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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 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: StefaanΒ Caenepeel

Copyright Β© 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 π‘š,π‘›βˆˆβ„•. Let 𝐾 be an infinite field and 𝑉 a 2π‘š-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 π‘‰βŠ—π‘›. Let 𝐡𝑛=𝐡𝑛(βˆ’2π‘š) be the Brauer algebra over 𝐾 with canonical generators 𝑠1,…,π‘ π‘›βˆ’1,𝑒1,…,π‘’π‘›βˆ’1 subject to the following relations: 𝑠2𝑖=1,𝑒2𝑖=(βˆ’2π‘š)𝑒𝑖,𝑒𝑖𝑠𝑖=𝑠𝑖𝑒𝑖=𝑒𝑖𝑠,βˆ€1β‰€π‘–β‰€π‘›βˆ’1,𝑖𝑠𝑗=𝑠𝑗𝑠𝑖,𝑠𝑖𝑒𝑗=𝑒𝑗𝑠𝑖,𝑒𝑖𝑒𝑗=𝑒𝑗𝑒𝑖𝑠,βˆ€1≀𝑖<π‘—βˆ’1β‰€π‘›βˆ’2,𝑖𝑠𝑖+1𝑠𝑖=𝑠𝑖+1𝑠𝑖𝑠𝑖+1,𝑒𝑖𝑒𝑖+1𝑒𝑖=𝑒𝑖,𝑒𝑖+1𝑒𝑖𝑒𝑖+1=𝑒𝑖+1𝑠,βˆ€1β‰€π‘–β‰€π‘›βˆ’2,𝑖𝑒𝑖+1𝑒𝑖=𝑠𝑖+1𝑒𝑖,𝑒𝑖+1𝑒𝑖𝑠𝑖+1=𝑒𝑖+1𝑠𝑖,βˆ€1β‰€π‘–β‰€π‘›βˆ’2.(1.1) Note that 𝐡𝑛 is a 𝐾-algebra with dimension (2π‘›βˆ’1)!!=(2π‘›βˆ’1)β‹…(2π‘›βˆ’3)β‹―3β‹…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 𝐡𝑛 on π‘‰βŠ—π‘› which we now recall. Let 𝛿𝑖𝑗 denote the Kronecker delta. For each integer 𝑖 with 1≀𝑖≀2π‘š, set π‘–β€²βˆΆ=2π‘š+1βˆ’π‘–. We fix an ordered basis {𝑣𝑖}2π‘šπ‘–=1 of 𝑉 such that 𝑣𝑖,𝑣𝑗𝑣=0=𝑖′,𝑣𝑗′,𝑣𝑖,𝑣𝑗′=𝛿𝑖𝑗𝑣=βˆ’π‘—β€²,𝑣𝑖,βˆ€1≀𝑖,π‘—β‰€π‘š.(1.2) For any 𝑖,π‘—βˆˆ{1,2,…,2π‘š}, let πœ€π‘–,π‘—βŽ§βŽͺ⎨βŽͺ⎩∢=1if𝑖=π‘—ξ…ž,𝑖<𝑗,βˆ’1if𝑖=π‘—ξ…ž,𝑖>𝑗,0otherwise.(1.3) For any simple tensor 𝑣𝑖1βŠ—β‹―βŠ—π‘£π‘–π‘›βˆˆπ‘‰βŠ—π‘›, the right action of 𝐡𝑛 on π‘‰βŠ—π‘› is defined on generators by 𝑣𝑖1βŠ—β‹―βŠ—π‘£π‘–π‘›ξ€Έπ‘ π‘—ξ‚€π‘£βˆΆ=βˆ’π‘–1βŠ—β‹―βŠ—π‘£π‘–π‘—βˆ’π‘–βŠ—π‘£π‘–π‘—+1βŠ—π‘£π‘–π‘—βŠ—π‘£π‘–π‘—+2βŠ—β‹―βŠ—π‘£π‘–π‘›ξ‚,𝑣𝑖1βŠ—β‹―βŠ—π‘£π‘–π‘›ξ€Έπ‘’π‘—βˆΆ=πœ€π‘–π‘—,𝑖𝑗+1𝑣𝑖1βŠ—β‹―βŠ—π‘£π‘–π‘—βˆ’1βŠ—ξƒ©π‘šξ“π‘˜=1ξ€·π‘£π‘˜β€²βŠ—π‘£π‘˜βˆ’π‘£π‘˜βŠ—π‘£π‘˜β€²ξ€ΈξƒͺβŠ—π‘£π‘–π‘—+2βŠ—β‹―βŠ—π‘£π‘–π‘›.(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(π‘‰βŠ—π‘›) is the image of 𝐡𝑛 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 𝐡𝑛(1) be the two-sided ideal of 𝐡𝑛 generated by 𝑒1. We set π‘Š1,π‘›ξ‚†βˆΆ=π‘£βˆˆπ‘‰βŠ—π‘›βˆ£π‘£π‘₯=0,βˆ€π‘₯βˆˆπ΅π‘›(1).(1.5) We call π‘Š1,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 𝐡𝑛/𝐡𝑛(1)≅𝐾𝔖𝑛, the group algebra of the symmetric group 𝔖𝑛. The right action of 𝐡𝑛 on π‘‰βŠ—π‘› gives rise to a right action of 𝐾𝔖𝑛 on π‘Š1,𝑛. We, therefore, have two natural 𝐾-algebra homomorphisms ξ€·πœ‘βˆΆπΎπ”–π‘›ξ€Έop⟢End𝐾Sp(𝑉)ξ€·π‘Š1,𝑛,πœ“βˆΆπΎSp(𝑉)⟢EndπΎπ”–π‘›ξ€·π‘Š1,𝑛.(1.6) In [4], De Concini and Strickland proved that the dimension of π‘Š1,𝑛 is independent of the field 𝐾 and πœ‘ is always surjective. Moreover, they showed that πœ‘ is an isomorphism if π‘šβ‰₯𝑛. When π‘š<𝑛, in [4, Theorem 3.5] they also described the kernel of πœ‘, that is, the annihilator of π‘Š1,𝑛 in the group algebra 𝐾𝔖𝑛. In this paper, we give another combinatorial characterization of Kerπœ‘.

For our aim, we need the notation of dual harmonic tensors. Maliakas in [6] proved that π‘Šβˆ—1,𝑛 has a good filtration when π‘šβ‰₯𝑛 by using the theory of rational representations of symplectic group. He claimed that it is also true for arbitrary π‘š. 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 π‘‰βŠ—π‘›π‘‰βŠ—π‘›π΅π‘›(1)β‰…π‘Šβˆ—1,𝑛,(1.7) and, thus, we call π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(1) the space of dual harmonic tensors. Therefore, we will only characterize the annihilator of π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(1) in the group algebra 𝐾𝔖𝑛.

2. The Main Results

In this section, we will give an elementary combinatorial characterization of the annihilator of π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(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 𝐡𝑛(βˆ’2π‘š) on 𝑛-tensor space π‘‰βŠ—π‘›.

For convenience, we set 𝐼𝑖(2π‘š,𝑛)∢=ξ€½ξ€·1,…,π‘–π‘›ξ€Έβˆ£π‘–π‘—βˆˆξ€Ύ.{1,2,…,2π‘š},βˆ€π‘—(2.1) For any 𝑖_=(𝑖1,β‹―,𝑖𝑛)∈𝐼(2π‘š,𝑛), we write 𝑣𝑖_=𝑣𝑖1βŠ—β‹―βŠ—π‘£π‘–π‘›. For 𝑖_∈𝐼(2π‘š,𝑛), an ordered pair (𝑠,𝑑)(1≀𝑠<𝑑≀𝑛) is called a symplectic pair in 𝑖_ if 𝑖𝑠=π‘–ξ…žπ‘‘. Two ordered pairs (𝑠,𝑑) and (𝑒,𝑣) are called disjoint if {𝑠,𝑑}∩{𝑒,𝑣}=βˆ…. We define the symplectic length ℓ𝑠(𝑣𝑖_) to be the maximal number of disjoint symplectic pairs (𝑠,𝑑) 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 HΓ€rterich, we have the following proposition.

Proposition 2.1. For any simple tensor 𝑣𝑖_βˆˆπ‘‰βŠ—π‘› there is 𝑣𝑖_π‘₯π‘š+1βˆˆπ‘‰βŠ—π‘›π΅π‘›(1), where π‘₯π‘š+1=βˆ‘π‘€βˆˆπ”–π‘š+1𝑀.

Proof. If we have proved the proposition over the base field β„š of rational numbers, it can be restated as a result in ℀𝔖𝑛 by restriction since π‘₯π‘š+1 is a β„€-linear combination of basis elements of ℀𝔖𝑛. Applying the specialization functor πΎβŠ—β„€, we obtain the present statement. Therefore, we now assume we work on the base field β„š.
By the actions of Brauer algebras on 𝑛-tensor spaces defined in Section 1, we know that π‘₯π‘š+1 only acts on the first π‘š+1 components of 𝑣𝑖_. Hence, we can set 𝑛=π‘š+1 without loss of the generality. Let 𝑣𝑖_=𝑣𝑖1βŠ—π‘£π‘–2βŠ—β‹―βŠ—π‘£π‘–π‘š+1. If the (π‘š+1)-tuple (𝑖1,𝑖2,…,π‘–π‘š+1) has a repeated number, for instance, 𝑖𝑠=π‘–π‘Ÿ with 𝑠<π‘Ÿ, then obviously 𝑣𝑖_π‘₯π‘š+1=𝑣𝑖_(𝑠,π‘Ÿ)π‘₯π‘š+1=βˆ’π‘£π‘–_π‘₯π‘š+1 and hence 𝑣𝑖_π‘₯π‘š+1=0, where (𝑠,π‘Ÿ) is a transposition.
Then, we assume that 𝑖1,𝑖2,…,π‘–π‘š+1 are different from each other. Noting that dimβ„šπ‘‰=2π‘š, there exists at least one symplectic pair in 𝑖_. We assume the symplectic length ℓ𝑠(𝑣𝑖_)=𝑠(1≀𝑠≀[(π‘š+1)/2]) and 𝑣𝑖_=𝑣1βŠ—π‘£2π‘šβŠ—π‘£2βŠ—π‘£2π‘šβˆ’1βŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£2π‘šβˆ’π‘ +1βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1 without loss of the generality. Then 𝑣𝑖_π‘₯π‘š+1=𝑣1βŠ—π‘£2π‘šβŠ—π‘£2βŠ—π‘£2π‘šβˆ’1βŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£2π‘šβˆ’π‘ +1βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1=12𝑣1βŠ—π‘£1β€²βˆ’π‘£1β€²βŠ—π‘£1ξ€ΈβŠ—π‘£2βŠ—π‘£2β€²βŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ β€²βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1≑12ξƒ©π‘šξ“π‘—=2π‘£π‘—β€²βŠ—π‘£π‘—βˆ’π‘£π‘—βŠ—π‘£π‘—β€²ξƒͺβŠ—π‘£2βŠ—π‘£2β€²βŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ β€²βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1ξ‚€modπ‘‰βŠ—(π‘š+1)𝐡(1)π‘š+1=ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2π‘£π‘—β€²βŠ—π‘£π‘—ξƒͺβŠ—π‘£2βŠ—π‘£2β€²βŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ β€²βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1.(2.2) In the following, the notation ≑ always means equivalence modπ‘‰βŠ—(π‘š+1)𝐡(1)π‘š+1. We abbreviate 𝑀𝑗 for π‘£π‘—β€²βŠ—π‘£π‘—, noting that 𝑀𝑗(1,2)=βˆ’π‘£π‘—βŠ—π‘£π‘—β€². By the same procedures, we obtain 𝑣𝑖_π‘₯π‘š+1≑𝑀1β€²βŠ—β‹―βŠ—π‘€(π‘˜βˆ’1)β€²βŠ—ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€(π‘˜+1)β€²βŠ—β‹―βŠ—π‘€π‘ β€²βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1,(2.3) where 1β‰€π‘˜β‰€π‘ .
Now we assume for 1<𝑙≀𝑠 that ((π‘™βˆ’1)!)𝑣𝑖_π‘₯π‘š+1≑𝑀1β€²ξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€π‘˜β€²1ξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€π‘˜β€²2ξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€π‘˜(π‘™βˆ’1)ξ…žβŠ—β‹―βŠ—π‘€π‘ β€²βŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1,(2.4) where the π‘™βˆ’1 summands βˆ‘π‘šπ‘—=π‘šβˆ’π‘ +2𝑀𝑗 appear at the (π‘˜1βˆ’1)-th, (π‘˜2βˆ’1)-th, …, (π‘˜π‘™βˆ’1βˆ’1)-th positions (1β‰€π‘˜1βˆ’1<π‘˜2βˆ’1<β‹―<π‘˜π‘™βˆ’1βˆ’1≀𝑠), respectively. We want to prove that (𝑙!)𝑣𝑖_π‘₯π‘š+1≑𝑀1β€²ξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€π‘˜β€²1ξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€π‘˜β€²2ξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘€π‘˜β€²π‘™βŠ—β‹―βŠ—π‘€π‘ ξ…žβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1,(2.5) where the 𝑙 summands βˆ‘π‘šπ‘—=π‘šβˆ’π‘ +2𝑀𝑗 appear at the (π‘˜1βˆ’1)-th, (π‘˜2βˆ’1)-th, …, (π‘˜π‘™βˆ’1)-th positions (1β‰€π‘˜1βˆ’1<π‘˜2βˆ’1<β‹―<π‘˜π‘™βˆ’1≀𝑠), respectively. Without loss of the generality, we only need to prove it for the case 1β‰€π‘˜1βˆ’1<π‘˜2βˆ’1<β‹―<π‘˜π‘™βˆ’1≀𝑙. In fact, we have (2(π‘™βˆ’1)!)𝑣𝑖_π‘₯π‘š+1≑𝑣1βŠ—π‘£1β€²βŠ—ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘£(𝑙+1)βŠ—π‘£(𝑙+1)ξ…žβŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ ξ…žβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1+ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘£π‘™βŠ—π‘£π‘™ξ…žβŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ ξ…žβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1=𝑣1βŠ—π‘£1β€²+π‘£π‘™βŠ—π‘£π‘™β€²ξ€ΈβŠ—ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘£(𝑙+1)βŠ—π‘£(𝑙+1)ξ…žβŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ ξ…žβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1β‰‘ξƒ©π‘™βˆ’1𝑗=2𝑀𝑗+π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘£(𝑙+1)βŠ—π‘£(𝑙+1)ξ…žβŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ ξ…žβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1β‰‘βˆ’(π‘™βˆ’2)((π‘™βˆ’1)!)𝑣𝑖_π‘₯π‘š+1+ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘£(𝑙+1)βŠ—π‘£(𝑙+1)ξ…žβŠ—β‹―βŠ—π‘£π‘ βŠ—π‘£π‘ ξ…žβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1,(2.6) where the last equivalence follows from the induction hypothesis and the fact 𝑀𝑗(1,2)=βˆ’π‘£π‘—βŠ—π‘£π‘—ξ…ž. Hence, we have proved what we desired.
As a consequence, we immediately get that (𝑠!)𝑣𝑖_π‘₯π‘š+1β‰‘ξƒ©π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺξƒ©βŠ—β‹―βŠ—π‘šξ“π‘—=π‘šβˆ’π‘ +2𝑀𝑗ξƒͺβŠ—π‘£π‘ +1βŠ—β‹―βŠ—π‘£π‘šβˆ’π‘ +1π‘₯π‘š+1.(2.7) However, π‘šβˆ’(π‘šβˆ’π‘ +1)=π‘ βˆ’1, there must exists a repeated 𝑀𝑗 in the right hand side of the above equivalence when written as a linear combination of simple tensors. Therefore, 𝑣𝑖_π‘₯π‘š+1≑0.

Theorem 2.2. The annihilator of the space π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(1) of dual harmonic tensors in the group algebra 𝐾𝔖𝑛 is the principal ideal ⟨π‘₯π‘š+1⟩.

Proof. We denote Ann(π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(1)) as the annihilator of the space π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(1) of dual harmonic tensors in the group algebra 𝐾𝔖𝑛. It follows from Proposition 2.1 that ⟨π‘₯π‘š+1ξƒ©π‘‰βŸ©βŠ†AnnβŠ—π‘›π‘‰βŠ—π‘›π΅π‘›(1)ξƒͺ.(2.8) On the other hand, by the work of [10], we know that π‘₯π‘š+1ξ¬ξ€½π‘š=πΎβˆ’Spanπœ†π”°,𝔱,βˆ£πœ†βŠ’π‘›,β„“(πœ†)>π‘š,𝔰,π”±βˆˆStd(πœ†)(2.9) where each π‘šπœ†π”°,𝔱 is the Murphy basis element in [10], and Std(πœ†) denotes the set of standard πœ†-tableaux with entries in {1,2,…,𝑛}.  In particular, [5, Theorem 1.8] shows that (see also [4]) dim𝐾π‘₯π‘š+1=ξ“πœ†βŠ’π‘›,β„“(πœ†)>π‘šξ€·dimπΎπ‘†πœ†ξ€Έ2=dimβ„šβ„šπ”–π‘›βˆ’Endβ„šSp(𝑉)ξƒ©π‘‰βŠ—π‘›π‘‰βŠ—π‘›π΅π‘›(1)ξƒͺ=dimπΎπΎπ”–π‘›βˆ’End𝐾Sp(𝑉)ξƒ©π‘‰βŠ—π‘›π‘‰βŠ—π‘›π΅π‘›(1)ξƒͺ=dim𝐾𝑉AnnβŠ—π‘›π‘‰βŠ—π‘›π΅π‘›(1)ξƒͺ,(2.10) where π‘†πœ† denotes the Specht module of 𝐾𝔖𝑛 associated to πœ†. This completes the proof of the theorem.

Let 𝐡𝑛(𝑓) be the two-sided ideal of 𝐡𝑛 generated by 𝑒1𝑒3⋯𝑒2π‘“βˆ’1 with 1≀𝑓≀[𝑛/2]. Let π‘‹π‘š+1βˆˆπ΅π‘› 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 π‘‰βŠ—π‘›/π‘‰βŠ—π‘›π΅π‘›(𝑓) of dual partially harmonic tensors of valence 𝑓 in the algebra 𝐡𝑛/𝐡𝑛(𝑓) is the principal ideal βŸ¨π‘‹π‘š+1+𝐡𝑛(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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