The center of the wreath product of symmetric group algebras
We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Far...
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| Zitieren: | The center of the wreath product of symmetric group algebras / O. Tout // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 302–322. — Бібліогр.: 12 назв. — англ. |
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| citation_txt | The center of the wreath product of symmetric group algebras / O. Tout // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 302–322. — Бібліогр.: 12 назв. — англ. |
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| description | We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra.
|
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© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 31 (2021). Number 2, pp. 302–322
DOI:10.12958/adm1338
The center of the wreath product
of symmetric group algebras∗
O. Tout
Communicated by D. Simson
Abstract. We consider the wreath product of two symmetric
groups as a group of blocks permutations and we study its conju-
gacy classes. We give a polynomiality property for the structure
coefficients of the center of the wreath product of symmetric group
algebras. This allows us to recover an old result of Farahat and
Higman about the polynomiality of the structure coefficients of the
center of the symmetric group algebra and to generalize our recent
result about the polynomiality property of the structure coefficients
of the center of the hyperoctahedral group algebra.
1. Introduction
The conjugacy classes of the symmetric group Sn can be indexed by
partitions of n. The conjugacy class associated to a partition λ is the
set of all permutations with cycle-type λ. The center of the symmetric
group algebra is the algebra over C generated by the conjugacy classes of
the symmetric group. Its structure coefficients have nice combinatorial
properties. In [1], Farahat and Higman, gave a polynomiality property for
the structure coefficients of the center of the symmetric group algebra.
By introducing partial permutations in [3], Ivanov and Kerov, gave a
combinatorial proof to this result.
∗This research is supported by Narodowe Centrum Nauki, grant number
2017/26/A/ST1/00189.
2020 MSC: 05E10, 05E16, 20C30.
Key words and phrases: symmetric groups, wreath products, structure coeffi-
cients, centers of finite groups algebras.
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O. Tout 303
We introduce in this paper the group Bk
kn which permutes n blocks
of k elements each. The permutation of the k elements in each block
is allowed. This group is the symmetric group Sn if k = 1 and in case
k = 2 it is the hyperoctahedral group Hn on 2n elements. In general, we
show that the group Bk
kn is isomorphic to the wreath product Sk ≀ Sn of
the symmetric group Sk by the symmetric group Sn. It is well known
that the conjugacy classes of Hn are indexed by pairs of partitions (λ, δ)
verifying |λ|+ |δ| = n, see [2] and [7]. We show that in general, for any
fixed integer k, the conjugacy classes of the group Bk
kn are indexed by
families of partitions λ = (λ(ρ))ρ⊢k indexed by the set of partitions of k
such that the sum of the sizes of all λ(ρ) equals n. This comes with no
surprise since it agrees with the description of the conjugacy classes of
the wreath product G ≀ Sn, where G is a finite group, given by Specht in
[6], see also [4] and [5]. However, our way to prove this fact for Sk ≀ Sn
seems to be easier.
Recently, in [10], we developed a framework in which the polynomiality
property for double-class algebras, and subsequently centers of groups
algebra, holds. In particular, we showed that our framework contains the
sequence of the symmetric groups and that of the hyperoctahedral groups.
Thus we obtained again the result of Farahat and Higman for the structure
coefficients of the center of the symmetric group algebra. In addition we
gave a polynomiality property for the structure coefficients of the center
of the hyperoctahedral group algebra in [10, Section 6.2].
In this paper we show that the general framework we gave in [10]
contains the sequence of groups (Bk
kn)n when k is a fixed integer. Thus, we
will be able to give a polynomiality property for the structure coefficients
of the center of the group Bk
kn algebra. This result can be obtained as
an application of Theorem 2.13 given by Wang in [12] which shows that
the structure coeffecients of the center of G ≀ Sn are polynomials in n for
any finite group G. However, the paper of Wang uses the Farahat-Higman
approach while the way taken in this paper is different and seems to
be easier to arrive to the result in the particular case when G = Sk. A
particular attention to the cases k = 2 (hyperoctahedral group) and k = 3
is given. In a separate paper [11] we generalise the concept of partial
permutation of Ivanov and Kerov to k-partial permutation and we give a
more combinatorial proof to this result.
The paper is organized as follows. In Section 2, we review all necessary
definitions of partitions and we describe the conjugacy classes of the
symmetric group. Then, we define explicitly the group Bk
kn in Section 3
and we study in details its conjugacy classes. After this, we show that it is
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304 Center of wreath product
isomorphic to the wreath product Sk ≀ Sn. In Sections 4.1 and 4.2 a special
treatment is given for the cases k = 2 and k = 3 respectively. The last
section contains our main result, that is a polynomiality property for the
structure coefficients of the center of the group Bk
kn algebra. In addition
some examples are given for the cases k = 2 and k = 3.
2. Partitions and conjugacy classes of the symmetric
group
If n is a positive integer, we denote by Sn the symmetric group of
permutations on the set [n] := {1, 2, . . . , n}. A partition λ is a list of
integers (λ1, . . . , λl) where λ1 > λ2 > . . . λl > 1. The λi are called the
parts of λ; the size of λ, denoted by |λ|, is the sum of all of its parts. If
|λ| = n, we say that λ is a partition of n and we write λ ⊢ n. The number
of parts of λ is denoted by l(λ). We will also use the exponential notation
λ = (1m1(λ), 2m2(λ), 3m3(λ), . . . ), where mi(λ) is the number of parts equal
to i in the partition λ. In case there is no confusion, we will omit λ from
mi(λ) to simplify our notation. If λ = (1m1(λ), 2m2(λ), 3m3(λ), . . . , nmn(λ))
is a partition of n then
∑n
i=1 imi(λ) = n. We will dismiss imi(λ) from
λ when mi(λ) = 0. For example, we will write λ = (12, 3, 62) instead of
λ = (12, 20, 3, 40, 50, 62, 70). If λ and δ are two partitions we define the
union λ ∪ δ and subtraction λ \ δ (if exists) as the following partitions:
λ ∪ δ = (1m1(λ)+m1(δ), 2m2(λ)+m2(δ), 3m3(λ)+m3(δ), . . . ),
λ \ δ = (1m1(λ)−m1(δ), 2m2(λ)−m2(δ), . . . ) if mi(λ) > mi(δ) for any i.
A partition is called proper if it does not have any part equal to 1. The
proper partition associated to a partition λ is the partition λ̄ := λ \
(1m1(λ)) = (2m2(λ), 3m3(λ), . . . ).
The cycle-type of a permutation of Sn is the partition of n obtained
from the lengths of the cycles that appear in its decomposition into product
of disjoint cycles. For example, the permutation
(2, 4, 1, 6)(3, 8, 10, 12)(5)(7, 9, 11)
of S12 has cycle-type (1, 3, 42). In this paper we will denote the cycle-type
of a permutation ω by ct(ω). It is well known that two permutations of
Sn belong to the same conjugacy class if and only if they have the same
cycle-type. Thus the conjugacy classes of the symmetric group Sn can be
indexed by partitions of n. If λ = (1m1(λ), 2m2(λ), 3m3(λ), . . . , nmn(λ)) is a
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O. Tout 305
partition of n, we will denote by Cλ the conjugacy class of Sn associated
to λ:
Cλ := {σ ∈ Sn | ct(σ) = λ}.
The cardinal of Cλ is given by:
|Cλ| =
n!
zλ
,
where
zλ := 1m1(λ)m1(λ)!2
m2(λ)m2(λ)! . . . n
mn(λ)mn(λ)!.
3. The conjugacy classes of the group B
k
kn
In this section we define the group Bk
kn then we study its conjugacy
classes. We show in Proposition 1 that these latter are indexed by families
of partitions indexed by all partitions of k. Then in Proposition 2 we
give an explicit formula for the size of any of its conjugacy classes. The
group Bk
kn is isomorphic to the wreath product Sk ≀ Sn as will be shown in
Proposition 3. However, we decided to work with this copy of the wreath
product in this paper since it seems very natural to present our main
result.
If i and k are two positive integers, we denote by pk(i) the following
set of size k :
pk(i) := {(i− 1)k + 1, (i− 1)k + 2, . . . , ik}.
The above set pk(i) will be called a k-tuple in this paper. We define
the group Bk
kn to be the subgroup of Skn formed by permutations that
send each set of the form pk(i) to another set with the same form:
Bk
kn := {w ∈ Skn; ∀ 1 6 r 6 n, ∃ 1 6 r′ 6 n | w(pk(r)) = pk(r
′)}.
Example 1.
(
1 2 3 | 4 5 6
1 3 2 | 6 5 4
)
∈ B3
6 but
(
1 2 3 | 4 5 6
1 3 6 | 2 4 6
)
/∈ B3
6.
Remark 1. The sign | will be added after k elements in the two-lines
notation of a permutation to let the reader easily see that we are working
in the case k. In the previous example k was 3.
The group Bk
kn as it is defined here appears in [10, Section 7.4]. When
k = 1, it is clear that B1
n is the symmetric group Sn. When k = 2, the
group B2
2n is the hyperoctahedral group Hn on 2n elements, see [9] where
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306 Center of wreath product
the author treats Hn as being the group B2
2n. It would be clear to see that
the order of the group Bk
kn is
|Bk
kn| = (k!)nn!
The decomposition of a permutation ω ∈ Bk
kn into product of disjoint
cycles has a remarkable pattern. To see this, fix a k-tuple pk(i) and a
partition ρ = (ρ1, ρ2, . . . , ρl) of k. Suppose now that while writing ω as a
product of disjoint cycles we get among the cycles the following pattern:
C1 = (a1, . . . , a2, . . . , aρ1 , . . . ), (1)
C2 = (aρ1+1, . . . , aρ1+2, . . . , aρ1+ρ2 , . . . ),
...
Cl = (aρ1+···+ρl−1+1, . . . , aρ1+···+ρl−1+2, . . . , aρ1+···+ρl−1+ρl , . . . ),
where
{a1, a2, . . . , aρ1+1, . . . , aρ1+···+ρl−1+ρl} = pk(i) for a certain i ∈ [n].
We should remark that since ω ∈ Bk
kn, if we consider bj = ω(aj) for any
1 6 j 6 |ρ| then there exists r ∈ [n] such that:
pk(r) = {b1, b2, . . . , bρ1+1, . . . , bρ1+···+ρl−1+ρl}.
This can be redone till we reach a2 which implies that in cycle C1, we have
the same number of elements between ai and ai+1 for any 1 6 i 6 ρ1 − 1.
Thus the size of the cycle C1 is a multiple of ρ1, say |C1| = mρ1. The
same can be done for all the other cycles Ci and in fact for any 1 6 i 6 l,
|Ci| = mρi. In addition, if we take the set of all the elements that figure
in the cycles Ci we will get a disjoint union of m k-tuples. That means:
l
∑
j=1
|Cj | =
l
∑
j=1
mρj = mk.
Now construct the partition ω(ρ) by grouping all the integers m as above.
Remark 2. We should pay the reader’s attention to two important
remarks after this construction. First of all, for any partition ρ of k there
are rk elements involved when adding the part r to ω(ρ). That means:
∑
ρ⊢k
∑
r>1
krmr(ω(ρ)) = kn which implies
∑
ρ⊢k
∑
r>1
rmr(ω(ρ)) = n.
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O. Tout 307
Now let pω denotes the blocks permutation of n associated to ω. That
is pω(i) = j whenever ω(pk(i)) = pk(j). Adding a part r to one of the
partitions ω(ρ) implies that r is the length of one of the cycles of pω. Thus
we have:
⋃
ρ⊢k
ω(ρ) = ct(pω).
In other words, one may first find the cycle-type of pω and then distribute all
its parts between the partitions ω(ρ) according to the above construction.
Example 2. Consider the following permutation, written in two-lines
notation, ω of B3
24
ω =
(
1 2 3 | 4 5 6 | 7 8 9 | 10 11 12 | 13 14 15 | 16 17 18 | 19 20 21 | 22 23 24
12 10 11 | 20 21 19 | 8 7 9 | 1 2 3 | 16 18 17 | 15 14 13 | 5 4 6 | 22 23 24
)
.
Its decomposition into product of disjoint cycles is:
ω = (1, 12, 3, 11, 2, 10)(4, 20)(5, 21, 6, 19)(7, 8)(9)(13, 16, 15, 17, 14, 18)
(22)(23)(24).
The first cycle (1, 12, 3, 11, 2, 10) contains all the elements of p3(1) thus it
contributes to ω(3). In it, there are two 3-tuples namely p3(1) and p3(4)
thus we should add 2 to the partition ω(3). For a similar reason, the cycle
(13, 16, 15, 17, 14, 18) contributes to ω(3) by 2 also, thus ω(3) becomes
(2, 2). By looking to the cycles (4, 20)(5, 21, 6, 19) we see that 5 and 6
belong to the same cycle while 4 belongs to the other, thus these cycles
will contribute to ω(2, 1). Since they are formed out of the two 3-tuples
p3(2) and p3(7), we should add the part 2 to the partition ω(2, 1). In the
same way, the cycles (7, 8)(9) contribute to ω(2, 1) by 1 to become the
partition (2, 1). The remaining cycles (22)(23)(24) give ω(1, 1, 1) = (1).
The reader should remark that pω is the permutation (1, 4)(2, 7)(3)(5, 6)(8)
of 8 and that:
ω(3) ∪ ω(2, 1) ∪ ω(13) = ct(pω) = (12, 23).
Definition 1. If ω ∈ Bk
kn, define type(ω) to be the following family of
partitions indexed by partitions of k
type(ω) := (ω(ρ))ρ⊢k.
Proposition 1. Two permutations α and β of Bk
kn are in the same
conjugacy class if and only if they both have the same type.
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308 Center of wreath product
Proof. Suppose α = γβγ−1 for some γ ∈ Bk
kn and fix a partition ρ of k.
Suppose that the cycles C1, . . . , Cl of Equation (1) contribute to β(ρ) then
the cycles
C
′
1 = (γ(a1), . . . , γ(a2), . . . , γ(aρ1), . . . ),
C
′
2 = (γ(aρ1+1), . . . , γ(aρ1+2), . . . , γ(aρ1+ρ2), . . . ),
...
C
′
l = (γ(aρ1+···+ρl−1+1), . . . , γ(aρ1+···+ρl−1+2), . . . , γ(aρ1+···+ρl−1+ρl), . . . ),
will contribute to α(ρ) and they have respectively the same lengths as
C1, . . . , C2. This proves the first implication.
Conversely, type(α) = type(β) means that α(ρ) = β(ρ) for any
partition ρ of k. In order to simplify, we will look at the elements of two fixed
k-tuples, say pk(1) = {1, 2, . . . , k} for α and pk(2) = {k+1, k+2, . . . , 2k}
for β, such that the distribution of the elements of pk(1) among the cycle
decomposition in α is similar to that of the elements of pk(2) in β. In
other words, the cycle decompositions of α and β are as follows:
α = (1, . . . , 2, . . . , ρ1, . . . )(ρ1 + 1, . . . , ρ1 + 2, . . . , ρ1 + ρ2, . . . )
· · · (ρ1 + · · ·+ ρl−1 + 1, . . . , ρ1 + · · ·+ ρl−1 + 2,
. . . , ρ1 + · · ·+ ρl−1 + ρl, . . . )
and
β = (k + 1, . . . , k + 2, . . . , k + ρ1, . . . )
(k + ρ1 + 1, . . . , k + ρ1 + 2, . . . , k + ρ1 + ρ2, . . . )
· · · (k + ρ1 + · · ·+ ρl−1 + 1, . . . , k + ρ1 + · · ·+ ρl−1 + 2,
. . . , k + ρ1 + · · ·+ ρl−1 + ρl, . . . )
Construct γ to be the permutation that orderly takes each element of the
above cycles of α to each element with the same order in β, that is:
γ(1) = k + 1, γ(α(1)) = β(k + 1), . . . γ(ρ1) = k + ρ1, . . .
It is then clear that α = γ−1βγ and γ ∈ Bk
kn.
Proposition 2. If ω is a permutation of Bk
kn then the size conj(ω) of
the conjugacy class of ω is given by
conj(ω) =
n!(k!)n
∏
ρ⊢k
zω(ρ)z
l(ω(ρ))
ρ
.
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O. Tout 309
Proof. Suppose that ct(pω) = λ, where λ is a partition of n. We will
explain how to obtain all the elements γ ∈ Bk
kn that have the same type
as ω. First of all, pγ must be decomposed in cycles with sizes equivalent
to those of pω. In other words, pγ should have the same cycle-type as pω
and there are
n!
zλ
choices to make pγ .
Now suppose that mr(λ) 6= 0 for some r ∈ [n]. That means that there
are exactly mr(λ) cycles of length r distributed in the family (ω(ρ))ρ⊢k.
Suppose they all appear in the partitions ω(ρ1), . . . , ω(ρm). In order to
have type(γ) = type(ω), we should make γ(ρ1) = ω(ρ1), γ(ρ2) = ω(ρ2)
and so on. That is γ(ρ1) should have parts equal to r as many as ω(ρ1)
has and so on. The total number of γ verifying these conditions is
mr(λ)!
∏
16j6m
mr(ω(ρj))!
=
mr(λ)!
∏
ρ⊢k
mr(ω(ρ))!
It remains to see how many γ ∈ Bk
kn one can make when he nows all
γ(ρ). For this fix a partition ρ1 of k such that γ(ρ1) 6= ∅ and suppose m1
is a part of γ(ρ1). There will be m1 k-tuples involved in the construction
of γ now. Choose one of them then distribute its elements according to ρ1
in k!
zρ1
ways. To complete γ, there will be (k!)m1−1 choices for the elements
between any two consecutive elements of the fixed k-tuples in any chosen
cycle.
By bringing together all the above arguments, the size conj(ω) of the
conjugacy class of ω is:
conj(ω) =
n!
zλ
∏
r,mr(λ) 6=0
mr(λ)!
∏
ρ⊢k
mr(ω(ρ))!
(k!)n
z
l(ω(ρ))
ρ
=
n!(k!)n
∏
ρ⊢k
zω(ρ)z
l(ω(ρ))
ρ
.
This equality is due to the fact that
l(λ) =
∑
r>1
mr(λ) =
∑
r>1
∑
ρ⊢k
mr(ω(ρ)).
We turn now to show that the group Bk
kn is isomorphic to the wreath
product Sk ≀ Sn. The wreath product Sk ≀ Sn is the group with underlying
set Sn
k × Sn and product defined as follows:
((σ1, . . . , σn); p).((ǫ1, . . . , ǫn); q) = ((σ1ǫp−1(1), . . . , σnǫp−1(n)); pq),
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310 Center of wreath product
for any ((σ1, . . . , σn); p), ((ǫ1, . . . , ǫn); q) ∈ Sn
k × Sn. The identity in this
group is (1; 1) := ((1k, 1k, . . . , 1k); 1n), where 1i denotes the identity
function of Si. The inverse of an element ((σ1, σ2, . . . , σn); p) ∈ Sk ≀ Sn is
given by
((σ1, σ2, . . . , σn); p)
−1 := ((σ−1
p(1), σ
−1
p(2), . . . , σ
−1
p(n)); p
−1).
For each permutation ω ∈ Bk
kn and each integer i ∈ [n], define ωi to
be the normalized restriction of ω on the block p−1
ω (i). That is:
ωi : [k] → [k]
b 7→ ωi(b) := ω
(
k(p−1
ω (i)− 1) + b
)
%k,
where % means that the integer is taken modulo k (for a multiple of k we
use k instead of 0).
Example 3. Consider the following permutation α of B3
18 :
α =
(
1 2 3 | 4 5 6 | 7 8 9 | 10 11 12 | 13 14 15 | 16 17 18
12 10 11 | 5 6 4 | 8 7 9 | 15 13 14 | 16 18 17 | 3 2 1
)
.
The blocks permutation associated to α is pα = (1, 4, 5, 6)(2)(3). In
addition, we have:
α1 = (1, 3), α2 = (1, 2, 3), α3 = (1, 2), α4 = (1, 3, 2), α5 = (1, 3, 2)
and
α6 = (2, 3).
Proposition 3. The map
ψ : Bk
kn → Sk ≀ Sn
ω 7→ ψ(ω) := ((ω1, . . . , ωn); pω),
is a group isomorphism.
Proof. ψ is clearly a bijection with inverse given by:
φ : Sk ≀ Sn → Bk
kn
((σ1, σ2, . . . , σn); p) 7→ σ,
where σ
(
k(a − 1) + b
)
= k(p(a) − 1) + σp(a)(b) for any a ∈ [n] and
any b ∈ [k]. It remains to show that if x = ((σ1, σ2, . . . , σn); p) and
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O. Tout 311
y = ((ǫ1, ǫ2, . . . , ǫn); q) are two elements of Sk ≀Sn then φ(x.y) = φ(x)φ(y).
To prove this let a ∈ [n] and b ∈ [k]. On the right hand we have:
φ(x)φ(y)
(
k(a− 1) + b
)
= φ(x)
(
k(q(a)− 1) + ǫq(a)(b)
)
= k(p(q(a))− 1) + σp(q(a))(ǫq(a)(b))
and on the left hand we have:
φ(xy)
(
k(a− 1) + b
)
= k((pq)(a)− 1) + σ(pq)(a)ǫp−1((pq)(a))(b).
This shows the equality φ(x.y) = φ(x)φ(y) and finishes the proof.
Example 4. Recall the permutation α ∈ B3
18 of Example 3 and consider
the following permutation β of the same group:
β =
(
1 2 3 | 4 5 6 | 7 8 9 | 10 11 12 | 13 14 15 | 16 17 18
4 5 6 | 18 17 16 | 8 9 7 | 1 2 3 | 12 11 10 | 15 14 13
)
.
We have:
αβ =
(
1 2 3 | 4 5 6 | 7 8 9 | 10 11 12 | 13 14 15 | 16 17 18
5 6 4 | 1 2 3 | 7 9 8 | 12 10 11 | 14 13 15 | 17 18 16
)
,
ψ(α) =
(
(
(1, 3), (1, 2, 3), (1, 2), (1, 3, 2), (1, 3, 2), (2, 3)
)
; (1, 4, 5, 6)(2)(3)
)
,
ψ(β) =
(
(
1, 1, (1, 2, 3), (1, 3), (1, 3), (1, 3)
)
; (1, 2, 6, 5, 4)(3)
)
and
ψ(αβ) =
(
(
1, (1, 2, 3), (2, 3), (1, 3, 2), (1, 2), (1, 2, 3)
)
; (1, 2)(3)(4)(5)(6)
)
.
Now we can easily verify that ψ(αβ) = ψ(α).ψ(β).
4. Special cases
In this section we treat two special cases, the case of k = 2 and that
of k = 3. We see that when k = 2, the group B2
2n is the hyperoctahedral
group on 2n elements. There are many papers in the literature, see [7] and
[2] for examples, that study the representations of the hyperoctahedral
group. Especially we are interested here in studying its conjugacy classes.
We recover some of its nice properties using our approach.
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312 Center of wreath product
4.1. Case k = 2 : The hyperoctahedral group
When k = 2 there are only two types of partitions of 2,mainly λ1 = (12)
and λ2 = (2). Now we are going to see how things go in the context of
B2
2n. We use the same arguments given in [10, Section 6.2]. If a ∈ p2(i),
we shall denote by a the element of the set p2(i) \ {a}. Therefore, we have,
a = a for any a ∈ [2n]. As seen in our general construction, the cycle
decomposition of a permutation of B2
2n will have two types of cycles. To
see this, suppose that ω is a permutation of B2
2n and take the following
cycle C of its decomposition:
C = (a1, . . . , al).
We distinguish two cases :
1) a1 appears in the cycle C, for example aj = a1. Since ω ∈ B2
2n
and ω(a1) = a2, we have ω(a1) = a2 = ω(aj). Likewise, since
ω(aj−1) = a1, we have ω(aj−1) = a1 which means that al = aj−1.
Therefore,
C = (a1, . . . aj−1, a1, . . . , aj−1)
and l = 2(j − 1) is even. We will denote such a cycle by (O,O).
2) a1 does not appear in the cycle C. Take the cycle C which contains
a1. Since ω(a1) = a2 and ω ∈ B2
2n, we have ω(a1) = a2 and so on.
That means that the cycle C has the following form,
C = (a1, a2, . . . , al)
and that C and C appear in the cycle decomposition of ω.
The cycles of the first case will contribute to ω(λ2) while those of the
second will contribute to ω(λ1). Suppose now that the cycle decomposition
of a permutation ω of B2
2n is as follows:
ω = C1C1C2C2 . . . CkCk(O
1,O1)(O2,O2) . . . (Ol,Ol),
where the cycles Ci (resp. (Oj ,Oj)) are written decreasingly according
to their sizes. From this decomposition, we obtain that the parts of the
partition ω(λ1) are the sizes of the sets Cj , while the parts of the partition
ω(λ2) are the sizes of the sets Oi :
ω(λ1) = (|C1|, . . . , |Ck|), ω(λ2) = (|O1|, . . . , |Ol|)
and
|ω(λ1)|+ |ω(λ2)| = n.
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O. Tout 313
Example 5. Consider the following permutation ω of B2
16
ω =
(
1 2 | 3 4 | 5 6 | 7 8 | 9 10 | 11 12 | 13 14 | 15 16
14 13 | 1 2 | 16 15 | 7 8 | 12 11 | 10 9 | 4 3 | 5 6
)
.
Its decomposition into product of disjoint cycles is as follows:
ω = (1, 14, 3)(2, 13, 4)(7)(8)(9, 12)(10, 11)(5, 16, 6, 15).
Then ω(λ1) = (3, 2, 1) and ω(λ2) = (2).
Apply Proposition 2 to obtain the following result.
Corollary 1. The size of the conjugacy class of a permutation ω ∈ B2
2n
is:
conj(ω) =
2nn!
2l(ω(λ1))+l(ω(λ2))zω(λ1)zω(λ2)
.
The above Corollary 1 is a well known formula for the sizes of the
conjugacy classes of the hyperoctahedral group, see [7] and [2].
4.2. Case k = 3
In a way similar to that of case k = 2, the fact that there are only
three partitions of 3 suggests that there are three types of cycles in the
decomposition into product of disjoint cycles of a permutation ω ∈ B3
3n.
Let C be a cycle of ω ∈ B3
3n, we distinguish the following three cases:
1) first case: all three elements of a certain p3(s) belong to C. For
simplicity, suppose s = 1 and
C = (a1 = 1, a2, a3, . . . , aj = 2, aj+1, . . . , al = 3, al+1, . . . ak).
Since ω ∈ B3
3n, the sets
{a2, aj+1, al+1}, {a3, aj+2, al+2}, . . . , {aj−1, al−1, ak}
all have the form p3(m) and thus C is a cycle of length 3(j − 1) that
contains a union of sets of the form p3(r).
2) second case: two and only two elements of a certain p3(s) belong
to C. Say,
C = (a1 = 1, a2, a3, . . . , aj = 2, aj+1, . . . , ak).
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314 Center of wreath product
Since ω ∈ B3
3n, there exists integers bi, 1 6 i 6 j − 1, such that
{a1, aj} = p3(b1) \ {c1},
{a2, aj+1} = p3(b2) \ {c2},
...
{aj−1, ak} = p3(bj−1) \ {cj−1},
and another cycle (c1, c2, . . . , cj−1) should thus appear in the de-
composition of ω.
3) third case: all three elements of a certain p3(s) belong to different
cycles. In this case, all three cycles will have the same lengths and
each one of them will contain elements belonging to different triplets.
Now for any permutation ω ∈ B3
3n, define γω to be the partition
obtained from the lengths divided by three of the cycles of the first case,
βω the partition obtained from the lengths of the cycles of the second
case divided by two and αω the partition obtained from the lengths of the
cycles of the third case. It would be clear that
ω(λ1) = αω, ω(λ2) = βω, ω(λ3) = γω and |γω|+ |βω|+ |αω| = n.
For example, for the permutation ω of Example 2, we have:
γω = (2, 2), βω = (2, 1) and αω = (1).
Using Proposition 2, we obtain the following result.
Corollary 2. The size conj(ω) of the conjugacy class of ω ∈ B3
3n is:
(3!)nn!
(3!)l(αω)zαω2
l(βω)zβω
3l(γω)zγω
= 2n−l(αω)−l(βω)3n−l(αω)−l(γω).
n!
zαωzβω
zγω
.
5. The center of the group B
k
kn
algebra
In this section, we present in Theorem 1 a polynomiality property
for the structure coefficients of the center of the group Bk
kn algebra. This
can be seen as a generalisation of the Farahat and Higman result in [1]
and our result in [10] that gave polynomiality properties for the structure
coefficients of the center of the symmetric group and the hyperoctahedral
group algebras respectively. A special treatment for the cases k = 2 and
k = 3 is given.
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O. Tout 315
5.1. The algebra Z(C[Bk
kn
])
The center of the group Bk
kn algebra will be denoted Z(C[Bk
kn]). It
is the algebra over C spanned by the "formal sum of elements of the"
conjugacy classes of Bk
kn. According to Proposition 1, these are indexed
by families of partitions x = (x(λ))λ⊢k satisfying the property:
|x| :=
∑
λ⊢k
|x(λ)| = n (2)
and for each such a family its associated conjugacy class Cx is
Cx = {t ∈ Bk
kn such that type(t) = x}
while its formal sum of elements is
Cx :=
∑
t∈Cx
t.
From now on and unless stated otherwise, x is a family of partition would
mean x = (x(λ))λ⊢k with the condition 2.
Let x and y be two family of partitions. In the algebra Z(C[Bk
kn]), the
product CxCy can be written as a linear combination as following
CxCy =
∑
z
czxyCz (3)
where z runs through all the families of partitions. The coefficients czxy
that appear in this equation are called the structure coefficients of the
center of the group Bk
kn algebra.
5.2. Polynomiality of the structure coefficients of Z(C[Bk
kn
])
When k = 1, Farahat and Higman were the first to give a polynomiality
property for the structure coefficients of the center of the symmetric group
algebra in [1]. In the case of the center of the hyperoctahedral group
algebra, we gave a polynomiality property for its structure coefficients in
[10]. The goal of this section is to generalize these results and show that
the structure coefficients of Z(C[Bk
kn]) have a polynomiality property for
any fixed k.
The most natural way to see a permutation ω ∈ Bk
kn as an element
of ω ∈ Bk
k(n+1) is by extending it by identity. By doing so, the new
permutation will have the same type as ω except that the partition ω(1k)
will become ω(1k) ∪ (1).
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316 Center of wreath product
Definition 2. A family of partitions x = (x(λ))λ⊢k is said to be proper
if and only if the partition x(1k) is proper. If x = (x(λ))λ⊢k is a proper
family of partitions such that |x| < n, we define Cx(n) to be the set of
elements t ∈ Bk
kn that have type equals to x except that x(1k) is replaced
by x(1k) ∪ (1n−|x|).
If x = (x(λ))λ⊢k is a proper family of partitions such that |x| = n0
then for any n > n0 we have by Proposition 2 the following result:
|Cx(n)| =
n!(k!)n
zx(1k)∪(1n−n0 )(k!)
l(x(1k))+n−n0
∏
λ⊢k,λ 6=(1k)
zx(λ)z
l(x(λ))
λ
=
n!(k!)n0−l(x(1k))
zx(1k)(n− n0)!
∏
λ⊢k,λ 6=(1k)
zx(λ)z
l(x(λ))
λ
.
Take x and y to be two proper families of partitions. For any integer
n > |x|, |y| we have the following equation in Z(C[Bk
kn])
Cx(n)Cy(n) =
∑
z
chxy(n)Cz(n), (4)
where h runs through all proper families of partitions verifying |z| 6 n.
In [10], under some conditions, a formula describing the form of the
structure coefficients of centers of finite group algebras is given. We show
below that the sequence (Bk
kn)n satisfies these conditions. This will allow
us to use [10, Corollary 6.3] in order to give a polynomiality property for
the structure coefficients chxy(n) described in Equation (4).
We will show below the conditions required in [10] for our sequence of
groups (Bk
kn)n. To avoid repetitions and confusing notations and since the
integer k will be fixed, we will use simply Gn to denote the group Bk
kn.
Hypothesis 1: For any integer 1 6 r 6 n, there exists a group Gr
n
isomorphic to Gn−r. Set
Gr
n := {ω ∈ Gn such that ω(i) = i for any 1 6 i 6 kr}.
for this reason.
Hypothesis 2: The elements of Gr
n and Gr commute between each
other which is normal since the permutations in these groups act on
disjoint sets.
Hypothesis 3: Gr
n+1 ∩Gn = Gr
n which is obvious.
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O. Tout 317
Hypothesis 4: For any z ∈ Gn, k(G
r1
n zG
r2
n ) := min{s|Gr1
n zG
r2
n ∩
Gs 6= ∅} 6 r1 + r2. To prove this, remark first that the size of the set
{1, . . . , kr1} ∩ {z(1), . . . , z(kr2)} is a multiple of k since z ∈ Gn, say it is
km. Suppose that
{h1, . . . , hkr1−km} = {1, . . . , kr1} \ {z(1), . . . , z(kr2)}.
We can find a permutation of the following form
(
1 | 2 | ... | kr2 | kr2+1 | ... | kr1+kr2−km | kr1+kr2−km+1 | ... | kn
z(1) | z(2) | ... | z(kr2) | h1 | ... | hkr1−km | ∗ | ... | ∗
)
in zGr2
n since it contains permutations that fixes the first kr2 images of
z. The stars are used to say that the images may not be fixed. Since the
multiplication by an element of Gr1
n to the left permutes the elements
greater than kr1 in the second line defining this permutation, the set
Gr1
n zG
r2
n contains thus a permutation of the following form
(
1 | 2 | ... | kr2 | kr2+1 | ... | kr1+kr2−km | kr1+kr2−km+1 | ... | kn
∗ | ∗ | ... | ∗ | h1 | ... | hkr1−km | kr1+kr2−km+1 | ... | kn
)
This permutation is also in Gr1+r2−m which ends the proof.
Hypothesis 5: If z ∈ Gn then we have zGr1
n z
−1 ∩Gr2
n = G
r(z)
n where
r(z) = |{z(1), z(2), . . . , z(kr1), 1, . . . , kr2}|
= kr1 + kr2 − |{z(1), z(2), . . . , z(kr1)} ∩ {1, . . . , kr2}|.
To prove this, let a = zbz−1 be an element of Gn which fixes the kr2 first
elements while b fixes the kr1 first elements. Then a also fixes the elements
z(1), . . . , z(kr1) which proves that zGr1
n z
−1∩Gr2
n ⊂ G
r(z)
n . In the opposite
direction, if p is a permutation of n which fixes the elements of the set
{z(1), z(2), . . . , z(kr1), 1, . . . , kr2} then p is in Gr2
n and in addition z−1pz
is in Gr1
n which implies that p = zz−1pzz−1 is in zGr1
n z
−1.
Now with all the necessary hypotheses verified we can apply the main
result in [10] to get the following theorem.
Theorem 1. Let x, y and h be three proper families of partitions. For
any n > |x|, |y|, |h|, the coefficients chxy(n) defined in Equation 4 are
polynomials in n with non-negative rational coefficients. In addition,
deg(chxy(n)) < |x|+ |y| − |h|.
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318 Center of wreath product
Proof. By [10, Corollary 6.3], if n > |x|, |y|, |h| then
chxy(n) =
|Cx(n)||Cy(n)||B
k
k(n−|x|)||B
k
k(n−|y|)|
|Bk
kn||Ch(n)|
∑
|h|6r6|x|+|y|
ahxy(r)
|Bk
k(n−r)|
where the ahxy(r) are positive, rational and independent numbers of n.
Since all the cardinals involved in this formula are known, we get after
simplification the following formula for chxy(n)
chxy(n) = (k!)l(h(1
k))−l(x(1k))−l(y(1k))
zh(1k)ch
zx(1k)zy(1k)cxcy
×
∑
|h|6r6|x|+|y|
(k!)r−|h|ahxy(r)(n− |h|)!
(n− r)!
,
where cx denotes
∏
λ⊢k,λ 6=(1k) zx(λ)z
l(x(λ))
λ . The result follows.
In [12, Theorem 2.13], Wang uses the Farahat-Higman approach to
prove that the structure coefficients of the center of the wreath product
G ≀ Sn are polynomials in n. His result recovers our Theorem 1 with the
special case when G = Sk. However, the way we prove Theorem 1 seems
to be easier and highlights the importance of the general framework given
in [10].
5.3. Special cases
In this section we revisit the two already known results of polynomiality
for the structure coefficients of the center of the symmetric group (k = 1)
algebra and the center of the hyperoctahedral (k = 2) group algebra. In
addition, as an application of our main theorem, we give a polynomiality
property in the case k = 3. In these three cases, we give explicit expressions
of products of conjugacy classes in the associated center algebra in order
to see our results.
k=1: The symmetric group. As seen in Section 2, the conjugacy
classes of the symmetric group Sn are indexed by partitions of n. If λ is a
partition of n the size of its associated conjugacy class is
|Cλ| =
n!
zλ
.
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O. Tout 319
If λ is a proper partition with |λ| < n, we define λn to be the partition
λ∪ (1n−|λ|). Now let λ and δ be two proper partitions with |λ|, |δ| < n. In
the center of the symmetric group algebra we have the following equation:
Cλn
Cδn
=
∑
γ
cγλδ(n)Cγ
n
(5)
where the sum runs through all proper partitions γ satisfying |γ| 6 |λ|+|δ|.
If we apply Theorem 1, we re-obtain the following result of Farahat and
Higman in [1, Theorem 2.2].
Theorem 2. Let λ, δ and γ be three proper partitions and let n >
|λ|, |δ|, |γ| be an integer. The structure coefficient cγλδ(n) of the center
of the symmetric group algebra defined by Equation (5) is a polynomial in
n with non-negative coefficients and
deg(cγλδ(n)) 6 |λ|+ |δ| − |γ|.
Example 6. The following two complete expressions in Z(C[Sn]) appear
in [8]. For any n > 4
C(1n−2,2)C(1n−2,2) =
n(n− 1)
2
C(1n) + 3C(1n−3,3) + 2C(1n−4,22)
and for any n > 5,
C(1n−2,2)C(1n−3,3) = 2(n− 2)C(1n−2,2) + 4C(1n−4,4) +C(1n−5,2,3).
k=2: The hyperoctahedral group. In Section 4.1, we showed that
the conjugacy classes of the hyperoctahedral group are indexed by pairs of
partitions (λ, δ) such that |λ|+ |δ| = n. The partition λ is that associated
to the partition (1, 1) of 2 while δ is associated to the partition (2). The
size of the class C(λ,δ) is given in Corollary 1
|C(λ,δ)| =
2nn!
2l(λ)+l(δ)zλzδ
.
By Definition 2, the pair (λ, δ) is proper if and only if the partition
λ is proper. For a proper pair (λ, δ) of partitions and for any integer
n > |λ|+ |δ|, we define the following pair of partitions:
(λ, δ)
n
:= (λ ∪ (1n−|λ|−|δ|), δ).
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320 Center of wreath product
Let (λ1, δ1) and (λ2, δ2) be two proper pairs of partitions. We have the
following equation in the center of the hyperoctahedral group algebra for
any integer n greater than |λ1|+ |δ1|, |λ2|+ |δ2|,
C(λ1,δ1)
n
C(λ2,δ2)
n
=
∑
(λ3,δ3)
c
(λ3,δ3)
(λ1,δ1)(λ2,δ2)
(n)C(λ3,δ3)
n
(6)
where the sum runs over all the proper pairs of partitions (λ3, δ3) satisfying
|λ3|+ |δ3| 6 |λ1|+ |δ1|+ λ2|+ |δ2|. As an application of Theorem 1, we
re-obtain the following result proved in [10, Corollary 6.11].
Corollary 3. Let (λ1, δ1), (λ2, δ2) and (λ3, δ3) be three proper pairs of
partitions, then for any n > |λ1|+ |δ1|, |λ2|+ |δ2|, |λ3|+ |δ3| the structure
coefficient c
(λ3,δ3)
(λ1,δ1)(λ2,δ2)
(n) of the center of the hyperoctahedral group algebra
defined in Equation (6) is a polynomial in n with non-negative coefficients
and we have
deg(c
(λ3,δ3)
(λ1,δ1)(λ2,δ2)
(n)) 6 |λ1|+ |δ1|+ |λ2|+ |δ2| − |λ3| − |δ3|.
Example 7. We give in this example the complete product of the class
C((1n−2),(2)) by itself whenever n > 4:
C((1n−2),(2))C((1n−2),(2)) = n(n− 1)C((1n),∅) + 2C((1n−4),(22))
+2C((1n−2),(12)) + 3C((1n−3,3),∅).
C((1n),∅) is the identity class and since any element in C((1n−2),(2))
has its inverse in C((1n−2),(2)), the coefficient of C((1n),∅) is the size of
the conjugacy class C((1n−2),(2)) which is n(n − 1). The coefficient of
C((1n−4),(22)) is 2 since if we fix a permutation of C((1n−4),(22)), say
(1, 3, 2, 4)(5, 7, 6, 8)(9)(10) . . . (2n),
then there exists only two pairs (α;β) ∈ C((1n−2),(2)) × C((1n−2),(2)) such
that
αβ = (1, 3, 2, 4)(5, 7, 6, 8)(9)(10) . . . (2n).
Mainly:
(α, β) = ((1, 3, 2, 4)(5) . . . (2n); (1)(2)(3)(4)(5, 7, 6, 8)(9) . . . (2n))
or
(α, β) = ((1)(2)(3)(4)(5, 7, 6, 8)(9) . . . (2n); (1, 3, 2, 4)(5) . . . (2n)).
There are only two permutations in B2
4, namely α = (1, 3, 2, 4) and
β = (1, 4, 2, 3), such that α, β ∈ C(∅,(2)) and α2 = β2 = (12)(34). Thus
the coefficient of C((1n−2),(12)) is 2. The last coefficient can be obtained
by identifying both sides.
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O. Tout 321
k=3: the group B
3
3k
Moving to the case k = 3, we showed in Section
4.2 that the conjugacy classes of the group B3
3n are indexed by triplet
of partitions (α, β, γ) such that |α| + |β| + |γ| = n. For a given triplet
(α, β, γ), the size of its associated conjugacy class is given in Corollary 2,
|C(α,β,γ)| = 2n−l(α)−l(β)3n−l(α)−l(γ).
n!
zαzβzγ
.
By Definition 2, the triplet (α, β, γ) is proper if and only if the partition
α is proper. Fix three proper triplet of partitions, (α1, β1, γ1), (α2, β2, γ2)
and (α3, β3, γ3), the structure coefficient associated to these triplets has
the following form according to the proof of Theorem 1,
c
(α3,β3,γ3)
(α1,β1,γ1)(α2,β2,γ2)
(n) =
(3!)l(α3)−l(α1)−l(α2)zα3
zβ3
2l(β3)zγ33
l(γ3)
zα1
zα2
zβ1
2l(β1)zγ13
l(γ1)zβ2
2l(β2)zγ23
l(γ2)
×
∑
r
(3!)r−|α3|−|β3|−|γ3|a
(α3,β3,γ3)
(α1,β1,γ1)(α2,β2,γ2)
(r)(n− |α3| − |β3| − |γ3|)!
(n− r)!
where the sum runs through all integers r with
|α3|+ |β3|+ |γ3| 6 r 6 |α1|+ |β1|+ |γ1|+ |α2|+ |β2|+ |γ2|.
Corollary 4. Let (α1, β1, γ1), (α2, β2, γ2) and (α3, β3, γ3) be three proper
triplets of partitions, then for any n > |α1| + |β1| + |γ1|, |α2| + |β2| +
|γ2|, |α3|+ |β3|+ |γ3| the coefficient c
(α3,β3,γ3)
(α1,β1,γ1)(α2,β2,γ2)
(n) is a polynomial
in n with non-negative coefficients and we have
deg(c
(α3,β3,γ3)
(α1,β1,γ1)(α2,β2,γ2)
(n)) 6 |α1|+ |β1|+ |γ1|+ |α2|+ |β2|+ |γ2|
−|α3| − |β3| − |γ3|.
Example 8. For n > 3, we leave it to the reader to verify the following
expressions in Z(C[B3
3n]) :
C((1n−2),(1),(1))C((1n−1),∅,(1)) = 2C((1n−3),(1),(12)) + 2(n− 1)C((1n−1),(1),∅)
+3C((1n−2),(1),(1))
and
C((1n−2),(1),(1))C((1n−1),(1),∅) = 2C((1n−3),(12),(1)) + 3(n− 1)C((1n−1),∅,(1))
+4C((1n−2),(12),∅) + 6C((1n−2),∅,(12)).
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322 Center of wreath product
Acknowledgements
The author is grateful to the Mathematical Institute of the Polish
Academy of Sciences branch in Toruń for their hospitality and financial
support during the time where this work was accomplished. Especially, he
would like to thank Prof. Piotr Śniady for many interesting discussions
about the topics presented in this paper.
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Contact information
Omar Tout Department of Mathematics, College of Science,
Sultan Qaboos University, P.O. Box 36, Al
Khod 123, Sultanate of Oman
E-Mail(s): o.tout@squ.edu.om
Received by the editors: 13.02.2019
and in final form 16.04.2019.
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| id | nasplib_isofts_kiev_ua-123456789-188713 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T15:15:00Z |
| publishDate | 2021 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Tout, O. 2023-03-11T16:20:08Z 2023-03-11T16:20:08Z 2021 The center of the wreath product of symmetric group algebras / O. Tout // Algebra and Discrete Mathematics. — 2021. — Vol. 31, № 2. — С. 302–322. — Бібліогр.: 12 назв. — англ. 1726-3255 DOI:10.12958/adm1338 2020 MSC: 05E10, 05E16, 20C30. https://nasplib.isofts.kiev.ua/handle/123456789/188713 We consider the wreath product of two symmetric groups as a group of blocks permutations and we study its conjugacy classes. We give a polynomiality property for the structure coefficients of the center of the wreath product of symmetric group algebras. This allows us to recover an old result of Farahat and Higman about the polynomiality of the structure coefficients of the center of the symmetric group algebra and to generalize our recent result about the polynomiality property of the structure coefficients of the center of the hyperoctahedral group algebra. This research is supported by Narodowe Centrum Nauki, grant number 2017/26/A/ST1/00189. The author is grateful to the Mathematical Institute of the Polish Academy of Sciences branch in Toruń for their hospitality and financial support during the time where this work was accomplished. Especially, he would like to thank Prof. Piotr Śniady for many interesting discussions about the topics presented in this paper. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics The center of the wreath product of symmetric group algebras Article published earlier |
| spellingShingle | The center of the wreath product of symmetric group algebras Tout, O. |
| title | The center of the wreath product of symmetric group algebras |
| title_full | The center of the wreath product of symmetric group algebras |
| title_fullStr | The center of the wreath product of symmetric group algebras |
| title_full_unstemmed | The center of the wreath product of symmetric group algebras |
| title_short | The center of the wreath product of symmetric group algebras |
| title_sort | center of the wreath product of symmetric group algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/188713 |
| work_keys_str_mv | AT touto thecenterofthewreathproductofsymmetricgroupalgebras AT touto centerofthewreathproductofsymmetricgroupalgebras |