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ISSN (print): 2251-7650, ISSN (on-line): 2251-7669 Vol. 3 No. 4 (2014), pp. 63-69.
c
2014 University of Isfahan
www.ui.ac.ir
SYMMETRY CLASSES OF POLYNOMIALS ASSOCIATED WITH THE DIRECT PRODUCT OF PERMUTATION GROUPS
E. BABAEI AND Y. ZAMANI∗
Communicated by Mark L. Lewis
Abstract. LetGi be a subgroup ofSmi, 1≤i≤k. Supposeχiis an irreducible complex character
ofGi. We consider G1× · · · ×Gk as subgroup of Sm, wherem=m1+· · ·+mk. In this paper, we give a formula for the dimension ofHd(G1× · · · ×Gk, χ1× · · · ×χk) and investigate the existence of an o-basis of this type of classes.
1. Introduction
The relative symmetric polynomials as a generalization of symmetric polynomials are introduced
in [7]. In [1, 10, 11], the authors studied the space of relative symmetric polynomials (symmetry class
of polynomials) with respect to the irreducible characters of certain groups. In this paper we study
the symmetry class of polynomials with respect to the direct product of permutation groups. First we
give a review of this notion (for more details, see [7]).
LetHd[x1, . . . , xm] be the complex space of homogeneous polynomials of degreedwith independent
commuting variables x1, . . . , xm. Let Γ+m,d be the set of all m-tuples of non-negative integers α =
(α1, . . . , αm), such that Pmi=1αi = d. For any α ∈ Γ+m,d, let Xα be the monomial xα11x
α2
2 · · ·xαmm .
Then the set {Xα| α ∈ Γ+m,d} is a basis of Hd[x1, . . . , xm]. An inner product on Hd[x1, . . . , xm] is
defined by
hXα, Xβi=δα,β.
MSC(2010): Primary: 05E05; Secondary: 15A69, 20C15.
Keywords: Symmetric polynomials, symmetry class of polynomials, orthogonal basis, permutaion groups, complex characters. Received: 30 January 2014, Accepted: 19 May 2014.
∗Corresponding author.
SupposeG is a subgroup of the symmetric groupSm. ThenG acts onHd[x1, . . . , xm] by
qσ(x1, . . . , xm) =q(xσ−1(1), . . . , xσ−1(m)),
and this action is extended linearly to the group algebraCG. Letχbe an irreducible complex character
of G. Consider the idempotent
T(G, χ) = χ(1)
|G| X
σ∈G
χ(σ)σ,
in the group algebraCG. The image ofHd[x1, . . . , xm] under the map T(G, χ) is called the symmetry
class of polynomials of degree d with respect to G and χ, and it is denoted by Hd(G, χ). For any
q ∈Hd[x1, . . . , xm],
q∗ =T(G, χ)(q)
is called asymmetrized polynomial with respect toGandχ. Forα∈Γ+m,d, we denote the symmetrized monomial (Xα)∗ byXα,∗. So
Hd(G, χ) =hXα,∗|α∈Γ+m,di.
The groupG also acts on Γ+m,d by
ασ= (ασ(1), . . . , ασ(m)).
Let ∆ be a set of representatives of orbits of Γ+m,d under the action G. For any α∈Γ+m,d, we have
kXα,∗k2 = χ(1)[χ,1]Gα
|G:Gα|
,
(1.1)
where Gα is the stabilizer subgroup of α under the action of G and [ , ]G is the inner product of
characters (see [5]). Hence, Xα,∗ 6= 0 if and only if [χ,1]Gα 6= 0.
Let Ω be the set of allα∈Γ+m,d with [χ,1]Gα 6= 0 and suppose ¯∆ = ∆∩Ω. We have
dimHd(G, χ) = χ(1) X
α∈∆¯
[χ,1]Gα.
(1.2)
An orthogonal basis of Hd(G, χ) of the form {Xα,∗|α ∈S},where S is a subset of Γ+m,d is called an
o-basis of Hd(G, χ). If χ is linear, then the set {Xα,∗|α ∈ ∆¯} is an o-basis of Hd(G, χ). If χ is not
linear, then Hd(G, χ) may have no o-basis.
In this paper, we considerG=G1× · · · ×Gk, whereGi is permutation group and we find a formula for
the dimension of Hd(G, χ). Then we investigate the existence of an o-basis for this symmetry class.
A similar result has been obtained for symmetry classes of tensors in [2, 3, 4, 6, 8, 9, 12].
2. Main Results
A partition of m is a non-increasing finite sequence of positive integers, whose sum is m. Ifπ is a
partition of m we denote π`m. Letπ = (π1, . . . , πk)`m. For any 1≤i≤k, suppose
Λi={t: i−1
X
j=1
πj < t≤ i X
j=1
The corresponding Young subgroup is defined by
Sπ =SΛ1× · · · ×SΛk,
where SΛi is the symmetric group on Λi.
Now, let Gi be a subgroup of Sπi, 1 ≤i ≤k. Let m = Pki=1πi. Rearranging the order, we may
assume π = (π1, . . . , πk) `m. LetG=G1× · · · ×Gk. Then G≤Sπ. In particular, we can consider
G as a subgroup of symmetric groupSm. Let χ be an irreducible character ofG. Then χ=Qki=1χi,
where χi is an irreducible character of Gi, 1≤i≤k.
Let Γ+Λi,ρi be the set of all πi-tuples of non-negative integers αi = (αr1, . . . , αrπi), such that
Pπi
j=1αrj = ρi, where rk = Pij−1=1πj +k, 1 ≤ k ≤ πi. Let Xi = {xr1, . . . , xrπi}, and suppose
Hρi[Xi] is the complex space of homogeneous polynomials of degree ρi with independent variables in
Xi. For anyαi ∈Γ+Λi,ρi, let Xα i
i be the monomial
xαr1
r1 · · ·x
αrπi rπi .
Then the set{Xiαi|αi∈Γ+Λi,ρi}is a basis ofHρi[Xi]. Naturally, we define an inner product onHρi[Xi]
by hXiαi, Xiβii =δαi,βi. We define Hρi(Gi, χi) =T(Gi, χi)(Hρi[Xi]) and Xα i,∗
i =T(Gi, χi)(Xα i i ). So
we have Hρi(Gi, χi) =hXα i,∗
i |αi∈Γ
+ Λi,ρii.
Letα= (α1, . . . , αm)∈Γ+m,d. Then (α|Λ1, . . . , α|Λk)∈Qki=1Γ +
Λi,ρi, whereρi = P
j∈Λiαj. There is a
bijection between Γ+m,d and the set
[
(ρ1,...,ρk)∈Γ+k,d k Y
i=1 Γ+Λi,ρi.
We denote α|Λi by αi. If g =g1· · ·gk, where gi ∈ Gi,1 ≤i ≤k, then αg = (α1g1, . . . , αkgk), so we
have
∆ = [
(ρ1,...,ρk)∈Γ+k,d
k Y
i=1
∆Λi,ρi,
where ∆Λi,ρi is a set of representatives of orbits of Γ+Λi,ρi under the action Gi.
Now, letα= (α1, . . . , αk)∈Qki=1Γ+Λi,ρi for some (ρ1, . . . , ρk)∈Γ+k,d. ThenGα =Qki=1(Gi)αi, where
(Gi)αi is the stabilizer subgroup ofαi∈Γ+Λi,ρi inGi. Hence we have [χ,1]Gα =Qki=1[χi,1](Gi)
αi. Thus
[χ,1]Gα 6= 0 if and only if [χi,1](Gi)αi 6= 0, for all i, 1 ≤ i ≤ k. We denote by ΩΛi,ρi, the set of all
αi∈Γ+
Λi,ρi with [χi,1](Gi)αi = 0 and define ¯6 ∆Λi,ρi = ΩΛi,ρi∩∆Λi,ρi. Summarizing previous statements,
Theorem 2.1. With respect to the above mentioned notations, the following equality holds.
¯
∆ = [
(ρ1,...,ρk)∈Γ+k,d k Y
i=1 ¯ ∆Λi,ρi.
In the following theorem, we give a formula for the dimension of Hd(G, χ).
Theorem 2.2. We have
dimHd(G, χ) =
X
(ρ1,...,ρk)∈Γ+k,d k Y
i=1
dimHρi(Gi, χi).
Proof. Applying (1.2) and Theorem 2.1, we have
dimHd(G, χ) = χ(1) X
α∈∆¯
[χ,1]Gα
=
k Y
i=1
χi(1) X
ρ∈ Γ+k,d
X
α∈Qki=1∆Λ¯ i,ρi
k Y
i=1
[χi,1](Gi)αi
= X
ρ∈ Γ+k,d
k Y
i=1
χi(1) X
αi∈ ∆Λ¯
i,ρi
[χi,1](Gi)αi
= X
ρ∈ Γ+k,d
k Y
i=1
dimHρi(Gi, χi),
where α= (α1, . . . , αk)∈Qki=1Γ+Λi,ρi and ρ= (ρ1, . . . , ρk)∈Γ+k,d. Corollary 2.3. If G=G1×G2 andχ=χ1×χ2 where χi ∈Irr(Gi), i= 1,2, then
dimHd(G, χ) = d X
i=0
dimHi(G1, χ1) dimHd−i(G2, χ2).
Proof. Since Γ+2,d={(i, d−i), 0≤i≤d}, the result is immediate by Theorem 2.2. LetDk,d+ be the set of all (ρ1, . . . , ρk)∈Γ+k,dsuch thatHρi(Gi, χi)6= 0,for any 1≤i≤k. We define
the subset Hρ1(G1, χ1)· · ·Hρk(Gk, χk) of Hd(G, χ) by
X
finite
f1· · ·fk|fi ∈Hρi(Gi, χi), 1≤i≤k .
It is easy to see thatHρ1(G1, χ1)· · ·Hρk(Gk, χk) is a vector space on Cand dim Hρ1(G1, χ1)· · ·Hρk(Gk, χk) =
k Y
i=1
dim Hρi(Gi, χi).
Theorem 2.4. Let G=G1× · · · ×Gk and χ=Qki=1χi∈Irr(G). Then
Hd(G, χ) =
M
(ρ1,...,ρk)∈D+k,d
Proof. Let α = (α1, . . . , αk) ∈ Qki=1ΓΛ+i,ρi, for some ρ = (ρ1, . . . , ρk) ∈ D+k,d. Then
Xα,∗=X1α1,∗· · ·Xkαk,∗. Hence
hXα,∗|α∈ k Y
i=1
Γ+Λi,ρii=Hρ1(G1, χ1)· · ·Hρk(Gk, χk).
If t = (t1, . . . , tk) ∈ D+k,d with ρ 6= t and suppose β = (β1, . . . , βk) ∈ Qk
i=1Γ +
Λi,ti, then hXα,∗, Xβ,∗i= 0, so we obtain
Hd(G, χ) = hXα,∗|α ∈Γ+m,di
= M
(ρ1,...,ρk)∈D+k,d
hXα,∗|α∈ k Y
i=1 Γ+Λi,ρii
= M
(ρ1,...,ρk)∈D+k,d
Hρ1(G1, χ1)· · ·Hρk(Gk, χk).
Let G=Sπ and χ = 1Sπ. Let Λiρi be the space of symmetric homogeneous polynomials of degree
ρi with independent variables in Xi. Using Theorem 2.4, we have
Hd(G, χ) =
M
(ρ1,...,ρk)∈D+k,d
Λ1ρ1· · ·Λkρk.
If α = (α1, . . . , αk), β = (β1, . . . , βk) ∈ Qki=1Γ+Λi,ρi for some (ρ1, . . . , ρk) ∈ Γ+k,d, then α = β if and
only if αi=βi for any 1≤i≤k. Thus
hXα, Xβi=δα,β= k Y
i=1
δαi,βi = k Y
i=1
hXiαi, Xiβii.
Therefore
hXα,∗, Xβ,∗i = χ(1) 2
|G|2
X
σ,θ∈G
χ(σ)χ(θ)hXασ, Xβθi
=
k Y
i=1
χi(1)2 |Gi|2
X
σi,θi∈Gi
χi(σi)χi(θi)hXα iσi i , Xβ
iθi i
=
k Y
i=1
hXiαi,∗, Xiβi,∗i.
(2.1)
In the following theorem we give a necessary and sufficient condition for existence of an o-basis of
Hd(G, χ).
Theorem 2.5. Symmetry class Hd(G, χ) has an o-basis if and only if for any ρ∈D+k,d and for any
Proof. Let ρ = (ρ1, . . . , ρk) ∈ Dk,d+ . Suppose {X α,∗
i | αi ∈ Sρi ⊆ Γ+Λi,ρi} is an o-basis of Hρi(Gi, χi),
1 ≤ i ≤ k. Let Sρ = {(α1, . . . , αk)|αi ∈ Sρi}. Then, by (2.1), {Xα,∗| α ∈ Sρ} is an o-basis of
Hρ1(G1, χ1)· · ·Hρk(Gk, χk). Now if we set S =
S
ρ∈D+k,dSρ, then {Xα,
∗| α ∈ S} is an o-basis of
Hd(G, χ), by Theorem 2.4.
Conversely, ifHd(G, χ) has an o-basis, then by Theorem 2.4, for any (ρ1, . . . , ρk)∈Dk,d+ ,
hXα,∗|α∈ k Y
i=1
Γ+Λi,ρii=Hρ1(G1, χ1)· · ·Hρk(Gk, χk),
has an o-basis, say {Xα,∗| α ∈ Sρ}, where Sρ ⊆ Qki=1Γ+Λi,ρi. Then there are subsets Si,ρi ⊆ Γ+Λi,ρi,
1 ≤ i ≤ k, such that Sρ = Qki=1Si,ρi. We show that Ei,ρi = {Xα i,∗
i | αi ∈ Si,ρi} is an o-basis of
Hρi(Gi, χi) for any 1≤i≤k.
Let qi∈Hρi[Xi], 1≤i≤k. Thenq =q1· · ·qk∈Hd[x1, . . . , xm] and we have
q∗ =T(G1, χ1)(q1)· · ·T(Gk, χk)(qk).
Since q∗∈ hXα,∗|α∈Qki=1Γ+Λi,ρii, we can supposeq∗ =P
α∈SρcαXα,
∗, wherec
α is complex number.
Let cα1 =· · ·=cαk =c
1/k α . Then
q∗ = X
α∈Sρ
cαXα,∗
= X
α1∈S 1,ρ1
cα1Xα 1,∗ 1 · · ·
X
αk∈S k,ρk
cαkXα k,∗ k .
Hence there are complex numbers λ1, . . . , λk such that T(Gi, χi)(qi) = λiPαi∈S
i,ρicαiX αi,∗ i ,
1≤i≤k. Since
|Sρ| = dimHρ1(G1, χ1)· · ·Hρk(Gk, χk)
=
k Y
i=1
dimHρi(Gi, χi)
≤ |S1,ρ1| × · · · × |Sk,ρk| = |Sρ|,
so, for any i, 1≤i≤k, the set Ei,ρi is a basis ofHρi(Gi, χi). It remains to show the orthogonality.
Let αi ∈Si,ρi. Then there is α = (α1, . . . , αk) ∈Qki=1Γ +
Λi,ρi such that α|Λi =α
i. Supposeβi ∈S i,ρi
such that αi 6=βi. Letβ = (α1, . . . , βi, . . . , αk). Then α, β∈S
ρand α6=β. Using (2.1), we have
0 = hXα,∗, Xβ,∗i
= Y
j6=i
kXjαj,∗k2
hXiαi,∗, Xiβi,∗i.
Since αi ∈ Si,ρi, 1 ≤ i ≤ k, we have kXα i,∗
i k 6= 0. Then hX αi,∗ i , X
βi,∗
i i = 0. This completes the
Acknowledgments
The authors thank the referee for his/her useful comments. This work was done while the first author
was in sabbatical leave at Instituto Nacional de Matemtica Pura e Aplicada (IMPA), Brazil. He would
like to express his thanks to Professor Hossein Movasati for this invitation. Also he would like to thank
the IMPA for some financial support.
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Esmaeil Babaei
Faculty of Sciences, Sahand University of Technology, P.O.Box 53317-11111, Tabriz, Iran
Email: e [email protected]
Yousef Zamani
Faculty of Sciences, Sahand University of Technology, P.O.Box 53317-11111, Tabriz, Iran
