International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 316389, 7 pages

http://dx.doi.org/10.1155/2012/316389

## 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 . Let be an infinite field and a -dimensional symplectic vector space over equipped with a skew bilinear form . The symplectic group acts naturally on from the left hand side, and hence on the -tensor space . Let be the Brauer algebra over with canonical generators subject to the following relations: Note that is a -algebra with dimension .

The Brauer algebra was first introduced by Brauer (see [1]) when he studied how the -tensor space decomposes into irreducible modules over the orthogonal group or the symplectic group. There is a right action of on which we now recall. Let denote the Kronecker delta. For each integer with , set . We fix an ordered basis of such that For any , let For any simple tensor , the right action of on is defined on generators by The acts as a signed transposition, and acts as a signed contraction. It is well known that the centralizer of the image of the group algebra in is the image of 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 be the two-sided ideal of generated by . We set
We call 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 , the group algebra of the symmetric group . The right action of on gives rise to a right action of on . We, therefore, have two natural -algebra homomorphisms
In [4], De Concini and Strickland proved that the dimension of is independent of the field and is always surjective. Moreover, they showed that is an isomorphism if . When , in [4, Theorem 3.5] they also described the kernel of , that is, the annihilator of in the group algebra . In this paper, we give another combinatorial characterization of .

For our aim, we need the notation of * dual harmonic tensors*. Maliakas in [6] proved that has a good filtration when by using the theory of rational representations of symplectic group. He claimed that it is also true for arbitrary . 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 the space of dual harmonic tensors. Therefore, we will only characterize the annihilator of in the group algebra .

#### 2. The Main Results

In this section, we will give an elementary combinatorial characterization of the annihilator of in the group algebra . 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 on -tensor space .

For convenience, we set
For any , we write . For , an ordered pair is called a * symplectic pair* in if . Two ordered pairs and are called disjoint if . We define the * symplectic length * to be the maximal number of disjoint symplectic pairs in (see [3, Page 198]). Without confusion, we will adopt the same symbol for the image of the canonical generator of the Brauer algebra in the group algebra . More or less motivated by the work [9] of Härterich, we have the following proposition.

Proposition 2.1. *For any simple tensor there is , where . *

*Proof. *If we have proved the proposition over the base field of rational numbers, it can be restated as a result in by restriction since is a -linear combination of basis elements of . Applying the specialization functor , we obtain the present statement. Therefore, we now assume we work on the base field .

By the actions of Brauer algebras on -tensor spaces defined in Section 1, we know that only acts on the first components of . Hence, we can set without loss of the generality. Let . If the -tuple has a repeated number, for instance, with , then obviously and hence , where is a transposition.

Then, we assume that are different from each other. Noting that , there exists at least one symplectic pair in . We assume the symplectic length and without loss of the generality. Then
In the following, the notation always means equivalence . We abbreviate for , noting that . By the same procedures, we obtain
where .

Now we assume for that
where the summands appear at the -th, -th, , -th positions , respectively. We want to prove that
where the summands appear at the -th, -th, , -th positions , respectively. Without loss of the generality, we only need to prove it for the case . In fact, we have
where the last equivalence follows from the induction hypothesis and the fact . Hence, we have proved what we desired.

As a consequence, we immediately get that
However, , there must exists a repeated in the right hand side of the above equivalence when written as a linear combination of simple tensors. Therefore, .

Theorem 2.2. *The annihilator of the space of dual harmonic tensors in the group algebra is the principal ideal . *

*Proof. *We denote as the annihilator of the space of dual harmonic tensors in the group algebra . It follows from Proposition 2.1 that
On the other hand, by the work of [10], we know that
where each is the Murphy basis element in [10], and denotes the set of standard -tableaux with entries in . In particular, [5, Theorem 1.8] shows that (see also [4])
where denotes the Specht module of associated to . This completes the proof of the theorem.

Let be the two-sided ideal of generated by with . Let 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 of dual partially harmonic tensors of valence in the algebra is the principal ideal .

#### 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).

#### References

- R. Brauer, “On algebras which are connected with the semisimple continuous groups,”
*Annals of Mathematics II*, vol. 38, no. 4, pp. 857–872, 1937. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. de Concini and C. Procesi, “A characteristic free approach to invariant theory,”
*Advances in Mathematics*, vol. 21, no. 3, pp. 330–354, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Dipper, S. Doty, and J. Hu, “Brauer algebras, symplectic Schur algebras and Schur-Weyl duality,”
*Transactions of the American Mathematical Society*, vol. 360, no. 1, pp. 189–213, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. de Concini and E. Strickland, “Traceless tensors and the symmetric group,”
*Journal of Algebra*, vol. 61, no. 1, pp. 112–128, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Hu, “Dual partially harmonic tensors and Brauer-Schur-Weyl duality,”
*Transformation Groups*, vol. 15, no. 2, pp. 333–370, 2010. View at Publisher · View at Google Scholar - M. Maliakas, “Traceless tensors and invariants,”
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 124, no. 1, pp. 73–80, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Hu and Z. Xiao, “On tensor spaces for Birman-Murakami-Wenzl algebras,”
*Journal of Algebra*, vol. 324, no. 10, pp. 2893–2922, 2010. View at Publisher · View at Google Scholar - J. Hu and Z.-K. Xiao, “Partially harmonic tensors and quantized Schur-Weyl duality,” in
*Proceedings of the International Workshop on Quantized Algebra and Physics*, M.-L. Ge, C. Bai, and N. Jing, Eds., pp. 109–137, World Scientific, 2011. View at Google Scholar - M. Härterich, “Murphy bases of generalized Temperley-Lieb algebras,”
*Archiv der Mathematik*, vol. 72, no. 5, pp. 337–345, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. E. Murphy, “The representations of Hecke algebras of type ${A}_{n}$,”
*Journal of Algebra*, vol. 173, no. 1, pp. 97–121, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Goodman and N. R. Wallach,
*Representations and Invariants of the Classical Groups*, vol. 68 of*Encyclopedia of Mathematics and Its Applications*, Cambridge University Press, Cambridge, UK, 1998. - H. Weyl,
*The Classical Groups, Their Invariants and Representations*, Princeton University Press, Princeton, NJ, USA, 1946.