A Combinatorial Note for Harmonic Tensors

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Volume 2012, Article ID 316389,7pages doi:10.1155/2012/316389

Research Article

A Combinatorial Note for Harmonic Tensors

Zhankui Xiao

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

Correspondence should be addressed to Zhankui Xiao,[email protected]

Received 25 March 2012; Accepted 16 May 2012

Academic Editor: Stefaan Caenepeel

Copyrightq2012 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.

We give another characterization of the annihilator of the space ofdualharmonic tensors in the

group algebra of symmetric group.

1. Introduction and Preliminaries

Let m, n ∈ N. LetK be an infinite field and V a 2m-dimensional symplectic vector space overKequipped with a skew bilinear form, . The symplectic group SpVacts naturally onV from the left hand side, and hence on then-tensor spaceV⊗n. Let Bn Bn−2mbe

the Brauer algebra over K with canonical generators s1, . . . , sn−1, e1, . . . , en−1 subject to the

following relations:

s2

i 1, ei2 −2mei, eisisieiei, ∀1≤i≤n−1,

sisj sjsi, siejejsi, eiej ejei, ∀1≤i < j−1≤n−2,

sisi1sisi1sisi1, eiei1eiei, ei1eiei1ei1, ∀1≤i≤n−2,

siei1eisi1ei, ei1eisi1ei1si, ∀1≤i≤n−2.

1.1

Note thatB_nis aK-algebra with dimension2n−1!! 2n−1·2n−3· · ·3·1.

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Kronecker delta. For each integeriwith 1≤i≤2m, seti:2m1−i. We fix an ordered basis

{vi}2mi1ofV such that

vi, vj0vi, v_j, v_i, v_jδ_ij −v_j, v_i, ∀1≤i, j≤m. 1.2

For anyi, j∈ {1,2, . . . ,2m}, let

εi,j : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

1 ifij, i < j,

−1 ifij, i > j, 0 otherwise.

1.3

For any simple tensorvi1⊗· · ·⊗vin ∈V⊗n, the right action ofBnonV⊗nis defined on generators

by

vi1⊗ · · · ⊗vinsj :−

vi1⊗ · · · ⊗vij−i⊗vij1⊗vij⊗vij2⊗ · · · ⊗vin ,

vi1⊗ · · · ⊗vinej :εij,ij1vi1⊗ · · · ⊗vij−1⊗

_m

k1

vk⊗v_k−v_k⊗v_k

⊗vij2⊗ · · · ⊗vin.

1.4

Thesjacts as a signed transposition, andejacts as a signed contraction. It is well known that

the centralizer of the image of the group algebraKSpVin End_KV⊗nis the image ofB_n and vice versa. This fact is called Schur-Weyl dualitysee1–3.

There is a variant of the above Schur-Weyl duality as we will describe. LetB1_n be the two-sided ideal ofBngenerated bye1. We set

W1,n:

v∈V⊗n_|_vx₀_, _∀x_∈_B1 n

. 1.5

We callW_1,nthe subspace of harmonic tensors or traceless tensors. It should be pointed out that this definition coincides with that given in4and11, Section 2.1by5, Corollary 2.6. Note thatB_n/B_n1∼KSn, the group algebra of the symmetric groupSn. The right action ofBnon

V⊗n _{gives rise to a right action of}_K_Sn _on_W_1,n_{. We, therefore, have two natural}_K_-algebra

homomorphisms

ϕ:KS_nop−→End_KSpVW_1,n, ψ:KSpV−→End_K_S_nW_1,n. 1.6

In4, De Concini and Strickland proved that the dimension ofW1,nis independent of the

fieldKandϕis always surjective. Moreover, they showed thatϕis an isomorphism ifm≥n. Whenm < n, in4, Theorem 3.5they also described the kernel ofϕ, that is, the annihilator of

W1,nin the group algebraKSn. In this paper, we give another combinatorial characterization

of Kerϕ.

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in5using representations of algebraic groups and canonical bases of quantized enveloping algebras. Furthermore,5, Corollary 1.6shows that

V⊗n

V⊗n_B1_n ∼W

∗

1,n, 1.7

and, thus, we callV⊗n/V⊗nB1_n the space of dual harmonic tensors. Therefore, we will only characterize the annihilator ofV⊗n/V⊗nB1_n in the group algebraKS_n.

2. The Main Results

In this section, we will give an elementary combinatorial characterization of the annihilator ofV⊗n/V⊗nB1_n in the group algebraKS_n. Besides4, Theorem 3.5, other characterizations of such annihilator can be found in7, Theorem 4.2and8, 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 algebraB_n−2monn-tensor space

V⊗n_.

For convenience, we set

I2m, n:i1, . . . , in|ij∈ {1,2, . . . ,2m},∀j. 2.1

For anyi i1,· · · , in∈I2m, n, we writevivi1⊗ · · · ⊗vin. Fori∈I2m, n, an ordered pair

s, t 1≤s < t≤nis called a symplectic pair iniifisit. Two ordered pairss, tandu, v

are called disjoint if{s, t}∩{u, v}∅. We define the symplectic length_sv_ito be the maximal number of disjoint symplectic pairss, tinisee3, Page 198. Without confusion, we will adopt the same symbol for the image of the canonical generatorsiof the Brauer algebra in

the group algebraKSn. More or less motivated by the work9of H¨arterich, we have the

following proposition.

Proposition 2.1. For any simple tensor vi ∈ V⊗n there is vixm1 ∈ V⊗nBn1, where xm1

w∈Sm1w.

Proof. If we have proved the proposition over the base fieldQof rational numbers, it can be

restated as a result inZS_nby restriction sincex_m1is aZ-linear combination of basis elements ofZSn. Applying the specialization functorK⊗Z, we obtain the present statement. Therefore, we now assume we work on the base fieldQ.

By the actions of Brauer algebras onn-tensor spaces defined inSection 1, we know that

xm1only acts on the firstm1 components ofvi. Hence, we can setnm1 without loss of

the generality. Letvi vi1⊗vi2⊗ · · · ⊗vim1. If them1-tuplei1, i2, . . . , im1has a repeated

number, for instance,i_s i_r withs < r, then obviouslyv_ix_m1 v_is, rx_m1 −v_ix_m1and hencevixm10, wheres, ris a transposition.

Then, we assume that i1, i2, . . . , im1 are diﬀerent from each other. Noting that

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svi s1≤s≤m1/2andviv1⊗v2m⊗v2⊗v2m−1⊗· · ·⊗vs⊗v2m−s1⊗vs1⊗· · ·⊗vm−s1

without loss of the generality. Then

vixm1v1⊗v2m⊗v2⊗v2m−1⊗ · · · ⊗vs⊗v2m−s1

⊗vs1⊗ · · · ⊗vm−s1xm1

1

2v1⊗v1−v1⊗v1⊗v2⊗v2⊗ · · · ⊗vs⊗vs

⊗vs1⊗ · · · ⊗vm−s1xm1

≡ 1

2

⎛

⎝m

j2

vj⊗v_j−v_j⊗v_j ⎞

⎠_⊗_v₂_⊗_v₂⊗ · · · ⊗v_s⊗v_s

⊗vs1⊗ · · · ⊗vm−s1xm1

modV⊗m1B_m11

⎛

⎝ m

jm−s2

vj⊗v_j ⎞

⎠_⊗_v₂_⊗_v₂⊗ · · · ⊗v_s⊗v_s

⊗vs1⊗ · · · ⊗vm−s1xm1.

2.2

In the following, the notation≡always means equivalence modV⊗m1B1_m1. We abbreviate

wjforvj⊗v_j, noting thatw_j1,2 −v_j⊗v_j. By the same procedures, we obtain

vixm1≡w1⊗ · · · ⊗wk−1⊗ ⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k1⊗ · · · ⊗w_s⊗v_s1⊗ · · · ⊗v_m₋_s1x_m1,

2.3

where 1≤k≤s.

Now we assume for 1< l≤sthat

l−1!vixm1 ≡w1⊗ · · · ⊗ ⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k

1⊗ · · · ⊗ ⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k

2

⊗ · · · ⊗

⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k

l−1⊗ · · · ⊗ws

⊗vs1⊗ · · · ⊗vm−s1xm1,

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where thel−1 summandsm_jm₋_s2wjappear at thek1−1-th,k2−1-th,. . .,kl−1−1-th

positions1≤k1−1< k2−1<· · ·< kl−1−1≤s, respectively. We want to prove that

l!vixm1≡w1⊗ · · · ⊗ ⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k

1⊗ · · · ⊗ ⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k

2

⊗ · · · ⊗

⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_w_k

l⊗ · · · ⊗ws

⊗vs1⊗ · · · ⊗vm−s1xm1,

2.5

where thelsummandsm_jm₋_s2wjappear at thek1−1-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

2l−1!vixm1≡v1⊗v1⊗ ⎛

⎝ m

jm−s2

wj ⎞ ⎠_{⊗ · · · ⊗}

⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_v_l1_⊗_v_l1

⊗ · · · ⊗vs⊗vs⊗v_s1⊗ · · · ⊗v_m₋_s1x_m1

⎛

⎝ m

jm−s2

wj ⎞ ⎠_{⊗ · · · ⊗}

⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_v_l_⊗_v_l

⊗ · · · ⊗vs⊗vs⊗v_s1⊗ · · · ⊗v_m₋_s1x_m1

v1⊗v1v_l⊗v_l⊗ ⎛

⎝ m

jm−s2

wj ⎞ ⎠_{⊗ · · · ⊗}

⎛

⎝ m

jm−s2

wj ⎞ ⎠

⊗vl1⊗vl1⊗ · · · ⊗v_s⊗v_s⊗v_s1⊗ · · · ⊗v_m₋_s1x_m1

≡

⎛

⎝l−1

j2

wj m

jm−s2

wj ⎞

⎠_⊗

⎛

⎝ m

jm−s2

wj ⎞ ⎠_{⊗ · · · ⊗}

⎛

⎝ m

jm−s2

wj ⎞ ⎠

⊗vl1⊗vl1⊗ · · · ⊗v_s⊗v_s⊗v_s1⊗ · · · ⊗v_m₋_s1x_m1

≡ −l−2l−1!vixm1 ⎛

⎝ m

jm−s2

wj ⎞ ⎠_{⊗ · · · ⊗}

⎛

⎝ m

jm−s2

wj ⎞ ⎠

⊗vl1⊗vl1⊗ · · · ⊗v_s⊗v_s⊗v_s1⊗ · · · ⊗v_m₋_s1x_m1,

2.6

where the last equivalence follows from the induction hypothesis and the fact wj1,2

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As a consequence, we immediately get that

s!v_ix_m1 ≡

⎛

⎝ m

jm−s2

wj ⎞ ⎠_{⊗ · · · ⊗}

⎛

⎝ m

jm−s2

wj ⎞

⎠_⊗_v_s1_{⊗ · · · ⊗}_v_m₋_s1_x_m1_. _2.7

However, m−m−s1 s−1, there must exists a repeatedw_j in the right hand side of the above equivalence when written as a linear combination of simple tensors. Therefore,

vixm1≡0.

Theorem 2.2. The annihilator of the space V⊗n_/V⊗n_B1

n of dual harmonic tensors in the group

algebraKSnis the principal idealxm1.

Proof. We denote AnnV⊗n/V⊗nB1n as the annihilator of the space V⊗n/V⊗nB1n of dual

harmonic tensors in the group algebraKS_n. It follows fromProposition 2.1that

xm1 ⊆Ann

V⊗n

V⊗n_B1_n

. 2.8

On the other hand, by the work of10, we know that

xm1K−Span

mλ

s,t|λn, λ> m,s,t∈Stdλ

, 2.9

where eachmλ_s_,_tis the Murphy basis element in10, and Stdλdenotes the set of standard

λ-tableaux with entries in {1,2, . . . , n}. In particular,5, Theorem 1.8 shows thatsee also 4

dimKxm1

λn,λ>m

dimKSλ 2

dimQQSn−EndQSpV

V⊗n

V⊗n_B_n1

dimKKSn−EndKSpV

V⊗n

V⊗n_B_n1

dimKAnn

V⊗n

V⊗n_B_n1

,

2.10

whereSλdenotes the Specht module ofKSnassociated toλ. This completes the proof of the

theorem.

LetB_nfbe the two-sided ideal ofBngenerated bye1e3· · ·e2f−1with 1≤f ≤n/2. Let

Xm1∈Bnbe the element defined in7, Page 2912. We end this note by a conjecture which is

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Conjecture 2.3. The annihilator of the space V⊗n_/V⊗n_Bf

n of dual partially harmonic tensors of

valencefin the algebraBn/Bfnis the principal idealXm1Bnf.

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 UniversityGrant no. 10BS323.

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