Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 316389, 7 pages
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, zhkxiao@gmail.com
Received 25 March 2012; Accepted 16 May 2012
Academic Editor: Stefaan Caenepeel
Copyright q 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.
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 ∈ 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 SpV acts naturally
on V from the left hand side, and hence on the n-tensor space V ⊗n . Let Bn Bn −2m be
the Brauer algebra over K with canonical generators s1 , . . . , sn−1 , e1 , . . . , en−1 subject to the
following relations:
s2i 1,
ei2 −2mei ,
si sj sj si ,
si ej ej si ,
si si1 si si1 si si1 ,
ei si si ei ei ,
ei ej ej ei ,
ei ei1 ei ei ,
si ei1 ei si1 ei ,
∀1 ≤ i ≤ n − 1,
∀1 ≤ i < j − 1 ≤ n − 2,
ei1 ei ei1 ei1 ,
ei1 ei si1 ei1 si ,
∀1 ≤ i ≤ n − 2,
1.1
∀1 ≤ i ≤ n − 2.
Note that Bn is a K-algebra with dimension 2n − 1!! 2n − 1 · 2n − 3 · · · 3 · 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 Bn on V ⊗n which we now recall. Let δij denote the
2
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
{vi }2m
i1 of V such that
vi , vj 0 vi , vj ,
vi , vj δij − vj , vi ,
∀1 ≤ i, j ≤ m.
1.2
For any i, j ∈ {1, 2, . . . , 2m}, let
εi,j
⎧
⎪
if i j , i < j,
⎪
⎨1
: −1 if i j , i > j,
⎪
⎪
⎩0
otherwise.
1.3
For any simple tensor vi1 ⊗· · ·⊗vin ∈ V ⊗n , the right action of Bn on V ⊗n is defined on generators
by
vi1 ⊗ · · · ⊗ vin sj : − vi1 ⊗ · · · ⊗ vij−i ⊗ vij1 ⊗ vij ⊗ vij2 ⊗ · · · ⊗ vin ,
vi1 ⊗ · · · ⊗ vin ej : εij ,ij1 vi1 ⊗ · · · ⊗ vij−1 ⊗
m
vk ⊗ vk − vk ⊗ vk k1
⊗ vij2 ⊗ · · · ⊗ vin .
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 KSpV in EndK V ⊗n is the image of Bn
and vice versa. This fact is called Schur-Weyl duality see 1–3.
1
There is a variant of the above Schur-Weyl duality as we will describe. Let Bn be the
two-sided ideal of Bn generated by e1 . We set
1
W1,n : v ∈ V ⊗n | vx 0, ∀x ∈ Bn .
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
1
that Bn /Bn ∼
KSn , the group algebra of the symmetric group Sn . The right action of Bn on
⊗n
V gives rise to a right action of KSn on W1,n . We, therefore, have two natural K-algebra
homomorphisms
ϕ : KSn op −→ EndKSpV W1,n ,
ψ : KSpV −→ EndKSn W1,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 m ≥ n.
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 Ker ϕ.
For our aim, we need the notation of dual harmonic tensors. Maliakas in 6 proved
∗
has a good filtration when m ≥ n by using the theory of rational representations of
that W1,n
symplectic group. He claimed that it is also true for arbitrary m. This claim was proved by Hu
International Journal of Mathematics and Mathematical Sciences
3
in 5 using representations of algebraic groups and canonical bases of quantized enveloping
algebras. Furthermore, 5, Corollary 1.6 shows that
V ⊗n
1
V ⊗n Bn
∗
∼
,
W1,n
1.7
1
and, thus, we call V ⊗n /V ⊗n Bn the space of dual harmonic tensors. Therefore, we will only
1
characterize the annihilator of V ⊗n /V ⊗n Bn in the group algebra KSn .
2. The Main Results
In this section, we will give an elementary combinatorial characterization of the annihilator
1
of V ⊗n /V ⊗n Bn 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
V ⊗n .
For convenience, we set
I2m, n : i1 , . . . , in | ij ∈ {1, 2, . . . , 2m}, ∀j .
2.1
For any i i1 , · · · , in ∈ I2m, n, we write vi vi1 ⊗ · · · ⊗ vin . For i ∈ I2m, n, an ordered pair
s, t 1 ≤ s < t ≤ n 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.
1
Proposition 2.1. For any simple tensor vi ∈ V ⊗n there is vi xm1 ∈ V ⊗n Bn , where xm1 w∈Sm1 w.
Proof. If we have proved the proposition over the base field Q of rational numbers, it can be
restated as a result in ZSn by restriction since xm1 is a Z-linear combination of basis elements
of ZSn . 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
xm1 only acts on the first m 1 components of vi . Hence, we can set n m 1 without loss of
the generality. Let vi vi1 ⊗ vi2 ⊗ · · · ⊗ vim1 . If the m 1-tuple i1 , i2 , . . . , im1 has a repeated
number, for instance, is ir with s < r, then obviously vi xm1 vi s, rxm1 −vi xm1 and
hence vi xm1 0, where s, r is a transposition.
Then, we assume that i1 , i2 , . . . , im1 are different from each other. Noting that
dimQ V 2m, there exists at least one symplectic pair in i. We assume the symplectic length
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International Journal of Mathematics and Mathematical Sciences
s vi s 1 ≤ s ≤ m1/2 and vi v1 ⊗v2m ⊗v2 ⊗v2m−1 ⊗· · ·⊗vs ⊗v2m−s1 ⊗vs1 ⊗· · ·⊗vm−s1
without loss of the generality. Then
vi xm1 v1 ⊗ v2m ⊗ v2 ⊗ v2m−1 ⊗ · · · ⊗ vs ⊗ v2m−s1
⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
1
v1 ⊗ v1 − v1 ⊗ v1 ⊗ v2 ⊗ v2 ⊗ · · · ⊗ vs ⊗ vs
2
⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
⎞
⎛
m
1 ⎝
≡
vj ⊗ vj − vj ⊗ vj ⎠ ⊗ v2 ⊗ v2 ⊗ · · · ⊗ vs ⊗ vs
2 j2
⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
⎛
⎝
1
mod V ⊗m1 Bm1
2.2
⎞
m
vj ⊗ vj ⎠ ⊗ v2 ⊗ v2 ⊗ · · · ⊗ vs ⊗ vs
jm−s2
⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1 .
1
In the following, the notation ≡ always means equivalence mod V ⊗m1 Bm1 . We abbreviate
wj for vj ⊗ vj , noting that wj 1, 2 −vj ⊗ vj . By the same procedures, we obtain
⎛
vi xm1 ≡ w1 ⊗ · · · ⊗ wk−1 ⊗ ⎝
⎞
m
wj ⎠ ⊗ wk1 ⊗ · · · ⊗ ws ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1 ,
jm−s2
2.3
where 1 ≤ k ≤ s.
Now we assume for 1 < l ≤ s that
⎛
m
l − 1!vi xm1 ≡ w1 ⊗ · · · ⊗ ⎝
⎞
⎛
wj ⎠ ⊗ wk1 ⊗ · · · ⊗ ⎝
jm−s2
⎛
⊗ ··· ⊗ ⎝
m
⎞
wj ⎠ ⊗ wkl−1
⊗ · · · ⊗ ws
jm−s2
⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1 ,
m
⎞
wj ⎠ ⊗ wk2
jm−s2
2.4
International Journal of Mathematics and Mathematical Sciences
5
where the l − 1 summands m
jm−s2 wj appear at the k1 − 1-th, k2 − 1-th, . . ., kl−1 − 1-th
positions 1 ≤ k1 − 1 < k2 − 1 < · · · < kl−1 − 1 ≤ s, respectively. We want to prove that
⎛
l!vi xm1 ≡ w1 ⊗ · · · ⊗ ⎝
⎛
⎞
m
wj ⎠ ⊗ wk1 ⊗ · · · ⊗ ⎝
jm−s2
⎛
m
⊗ ··· ⊗ ⎝
m
⎞
wj ⎠ ⊗ wk2
jm−s2
⎞
2.5
wj ⎠ ⊗ wkl ⊗ · · · ⊗ ws
jm−s2
⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1 ,
where the l summands m
jm−s2 wj appear at the k1 −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!vi xm1 ≡ v1 ⊗ v1 ⊗ ⎝
m
⎞
⎛
m
wj ⎠ ⊗ · · · ⊗ ⎝
jm−s2
⎞
wj ⎠ ⊗ vl1 ⊗ vl1
jm−s2
⊗ · · · ⊗ vs ⊗ vs ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
⎛
⎛
⎞
⎞
m
m
⎝
wj ⎠ ⊗ · · · ⊗ ⎝
wj ⎠ ⊗ vl ⊗ vl
jm−s2
jm−s2
⊗ · · · ⊗ vs ⊗ vs ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
⎛
⎛
⎞
⎞
m
m
v1 ⊗ v1 vl ⊗ vl ⊗ ⎝
wj ⎠ ⊗ · · · ⊗ ⎝
wj ⎠
jm−s2
jm−s2
2.6
⊗ vl1 ⊗ vl1 ⊗ · · · ⊗ vs ⊗ vs ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
⎛
⎛
⎞ ⎛
⎞
⎞
l−1
m
m
m
≡ ⎝ wj wj ⎠ ⊗ ⎝
wj ⎠ ⊗ · · · ⊗ ⎝
wj ⎠
j2
jm−s2
jm−s2
jm−s2
⊗ vl1 ⊗ vl1 ⊗ · · · ⊗ vs ⊗ vs ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1
⎛
⎛
⎞
⎞
m
m
≡ − l − 2l − 1!vi xm1 ⎝
wj ⎠ ⊗ · · · ⊗ ⎝
wj ⎠
jm−s2
jm−s2
⊗ vl1 ⊗ vl1 ⊗ · · · ⊗ vs ⊗ vs ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1 ,
where the last equivalence follows from the induction hypothesis and the fact wj 1, 2 −vj ⊗ vj . Hence, we have proved what we desired.
6
International Journal of Mathematics and Mathematical Sciences
As a consequence, we immediately get that
⎛
s!vi xm1 ≡ ⎝
m
⎛
⎞
m
wj ⎠ ⊗ · · · ⊗ ⎝
jm−s2
⎞
wj ⎠ ⊗ vs1 ⊗ · · · ⊗ vm−s1 xm1 .
2.7
jm−s2
However, m − m − s 1 s − 1, 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 xm1 ≡ 0.
1
Theorem 2.2. The annihilator of the space V ⊗n /V ⊗n Bn of dual harmonic tensors in the group
algebra KSn is the principal ideal xm1 .
1
1
Proof. We denote AnnV ⊗n /V ⊗n Bn as the annihilator of the space V ⊗n /V ⊗n Bn
harmonic tensors in the group algebra KSn . It follows from Proposition 2.1 that
xm1 ⊆ Ann
of dual
V ⊗n
.
1
V ⊗n Bn
2.8
On the other hand, by the work of 10, we know that
xm1 K − Span mλs,t | λ n, λ > m, s, t ∈ Stdλ ,
2.9
where each mλs,t is the Murphy basis element in 10, and Stdλ denotes the set of standard
λ-tableaux with entries in {1, 2, . . . , n}. In particular, 5, Theorem 1.8 shows that see also
4
dimK xm1 dimK Sλ
2
λn,λ>m
dimQ QSn − EndQSpV dimK KSn − EndKSpV dimK Ann
V ⊗n
1
V ⊗n Bn
V ⊗n
1
V ⊗n Bn
V ⊗n
2.10
1
V ⊗n Bn
,
where Sλ denotes the Specht module of KSn associated to λ. This completes the proof of the
theorem.
f
Let Bn be the two-sided ideal of Bn generated by e1 e3 · · · e2f−1 with 1 ≤ f ≤ n/2. Let
Xm1 ∈ Bn 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.
International Journal of Mathematics and Mathematical Sciences
7
f
Conjecture 2.3. The annihilator of the space V ⊗n /V ⊗n Bn of dual partially harmonic tensors of
f
f
valence f in the algebra Bn /Bn is the principal ideal Xm1 Bn .
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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