑖=𝑠𝑖+1𝑒𝑖,𝑒𝑖+1𝑒𝑖𝑠𝑖+1=𝑒𝑖+1𝑠𝑖,1𝑖𝑛2.(1.1) Note that 𝐵𝑛 is a 𝐾-algebra with dimension (2𝑛1)!!=(2𝑛1)(2𝑛3)31.

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 𝑉𝑛 which we now recall. Let 𝛿𝑖𝑗 denote the Kronecker delta. For each integer 𝑖 with 1𝑖2𝑚, set 𝑖=2𝑚+1𝑖. We fix an ordered basis {𝑣𝑖}2𝑚𝑖=1 of 𝑉 such that 𝑣𝑖,𝑣𝑗𝑣=0=𝑖,𝑣𝑗,𝑣𝑖,𝑣𝑗=𝛿𝑖𝑗𝑣=𝑗,𝑣𝑖,1𝑖,𝑗𝑚.(1.2) For any 𝑖,𝑗{1,2,,2𝑚}, let 𝜀𝑖,𝑗=1if𝑖=𝑗,𝑖<𝑗,1if𝑖=𝑗,𝑖>𝑗,0otherwise.(1.3) For any simple tensor 𝑣𝑖1𝑣𝑖𝑛𝑉𝑛, the right action of 𝐵𝑛 on 𝑉𝑛 is defined on generators by 𝑣𝑖1𝑣𝑖𝑛𝑠𝑗𝑣=𝑖1𝑣𝑖𝑗𝑖𝑣𝑖𝑗+1𝑣𝑖𝑗𝑣𝑖𝑗+2𝑣𝑖𝑛,𝑣𝑖1𝑣𝑖𝑛𝑒𝑗=𝜀𝑖𝑗,𝑖𝑗+1𝑣𝑖1𝑣𝑖𝑗1𝑚𝑘=1𝑣𝑘𝑣𝑘𝑣𝑘𝑣𝑘𝑣𝑖𝑗+2𝑣𝑖𝑛.(1.4) 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(𝑉𝑛) is the image of 𝐵𝑛 and vice versa. This fact is called Schur-Weyl duality (see [13]).

There is a variant of the above Schur-Weyl duality as we will describe. Let 𝐵𝑛(1) be the two-sided ideal of 𝐵𝑛 generated by 𝑒1. We set 𝑊1,𝑛=𝑣𝑉𝑛𝑣𝑥=0,𝑥𝐵𝑛(1).(1.5) 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 𝑉𝑛 gives rise to a right action of 𝐾𝔖𝑛 on 𝑊1,𝑛. We, therefore, have two natural 𝐾-algebra homomorphisms 𝜑𝐾𝔖𝑛opEnd𝐾Sp(𝑉)𝑊1,𝑛,𝜓𝐾Sp(𝑉)End𝐾𝔖𝑛𝑊1,𝑛.(1.6) 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 𝑚𝑛. 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 of dual harmonic tensors. Maliakas in [6] proved that 𝑊1,𝑛 has a good filtration when 𝑚𝑛 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 𝑉𝑛𝑉𝑛𝐵𝑛(1)𝑊1,𝑛,(1.7) and, thus, we call 𝑉𝑛/𝑉𝑛𝐵𝑛(1) the space of dual harmonic tensors. Therefore, we will only characterize the annihilator of 𝑉𝑛/𝑉𝑛𝐵𝑛(1) in the group algebra 𝐾𝔖𝑛.

2. The Main Results

In this section, we will give an elementary combinatorial characterization of the annihilator of 𝑉𝑛/𝑉𝑛𝐵𝑛(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 𝑉𝑛.

For convenience, we set 𝐼𝑖(2𝑚,𝑛)=1,,𝑖𝑛𝑖𝑗.{1,2,,2𝑚},𝑗(2.1) For any 𝑖_=(𝑖1,,𝑖𝑛)𝐼(2𝑚,𝑛), we write 𝑣𝑖_=𝑣𝑖1𝑣𝑖𝑛. For 𝑖_𝐼(2𝑚,𝑛), an ordered pair (𝑠,𝑡)(1𝑠<𝑡𝑛) 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 length 𝑠(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 tensor 𝑣𝑖_𝑉𝑛 there is 𝑣𝑖_𝑥𝑚+1𝑉𝑛𝐵𝑛(1), where 𝑥𝑚+1=𝑤𝔖𝑚+1𝑤.

Proof. If we have proved the proposition over the base field of rational numbers, it can be restated as a result in 𝔖𝑛 by restriction since 𝑥𝑚+1 is a -linear combination of basis elements of 𝔖𝑛. Applying the specialization functor 𝐾, we obtain the present statement. Therefore, we now assume we work on the base field .
By the actions of Brauer algebras on 𝑛-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_=𝑣𝑖1𝑣𝑖2𝑣𝑖𝑚+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 that 𝑖1,𝑖2,,𝑖𝑚+1 are different from each other. Noting that dim𝑉=2𝑚, there exists at least one symplectic pair in 𝑖_. We assume the symplectic length 𝑠(𝑣𝑖_)=𝑠(1𝑠[(𝑚+1)/2]) and 𝑣𝑖_=𝑣1𝑣2𝑚𝑣2𝑣2𝑚1𝑣𝑠𝑣2𝑚𝑠+1𝑣𝑠+1𝑣𝑚𝑠+1 without loss of the generality. Then 𝑣𝑖_𝑥𝑚+1=𝑣1𝑣2𝑚𝑣2𝑣2𝑚1𝑣𝑠𝑣2𝑚𝑠+1𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1=12𝑣1𝑣1𝑣1𝑣1𝑣2𝑣2𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+112𝑚𝑗=2𝑣𝑗𝑣𝑗𝑣𝑗𝑣𝑗𝑣2𝑣2𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1mod𝑉(𝑚+1)𝐵(1)𝑚+1=𝑚𝑗=𝑚𝑠+2𝑣𝑗𝑣𝑗𝑣2𝑣2𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1.(2.2) In the following, the notation always means equivalence mod𝑉(𝑚+1)𝐵(1)𝑚+1. We abbreviate 𝑤𝑗 for 𝑣𝑗𝑣𝑗, noting that 𝑤𝑗(1,2)=𝑣𝑗𝑣𝑗. By the same procedures, we obtain 𝑣𝑖_𝑥𝑚+1𝑤1𝑤(𝑘1)𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤(𝑘+1)𝑤𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1,(2.3) where 1𝑘𝑠.
Now we assume for 1<𝑙𝑠 that ((𝑙1)!)𝑣𝑖_𝑥𝑚+1𝑤1𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤𝑘1𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤𝑘2𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤𝑘(𝑙1)𝑤𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1,(2.4) where the 𝑙1 summands 𝑚𝑗=𝑚𝑠+2𝑤𝑗 appear at the (𝑘11)-th, (𝑘21)-th, , (𝑘𝑙11)-th positions (1𝑘11<𝑘21<<𝑘𝑙11𝑠), respectively. We want to prove that (𝑙!)𝑣𝑖_𝑥𝑚+1𝑤1𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤𝑘1𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤𝑘2𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑤𝑘𝑙𝑤𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1,(2.5) where the 𝑙 summands 𝑚𝑗=𝑚𝑠+2𝑤𝑗 appear at the (𝑘11)-th, (𝑘21)-th, , (𝑘𝑙1)-th positions (1𝑘11<𝑘21<<𝑘𝑙1𝑠), respectively. Without loss of the generality, we only need to prove it for the case 1𝑘11<𝑘21<<𝑘𝑙1𝑙. In fact, we have (2(𝑙1)!)𝑣𝑖_𝑥𝑚+1𝑣1𝑣1𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑣(𝑙+1)𝑣(𝑙+1)𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1+𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑣𝑙𝑣𝑙𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1=𝑣1𝑣1+𝑣𝑙𝑣𝑙𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑣(𝑙+1)𝑣(𝑙+1)𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1𝑙1𝑗=2𝑤𝑗+𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑣(𝑙+1)𝑣(𝑙+1)𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1(𝑙2)((𝑙1)!)𝑣𝑖_𝑥𝑚+1+𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑣(𝑙+1)𝑣(𝑙+1)𝑣𝑠𝑣𝑠𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1,(2.6) where the last equivalence follows from the induction hypothesis and the fact 𝑤𝑗(1,2)=𝑣𝑗𝑣𝑗. Hence, we have proved what we desired.
As a consequence, we immediately get that (𝑠!)𝑣𝑖_𝑥𝑚+1𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑚𝑗=𝑚𝑠+2𝑤𝑗𝑣𝑠+1𝑣𝑚𝑠+1𝑥𝑚+1.(2.7) However, 𝑚(𝑚𝑠+1)=𝑠1, 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+10.

Theorem 2.2. The annihilator of the space 𝑉𝑛/𝑉𝑛𝐵𝑛(1) of dual harmonic tensors in the group algebra 𝐾𝔖𝑛 is the principal ideal 𝑥𝑚+1.

Proof. We denote Ann(𝑉𝑛/𝑉𝑛𝐵𝑛(1)) as the annihilator of the space 𝑉𝑛/𝑉𝑛𝐵𝑛(1) of dual harmonic tensors in the group algebra 𝐾𝔖𝑛. It follows from Proposition 2.1 that 𝑥𝑚+1𝑉Ann𝑛𝑉𝑛𝐵𝑛(1).(2.8) On the other hand, by the work of [10], we know that 𝑥𝑚+1𝑚=𝐾Span𝜆𝔰,𝔱,𝜆𝑛,(𝜆)>𝑚,𝔰,𝔱Std(𝜆)(2.9) where each 𝑚𝜆𝔰,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]) dim𝐾𝑥𝑚+1=𝜆𝑛,(𝜆)>𝑚dim𝐾𝑆𝜆2=dim𝔖𝑛EndSp(𝑉)𝑉𝑛𝑉𝑛𝐵𝑛(1)=dim𝐾𝐾𝔖𝑛End𝐾Sp(𝑉)𝑉𝑛𝑉𝑛𝐵𝑛(1)=dim𝐾𝑉Ann𝑛𝑉𝑛𝐵𝑛(1),(2.10) where 𝑆𝜆 denotes the Specht module of 𝐾𝔖𝑛 associated to 𝜆. This completes the proof of the theorem.

Let 𝐵𝑛(𝑓) be the two-sided ideal of 𝐵𝑛 generated by 𝑒1𝑒3𝑒2𝑓1 with 1𝑓[𝑛/2]. Let 𝑋𝑚+1𝐵𝑛 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 𝑉𝑛/𝑉𝑛𝐵𝑛(𝑓) of dual partially harmonic tensors of valence 𝑓 in the algebra 𝐵𝑛/𝐵𝑛(𝑓) is the principal ideal 𝑋𝑚+1+𝐵𝑛(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).

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