# Irreducible Modular Representations of the Reflection GroupG(m,1,n)

Authors: José O. Araujo; Tim Bratten; Cesar L. Maiarú
Journal: Journal of Mathematics
 (2015)

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## Abstract

In an article published in 1980, Farahat and Peel realized the irreducible modular representations of the symmetric group. One year later, Al-Aamily, Morris, and Peel constructed the irreducible modular representations for a Weyl group of typeBn. In both cases, combinatorial methods were used. Almost twenty years later, using a geometric construction based on the ideas of Macdonald, first Aguado and Araujo and then Araujo, Bigeón, and Gamondi also realized the irreducible modular representations for the Weyl groups of types An and Bn. In this paper, we extend the geometric construction based on the ideas of Macdonald to realize the irreducible modular representations of the complex reflection group of type G(m,1,n).


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## Body

## 1. Introduction

The irreducible modular representations for the symmetric group were first realized by Farahat and Peel in the manuscript [1] and one year later Al-Aamily et al. [2], using similar methods, constructed the irreducible modular representations for a Weyl group of type Bn.In an article published in 1972 [3], Macdonald introduced a geometric construction of some irreducible representations for the Weyl groups built from the action of a Weyl group on the associated root systems. Every irreducible representation of a Weyl group of type An or Bn can be realized as Macdonald’s representation (see Carter [4] or Lusztig [5]). Following Macdonald’s ideas, the irreducible modular representations for the symmetric group were constructed in [6] and in [7]; a similar circle of ideas produces the irreducible modular representations for a Weyl group of type Bn. Along these lines, a construction of a Gelfand model for the complex reflection group Gm,1,n was given in [8], also based on Macdonald’s ideas (see [9]).The main result of this study is an extension of the construction given in [6] to obtain the irreducible modular representations of the group Gm,1,n (see Theorem 25).


## 2. Notation and Preliminary Results

In this section, we introduce the main ingredients used in our construction and their respective notations. In particular,Snm will denote the group Gm,1,n, which is also known as the generalized symmetric group. K will denote a field of characteristic p≠2. A will denote the polynomial ring K[x1,…,xn], and, for every j≥0, Aj denotes the subspace of homogenous polynomials of degree j. We let In denote the set {1,2,…,n} and Sn indicates the permutation group of In.Initially we assume thatK contains all mth roots of the unity; in particular, we assume that m and p are coprime. In order to deal with the general case, this hypothesis will be reconsidered at the end of the paper.For convenience, the groupSnm will be presented as the semidirect product:(1)Snm=Cmn⋉Sn,where Cm⊂K is the group of the mth roots of the unity.Each elementσ∈Snm has a unique decomposition(2)σ=d,τ,where d=(d1,…,dn)∈Cmn and τ∈Sn.In order to simplify notation, in certain cases,Cmn will be identified with Cmn,1 and Sn with 1,Sn, that is, d∈Cmn with d,1 and τ∈Sn with 1,τ.


### 2.1. The Character

The following character will be utilized for defining projectors associated with subgroups ofSnm. Let χ:Snm→K be the linear character given by(3)χσ=∏i=1ndisgn⁡τ,where sgn is the sign map.Remark 1.
In the case whereK=C, the field of complex numbers, this character is, precisely, the determinant of the geometric representation of Smn, realized as a group generated by unitary reflections (see [10]). Thus, we have(4)χd=∏i=1ndiifd∈Cmn,χτ=sgn⁡τifτ∈Sn.


### 2.2. Subgroups

As in the case of characteristic zero, the irreducible representations are associated with subgroups ofSnm. We now introduce the type of subgroup that will be used for building irreducible modules.IfJ⊂In, let SnmJ denote the subgroup of Snm given by(5)SnmJ=d,τ∈Snm:dk=1,τk=k,∀k∈In-J.Note that(6)SnmJ=CmnJ×sSnJ,where SnJ is the group of all permutations which fix all elements in In-J and CmnJ is the subgroup of Cmn given by(7)CmnJ=d∈Cmn:dk=1,∀k∈In-J.


### 2.3. The Action of theSnm in A and the Bilinear Form

From now on,M={α:In→N0} will denote the set of multi-indexes and we use the notation(8)xα=x1α1⋯xnαnifα∈M.There is a natural action ofSnm in the polynomial ring A as well as in its dual space A∗, which gives them Snm-module structure. This action is described as follows: consider Sn acting on the set multi-indexes M by(9)τ·α=α∘τ-1ifτ∈Sn.This action, in turn, naturally induces an action of Snm on the polynomial ring A given by(10)σ·xα=∏i=1ndixiτ·αi,where α∈M. On the other hand, for φ∈A∗, we have(11)σ·φP=φσ-1P∀P∈A.IfSn is identified with the subgroup 1×Sn of Snm, it is clear that the defined action extends the natural action of the symmetric group on the polynomial ring A.We will use theK-bilinear form, -,-, defined on A determined by(12)xα,xβ=δα,β,where α,β∈M and δ is the Kronecker function.


### 2.4.  Multipartitions

The multipartitions are introduced as the natural parameters for enumerating the conjugacy classes ofSnm (see Proposition 4).Ifλ is a partition of n, it can be denoted as λ=λ1,…,λl with λ1≥⋯≥λl or as λ=(1m1,2m2,…,nmn), where(13)mi=j:λj=i.A partition λ=(1m1,2m2,…,nmn) of n is called p-regular if(14)mj<pforj=1,…,n.A conjugacy class of a finite group is calledp-regular if p does not divide the order of an element of the class.Remark 2.
For the symmetric groupSn, the number of nonequivalent irreducible modular representations coincides with the number of p-regular conjugacy classes of Sn [11, Theorem 11.5]. For a general finite group this is true if K is a decomposition field for G (see [12, Theorem 21.25 and p. 492]).Definition 3.
m-partition Λ of n is m-upleμ0,…,μm-1 where each μi is a partition and(15)∑i=0m-1μi=n.The m-partition is called p-regular if all partitions μi are p-regular.The following results will be used later on. They are well known and will be presented without proof.Proposition 4.
There is bijective correspondence among the conjugacy classes ofSnm and the m-partitions of n.Proof.
See [13].Lemma 5.
The number ofp-regular partitions of n coincides with the number of p-regular conjugacy classes of Sn.Proof.
See [11].Remark 6.
Each element inSnm can be uniquely factorized, except for reordering, as a product of disjoint cycles, where there are m distinct classes of these cycles. This is the reason for which the conjugacy classes of Snm are indexed by m-partitions of n and, consequently, the p-regular conjugacy classes are indexed by p-regular m-partitions; see [13].


## 2.1. The Character

The following character will be utilized for defining projectors associated with subgroups ofSnm. Let χ:Snm→K be the linear character given by(3)χσ=∏i=1ndisgn⁡τ,where sgn is the sign map.Remark 1.
In the case whereK=C, the field of complex numbers, this character is, precisely, the determinant of the geometric representation of Smn, realized as a group generated by unitary reflections (see [10]). Thus, we have(4)χd=∏i=1ndiifd∈Cmn,χτ=sgn⁡τifτ∈Sn.


## 2.2. Subgroups

As in the case of characteristic zero, the irreducible representations are associated with subgroups ofSnm. We now introduce the type of subgroup that will be used for building irreducible modules.IfJ⊂In, let SnmJ denote the subgroup of Snm given by(5)SnmJ=d,τ∈Snm:dk=1,τk=k,∀k∈In-J.Note that(6)SnmJ=CmnJ×sSnJ,where SnJ is the group of all permutations which fix all elements in In-J and CmnJ is the subgroup of Cmn given by(7)CmnJ=d∈Cmn:dk=1,∀k∈In-J.


## 2.3. The Action of theSnm in A and the Bilinear Form

From now on,M={α:In→N0} will denote the set of multi-indexes and we use the notation(8)xα=x1α1⋯xnαnifα∈M.There is a natural action ofSnm in the polynomial ring A as well as in its dual space A∗, which gives them Snm-module structure. This action is described as follows: consider Sn acting on the set multi-indexes M by(9)τ·α=α∘τ-1ifτ∈Sn.This action, in turn, naturally induces an action of Snm on the polynomial ring A given by(10)σ·xα=∏i=1ndixiτ·αi,where α∈M. On the other hand, for φ∈A∗, we have(11)σ·φP=φσ-1P∀P∈A.IfSn is identified with the subgroup 1×Sn of Snm, it is clear that the defined action extends the natural action of the symmetric group on the polynomial ring A.We will use theK-bilinear form, -,-, defined on A determined by(12)xα,xβ=δα,β,where α,β∈M and δ is the Kronecker function.


## 2.4.  Multipartitions

The multipartitions are introduced as the natural parameters for enumerating the conjugacy classes ofSnm (see Proposition 4).Ifλ is a partition of n, it can be denoted as λ=λ1,…,λl with λ1≥⋯≥λl or as λ=(1m1,2m2,…,nmn), where(13)mi=j:λj=i.A partition λ=(1m1,2m2,…,nmn) of n is called p-regular if(14)mj<pforj=1,…,n.A conjugacy class of a finite group is calledp-regular if p does not divide the order of an element of the class.Remark 2.
For the symmetric groupSn, the number of nonequivalent irreducible modular representations coincides with the number of p-regular conjugacy classes of Sn [11, Theorem 11.5]. For a general finite group this is true if K is a decomposition field for G (see [12, Theorem 21.25 and p. 492]).Definition 3.
m-partition Λ of n is m-upleμ0,…,μm-1 where each μi is a partition and(15)∑i=0m-1μi=n.The m-partition is called p-regular if all partitions μi are p-regular.The following results will be used later on. They are well known and will be presented without proof.Proposition 4.
There is bijective correspondence among the conjugacy classes ofSnm and the m-partitions of n.Proof.
See [13].Lemma 5.
The number ofp-regular partitions of n coincides with the number of p-regular conjugacy classes of Sn.Proof.
See [11].Remark 6.
Each element inSnm can be uniquely factorized, except for reordering, as a product of disjoint cycles, where there are m distinct classes of these cycles. This is the reason for which the conjugacy classes of Snm are indexed by m-partitions of n and, consequently, the p-regular conjugacy classes are indexed by p-regular m-partitions; see [13].


## 3. The ModulesNλ and NΛ

As has been indicated above, it will be assumed thatp does not divide m until the last section. After that, the extension to the general case does not present a major difficulty and will be treated at the end.In this section two modules will be introduced:Mλ, associated with the symmetric group Sn, and MΛ, associated with the group Snm. Then, we will define spaces Nλ and NΛ of linear functionals in Mλ and MΛ, respectively, on which we will realize the irreducible modular representations of Sn and Snm. It is convenient to have in account what happens in the case of the symmetric group, as it can be extended to the construction for the group Snm, from the key fact given in Theorem 12(i).WhenK is field of characteristic 0, it is not difficult to show that Mλ≃Nλ as Sn-modules and MΛ≃NΛ as Snm-modules.For each nonempty setC=c1,…,ck of In, where c1<⋯<ck, VC(x1,…,xn) will denote the usual Vandermonde determinant(16)VC=∏1≤i<j≤kxcj-xci.To each partition λ=λ1,…,λr of n, we associate the subset Oλ of M given by(17)α∈Oλ⟺α-10=λ1,α-11=λ2,…,α-1r-1=λr.It is clear that Oλ is Sn-orbit in M with the action given in (9).For eachl, 1≤l≤λ1=u, we consider the subset Cλ,l given by(18)Cλ,l=i∈In:∑j=1l-1λj′+1≤i≤∑j=1lλj′,where λ′=λ1′,…,λu′ is theconjugate partition of λ.Now, we defineeλ∈A and αλ∈M as follows:(19)eλ=∏l=1λ1VCλ,lx1,…,xn,(20)αλi=i-1-∑j=1l-1λj′ifi∈Cλ,l.Proposition 7.
LetSλ be the subgroup of Sn defined as(21)Sλ=Sλ1′Cλ1,1×⋯×Sλu′Cλu,u;then(22)∑τ∈Sλsgn⁡ττxαλ=eλ.Proof.
The result is implied by following fact: given a nonempty subsetC=c1,…,ck of In, where c1<⋯<ck, αC∈M is defined by(23)αCj=l-1ifj=cl0ifj≠cl;then(24)∑τ∈Sλsgn⁡ττxαC=VCx1,…,xn.To eachSn-orbit γ in M, we associate the subspace Sγ of A defined by(25)Sγ=∑α∈γaαxα,aα∈K.Definition 8.
Ifλ is partition of n, one defines Mλ to be the subspace of A generated by the Sn-orbit of eλ. This means(26)Mλ=τeλ:τ∈Sn.Remark 9.
IfK is a field of characteristic 0 and λ′=λ1′,…,λu′, then eλ coincides with the product of linear functionals associated with a set of positive roots of a subgroup of Gelfand of type(27)Aλ1′-1×⋯×Aλu′-1.In this case, according to [3], Mλ gives a Macdonald representation for Sn, considered as a Weyl group of type An-1, which is associated with a subgroup of the type given in (27) (see [4] or [5]). The Sn-module Mλ can also be realized in the following way. Let δ be the differential operator defined by(28)δ=∑i=1n∂∂xi;then(29)Mλ=P∈Sγ∣δP=0conforming with the statement of Theorem 4.2 of [14].LetΛ=μ0,…,μm-1 be m-partition of n and, for each i, 0≤i<m, qi=μi and suppose that(30)p0=0,pi=∑j<iqj. Define eΛ∈A as(31)eΛx1,…,xn=∏i=0m-1xpi+1⋯xpi+qiieμixpi+1m⋯xpi+qimand define αΛ∈M by(32)αΛj=mαμij-pi+iifpi+1≤j≤pi+qi,where αμi is as in (20).Now, ifΓ=OαΛ is the Sn-orbit of αΛ in M, it is clear that eΛ∈SΓ, where SΓ is as in (25).Remark 10.
Provided that the mappingλ→αλ which sends a partition to a multi-index is injective, the mapping Λ→αΛ which sends m-partition to a multi-index is also injective.Definition 11.
IfΛ is m-partition of n, one defines MΛ as the subspace of A generated by the Sn-orbit of eΛ. This means that(33)MΛ=σeΛ:σ∈Sn.LetΔ be the differential operator(34)Δ=∑i=1n∂m∂xim.As in the case of the symmetric group, by Theorem  2.5 in [8] it turns out that(35)MΛ=P∈SΓ∣ΔP=0.


### 3.1. The SubspacesNλ and NΛ

We identify the symmetric groupSn with G1,1,n=Sn1.For each partitionλ of n, consider fλ∈Mλ∗ (the dual subspace of Mλ) defined by(36)fλP=P,eλP∈Mλ.Using the canonical action of Sn on Mλ∗, we define Nλ as the subspace of Mλ∗ generated by the Sn-orbit of fλ. This means that(37)Nλ=τfλ:τ∈Sn.Analogously, for each m-partition Λ=μ0,…,μm-1 of n, we consider fΛ∈MΛ∗ given by(38)fΛP=P,eΛP∈MΛ.With respect to the canonical action of Snm on MΛ∗, we define NΛ as the subspace of MΛ∗ generated by the Sn-orbit of fΛ. This means that(39)NΛ=σfΛ:σ∈Sn.With a different notation, the following result can be found in [6] Lemma 2.2 and Theorem 2.3.Theorem 12.
Keeping the previous notation one has the following:(i)
Ifλ is p-regular, then Nλ≠0.(ii)
IfNλ≠0, Nλ is an irreducible Sn-module.(iii)
Ifλ and μ are p-regular partitions and λ≠μ, then Nλ≄Nμ.(iv)
Every simpleSn-module is isomorphic to Nλ for some p-regular partition λ of n.In the proof of Theorem12, an idempotent in KSn plays a central role. Analogously, in order to establish a similar result in the case of Snm, an idempotent in KSnm is defined as follows.GivenΛ=μ0,…,μm-1, m-partition of n, with(40)μi=μ1i,…,μri,μi=qi,p0=0,pi=∑j<iqjand as in (18), we define for each l, 1≤l≤μ1i=ri, the subset CΛ,i,l given by(41)CΛ,i,l=k∈pi+1,…,pi+qi:∑j=1l-1μij′+1≤k-pi≤∑j=1lμij′.Then, we have(42)eΛx1,…,xn=∏i=0m-1xpi+1⋯xpi+qii×∏l=1riVCΛ,i,lxpi+1m,…,xpi+qim.Let HΛ⊆Snm be the subgroup defined by(43)HΛ=×i=0m-1×l=1riSmnCΛ,i,l,where SnmCΛ,i,l is defined in (5). Since each element σ∈Snm has a unique decomposition as σ=d,τ with d∈Cmn and τ∈Sn, the subgroup HΛ also has a decomposition as(44)HΛ=CΛ×sSΛ,where(45)SΛ=∏i=0m-1∏l=1riSn1CΛ,i,l,CΛ=∏i=0m-1∏l=1riCmnCΛ,i,l.Each d∈CΛ can be factorized uniquely as(46)d=d0d1⋯dm-1,where di is in ∏l=1riCmnCΛ,i,l. We let χΛ denote the linear character of HΛ given by(47)χΛd,τ=∏i=0m-1χdiisgn⁡τ.Finally we define the operators ΩΛ and ΩΛ∗ in KSnm by (48)ΩΛ=1CΛ∑σ∈HΛχΛσ-1σ,ΩΛ∗=1CΛ∑σ∈HΛχΛσσ.Remark 13.
A reflections in a vector space is a diagonalizable endomorphism such that the space of fixed points of s is a hyperplane. Acting on A1, HΛ is realized as a group of reflections (this means it is generated by reflections). If K is the field of complex numbers, from (42) and (43), the polynomial eΛ can be factorized as a product of linear functionals associated with a system of roots of the subgroup HΛ of Snm where some of them, precisely the ones associated with the roots ei, are taken with certain multiplicities, in a way such that(49)eΛx1,…,xn=∏i=0m-1xpi+1⋯xpi+qii×∏l=1ri∏j,k∈CΛ,i,l,j<kxkm-xjm.Proposition 14.
With the previous notations, for eachm-partition Λ=μ0,…,μm-1 of n, we have(i)
(50)
τ
Ω
Λ
=
Ω
Λ
τ
=
χ
Λ
τ
Ω
Λ
∀
τ
∈
H
Λ
,
(ii)
(51)
Ω
Λ
=
1
C
Λ
∑
d
∈
C
Λ
χ
Λ
d
-
1
d
∑
τ
∈
S
Λ
sgn
⁡
τ
τ
,
Ω
Λ
∗
=
1
C
Λ
∑
d
∈
C
Λ
χ
Λ
d
d
∑
τ
∈
S
Λ
sgn
⁡
τ
τ
,
(iii)
(52)
Ω
Λ
x
α
Λ
=
e
Λ
.Proof.
(i) This identity is clear from the definition ofΩΛ.
(ii) It is a consequence of the decomposition (44).
(iii) By (22), we have(53)∑t∈SΛsgn⁡ττxαΛ=eΛ.Since each element d∈CΛ can be factorized as d=d0,…,dm-1, where(54)di∈∏l=1riCm,inCΛ,i,l,we have(55)deΛ=∏i=0m-1χdieΛ=χdeΛ.Hence,(56)∑d∈CΛχd-1deΛ=CΛeΛand (iii) is proved.Using the same notation as in (42) we have the following lemma.Lemma 15.
Letτ be a reflection of order k in A1 and r a root of τ. If P∈A is such that(57)τP=detτjPfor some j∈Z, then rk is a factor of P.Proof.
From the hypothesis, we haveτr=ζr with ζ a primitive kth root of unity. Fix φ1=r,φ2,…,φn, a base of A1, where φ2,…,φn is a base of the reflective hyperplane of τ. Thus, rφi=φi for i≥2. We can express P as(58)P=∑α∈Mλαφα,where φα=φ1α1⋯φnαn.
Asdetτ=ζ, in the identity(59)τP=detτjP, it can be assumed that j∈0,1,…,k-1. It follows that(60)∑α∈Mζα1λαφα=ζj∑α∈Mλαφα,where(61)α1≡jmodk,∀λα≠0.If α1 has the form j+hk, with h≥0, so that α1≥j whenever λα≠0, then rj=φ1j is a factor of P.Corollary 16.
IfP∈M, then eΛ is a factor of ΩΛP.Proof.
Because of the result established in Proposition14(i), it follows that(62)σΩΛP=χΛσΩΛP∀σ∈HΛ.Taking into account the reflections in HΛ and the expression of eΛ in (49), from Lemma 15 it follows that all distinct factors of eΛ in the set(63)xj-ζxkj,k∈CΛ,i,l,j<k∪xjipi+1≤j≤pi+qiare factors of P. But as they are irreducible factors not associated with A, the fact that eΛ is factor of P results.We letgrP denote the degree of the polynomial P.Proposition 17.
LetΛ=μ0,…,μm-1 be m-partition of n and P∈A. Considering the action of ΩΛ on A given by (10) and on NΛ given by the restriction of the usual action on MΛ∗, we have the following:(i)
ΩΛP=eΛQ, where Q is a polynomial HΛ-invariant.(ii)
Ifgr(P)=gr(eΛ), then(64)ΩΛP=P,eΛeΛ.(iii)
For eachφ∈NΛ(65)ΩΛ∗φ=φeΛfΛ.Proof.
(i) By Proposition14, for each τ∈HΛ we have(66)τΩΛ=χΛτΩΛ,ΩΛxαΛ=eΛ.Therefore,(67)τΩΛP=χΛτΩΛP.In particular, this identity is valid for a reflection in HΛ, so that by Lemma 15 it follows that(68)ΩΛP=eΛQ.On the other hand, if we apply τ∈HΛ to both sides of the former identity, we have (69)χΛτeΛQ=χΛτΩΛP=τΩΛP=τeΛQ=τeΛτQ=χΛτeΛτQso that Q is HΛ-invariant.
(ii) By the linearity ofΩΛ, it is sufficient to prove that for each α∈M such that α=greΛ we have(70)ΩΛxα=xα,eΛeΛ.By the identity given in Proposition 14,(71)ΩΛxαΛ=eΛ.Thus, if α does not belong to the Sn-orbit of αΛ, this means that(72)xα,eΛ=0.Moreover, given that grΩΛxα=gr(eΛ), by (i) there exists λ∈K such that(73)ΩΛxα=λeΛ.As xα and xαΛ are in different Sn-orbits, it must be the case that λ=0 and so (70) is proved when α does not belong to SΛ-orbit of αΛ.
Ifα is in Sn-orbit of αΛ, there is τ∈Sn such that(74)xα=τxαΛ.Then,(75)ΩΛxα=ΩΛτxαΛ=sgn⁡τeΛ.Since the coefficient of τxαΛ in the monomial decomposition of eΛ is precisely sgn⁡τ, this completes the proof of the identity (70).
(iii) Ifφ∈NΛ and P∈MΛ, we have(76)ΩΛ∗φP=1CΛ∑σ∈HΛχΛσσφP=1CΛ∑σ∈HΛχΛσφσ-1P=φ1CΛ∑σ∈HΛχΛσσ-1P=φΩΛP=φP,eΛeΛ=φeΛfΛP.


## 3.1. The SubspacesNλ and NΛ

We identify the symmetric groupSn with G1,1,n=Sn1.For each partitionλ of n, consider fλ∈Mλ∗ (the dual subspace of Mλ) defined by(36)fλP=P,eλP∈Mλ.Using the canonical action of Sn on Mλ∗, we define Nλ as the subspace of Mλ∗ generated by the Sn-orbit of fλ. This means that(37)Nλ=τfλ:τ∈Sn.Analogously, for each m-partition Λ=μ0,…,μm-1 of n, we consider fΛ∈MΛ∗ given by(38)fΛP=P,eΛP∈MΛ.With respect to the canonical action of Snm on MΛ∗, we define NΛ as the subspace of MΛ∗ generated by the Sn-orbit of fΛ. This means that(39)NΛ=σfΛ:σ∈Sn.With a different notation, the following result can be found in [6] Lemma 2.2 and Theorem 2.3.Theorem 12.
Keeping the previous notation one has the following:(i)
Ifλ is p-regular, then Nλ≠0.(ii)
IfNλ≠0, Nλ is an irreducible Sn-module.(iii)
Ifλ and μ are p-regular partitions and λ≠μ, then Nλ≄Nμ.(iv)
Every simpleSn-module is isomorphic to Nλ for some p-regular partition λ of n.In the proof of Theorem12, an idempotent in KSn plays a central role. Analogously, in order to establish a similar result in the case of Snm, an idempotent in KSnm is defined as follows.GivenΛ=μ0,…,μm-1, m-partition of n, with(40)μi=μ1i,…,μri,μi=qi,p0=0,pi=∑j<iqjand as in (18), we define for each l, 1≤l≤μ1i=ri, the subset CΛ,i,l given by(41)CΛ,i,l=k∈pi+1,…,pi+qi:∑j=1l-1μij′+1≤k-pi≤∑j=1lμij′.Then, we have(42)eΛx1,…,xn=∏i=0m-1xpi+1⋯xpi+qii×∏l=1riVCΛ,i,lxpi+1m,…,xpi+qim.Let HΛ⊆Snm be the subgroup defined by(43)HΛ=×i=0m-1×l=1riSmnCΛ,i,l,where SnmCΛ,i,l is defined in (5). Since each element σ∈Snm has a unique decomposition as σ=d,τ with d∈Cmn and τ∈Sn, the subgroup HΛ also has a decomposition as(44)HΛ=CΛ×sSΛ,where(45)SΛ=∏i=0m-1∏l=1riSn1CΛ,i,l,CΛ=∏i=0m-1∏l=1riCmnCΛ,i,l.Each d∈CΛ can be factorized uniquely as(46)d=d0d1⋯dm-1,where di is in ∏l=1riCmnCΛ,i,l. We let χΛ denote the linear character of HΛ given by(47)χΛd,τ=∏i=0m-1χdiisgn⁡τ.Finally we define the operators ΩΛ and ΩΛ∗ in KSnm by (48)ΩΛ=1CΛ∑σ∈HΛχΛσ-1σ,ΩΛ∗=1CΛ∑σ∈HΛχΛσσ.Remark 13.
A reflections in a vector space is a diagonalizable endomorphism such that the space of fixed points of s is a hyperplane. Acting on A1, HΛ is realized as a group of reflections (this means it is generated by reflections). If K is the field of complex numbers, from (42) and (43), the polynomial eΛ can be factorized as a product of linear functionals associated with a system of roots of the subgroup HΛ of Snm where some of them, precisely the ones associated with the roots ei, are taken with certain multiplicities, in a way such that(49)eΛx1,…,xn=∏i=0m-1xpi+1⋯xpi+qii×∏l=1ri∏j,k∈CΛ,i,l,j<kxkm-xjm.Proposition 14.
With the previous notations, for eachm-partition Λ=μ0,…,μm-1 of n, we have(i)
(50)
τ
Ω
Λ
=
Ω
Λ
τ
=
χ
Λ
τ
Ω
Λ
∀
τ
∈
H
Λ
,
(ii)
(51)
Ω
Λ
=
1
C
Λ
∑
d
∈
C
Λ
χ
Λ
d
-
1
d
∑
τ
∈
S
Λ
sgn
⁡
τ
τ
,
Ω
Λ
∗
=
1
C
Λ
∑
d
∈
C
Λ
χ
Λ
d
d
∑
τ
∈
S
Λ
sgn
⁡
τ
τ
,
(iii)
(52)
Ω
Λ
x
α
Λ
=
e
Λ
.Proof.
(i) This identity is clear from the definition ofΩΛ.
(ii) It is a consequence of the decomposition (44).
(iii) By (22), we have(53)∑t∈SΛsgn⁡ττxαΛ=eΛ.Since each element d∈CΛ can be factorized as d=d0,…,dm-1, where(54)di∈∏l=1riCm,inCΛ,i,l,we have(55)deΛ=∏i=0m-1χdieΛ=χdeΛ.Hence,(56)∑d∈CΛχd-1deΛ=CΛeΛand (iii) is proved.Using the same notation as in (42) we have the following lemma.Lemma 15.
Letτ be a reflection of order k in A1 and r a root of τ. If P∈A is such that(57)τP=detτjPfor some j∈Z, then rk is a factor of P.Proof.
From the hypothesis, we haveτr=ζr with ζ a primitive kth root of unity. Fix φ1=r,φ2,…,φn, a base of A1, where φ2,…,φn is a base of the reflective hyperplane of τ. Thus, rφi=φi for i≥2. We can express P as(58)P=∑α∈Mλαφα,where φα=φ1α1⋯φnαn.
Asdetτ=ζ, in the identity(59)τP=detτjP, it can be assumed that j∈0,1,…,k-1. It follows that(60)∑α∈Mζα1λαφα=ζj∑α∈Mλαφα,where(61)α1≡jmodk,∀λα≠0.If α1 has the form j+hk, with h≥0, so that α1≥j whenever λα≠0, then rj=φ1j is a factor of P.Corollary 16.
IfP∈M, then eΛ is a factor of ΩΛP.Proof.
Because of the result established in Proposition14(i), it follows that(62)σΩΛP=χΛσΩΛP∀σ∈HΛ.Taking into account the reflections in HΛ and the expression of eΛ in (49), from Lemma 15 it follows that all distinct factors of eΛ in the set(63)xj-ζxkj,k∈CΛ,i,l,j<k∪xjipi+1≤j≤pi+qiare factors of P. But as they are irreducible factors not associated with A, the fact that eΛ is factor of P results.We letgrP denote the degree of the polynomial P.Proposition 17.
LetΛ=μ0,…,μm-1 be m-partition of n and P∈A. Considering the action of ΩΛ on A given by (10) and on NΛ given by the restriction of the usual action on MΛ∗, we have the following:(i)
ΩΛP=eΛQ, where Q is a polynomial HΛ-invariant.(ii)
Ifgr(P)=gr(eΛ), then(64)ΩΛP=P,eΛeΛ.(iii)
For eachφ∈NΛ(65)ΩΛ∗φ=φeΛfΛ.Proof.
(i) By Proposition14, for each τ∈HΛ we have(66)τΩΛ=χΛτΩΛ,ΩΛxαΛ=eΛ.Therefore,(67)τΩΛP=χΛτΩΛP.In particular, this identity is valid for a reflection in HΛ, so that by Lemma 15 it follows that(68)ΩΛP=eΛQ.On the other hand, if we apply τ∈HΛ to both sides of the former identity, we have (69)χΛτeΛQ=χΛτΩΛP=τΩΛP=τeΛQ=τeΛτQ=χΛτeΛτQso that Q is HΛ-invariant.
(ii) By the linearity ofΩΛ, it is sufficient to prove that for each α∈M such that α=greΛ we have(70)ΩΛxα=xα,eΛeΛ.By the identity given in Proposition 14,(71)ΩΛxαΛ=eΛ.Thus, if α does not belong to the Sn-orbit of αΛ, this means that(72)xα,eΛ=0.Moreover, given that grΩΛxα=gr(eΛ), by (i) there exists λ∈K such that(73)ΩΛxα=λeΛ.As xα and xαΛ are in different Sn-orbits, it must be the case that λ=0 and so (70) is proved when α does not belong to SΛ-orbit of αΛ.
Ifα is in Sn-orbit of αΛ, there is τ∈Sn such that(74)xα=τxαΛ.Then,(75)ΩΛxα=ΩΛτxαΛ=sgn⁡τeΛ.Since the coefficient of τxαΛ in the monomial decomposition of eΛ is precisely sgn⁡τ, this completes the proof of the identity (70).
(iii) Ifφ∈NΛ and P∈MΛ, we have(76)ΩΛ∗φP=1CΛ∑σ∈HΛχΛσσφP=1CΛ∑σ∈HΛχΛσφσ-1P=φ1CΛ∑σ∈HΛχΛσσ-1P=φΩΛP=φP,eΛeΛ=φeΛfΛP.


## 4. The Bilinear Form

In this section, some properties of the linear form introduced inA will be proven, in order to extend the results in [6] to the group Snm.Definition 18.
IfJ is a subset of In, and α∈M one says that α is supported in J if αi=0 for each i∈In-J. One also says that the monomial xα is supported in J if α is supported in J. If P∈A, one says that P is supported in J if each monomial in P is.Ifα∈M we associate αJ∈M supported in J given by(77)αiJ=αi,ifi∈J,0,ifi∉J.It is clear that α is supported in J if, and only if, αJ=α.IfIn=⋃l=0hJl is a partition of In and α∈M, α can be decomposed in unique way as (78)α=∑l=0hαJlsuch that if α,β∈M, then(79)xα,xβ=∏l=0hxαJl,xβJl.Proposition 19.
LetP,Q∈A and In=⋃l=0hJl is a partition of In. If P and Q are factorized as(80)P=∏l=0hPJl,Q=∏l=0hQJl,where PJl and QJl are both supported in Jl for each l, then(81)P,Q=∏l=0hPJl,QJl.Proof.
The proof can be obtained using induction inh, so it is sufficient to treat the case of a partition of two terms. For simplicity, we express In=I∪J. If we write(82)P=∑aαxα,Q=∑bαxα,then(83)P,Q=∑α,βδα,βaαbβ,where δ is Kronecker’s function. On the other hand, it follows from the factorization that(84)P=PIPJ,Q=QIQJand this decomposition is unique as was observed in (78); therefore,(85)aα=aαIaαJ,bβ=bβIbβJ,δα,β=δαI,βIδαJ,βJ,where aαIaαJ are the coefficients of xαI and xαJ in PI and PJ, respectively, and, analogously, bβIbβJ are the coefficient of xβI and xβJ in QI and QJ. Then (86)∑α,βδα,βaαbβ=∑αI,βIδαI,βIaαIbβI∑δαJ,βJδαJ,βJaαJbβJ=PI,QIPJ,QJfrom which the following proposition holds.Proposition 20.
IfP,Q∈A and k is a natural number, then(87)Px1k,…,xnk,Qx1k,…,xnk=Px1,…,xn,Qx1,…,xn.Proof.
This is a consequence of the identity(88)xkα,xkβ=xα,xβfor each pair α,β∈M.Given a decompositionP=(P0,…,Pm-1) of In, we have(89)In=⋃l=0m-1Pl,where the subsets Pl are pairwise disjoint. We indicate by NP the set of all α∈M such that, for each l, 0≤l<m, αPl is congruent to l module m.Definition 21.
IfP∈A, one will say that P has m-type P if(90)P=∑α∈NPλαxα.Proposition 22.
LetP and Q∈A be P with m-type P and Q with m-type Q. If P≠Q, then P,Q=0.Proof.
IfP and Q are different decompositions of In, it is clear that NP∩NQ=∅ and consequently P,Q=0.Lemma 23.
IfΛ is a p-regular m-partition, the bilinear form , when restricted to MΛ×MΛ is nonzero.Proof.
LetΛ=μ0,…,μm-1 be the m-partition of n. For each i, 0≤i≤m-1, let qi=μi and suppose that(91)p0=0,pi=∑j<iqj.Since Λ is p-regular, each μi is a p-regular partition. By Lemma 2.2 (iv) in [6], there are permutations σi in Sqipi+1,…,pi+qi such that for each i(92)eαμi,σieαμi≠0.If we put τ=σ0⋯σm-1∈Sn from (31) and Propositions 19 and 20 we have(93)eΛ,τeΛ=∏i=0m-1eαμi,σeαμi≠0.


## 5. The Modular Representations

Theorem 24.
Using the previous notation, we have the following:(i)
IfΛ is a p-regular m-partition of n, then NΛ≠0.(ii)
IfNΛ≠0, NΛ is an irreducible Snm-module.(iii)
IfΛ and Υ are p-regular m-partitions of n and Λ≠Υ, then NΛ≄NΩ.(iv)
Every simpleSnm-module is isomorphic to NΛ for some p-regular m-partition Λ of n.Proof.
(i) This is a consequence of Lemma23.
(ii) LetΛ=μ0,…,μm-1 be an m-partition of n, L a Snm-submodule of NΛ, and φ∈L, φ nonzero. Given that φ∈MΛ∗, there exists P∈MΛ such that(94)φQ=P,Q∀Q∈MΛ.If φσeΛ=0 for every σ∈Sn, the linear functional φ should be zero, but we have assumed that φ≠0. Hence, there exists π∈Sn such that gπeΛ≠0. We can suppose, without lack of generality, that φeΛ≠0, changing φ by π-1φ if that was the case. Now, if Q∈MΛ, gr(Q)=gr(eΛ) and by Proposition 17(95)φeΛfΛ=ΩΛ∗φ∈L.Since φeΛ≠0, it follows that fΛ∈L, which means that L=NΛ.
(iii) LetΛ and Υ be p-regular m-partitions of n such that NΛ≃NΥ. We prove in Lemma 23 that there exists τ∈Sn such that(96)eΛ,τeΛ≠0.If in Proposition 17(iii) we change φ by τ-1fΛ, we have(97)ΩΛ∗τ-1fΛ=τ-1fΛeΛfΛ=eΛ,τeΛfΛ≠0.Since the operator ΩΛ∗ is nonzero in NΛ and it is a linear combination of elements in Snm, ΩΛ∗, it must be nonzero in NΥ. Hence, there exists π∈Sn such that(98)ΩΛ∗π-1fΥ≠0.Thus,(99)0≠ΩΛ∗π-1fΥ=eΥ,πeΛfΥ.Hence,(100)eΥ,πeΛ≠0but this only occurs if αΥ and αΛ belong to the same Sn-orbit, and, from Remark 10, it must be Υ=Λ.
(iv) It is sufficient to prove that the number ofp-regularm-partitions agrees with the number of p-regular conjugacy classes in Snm. But this is a consequence of the fact that there is a bijection between the m-partitions and the conjugacy classes of Snm which put in correspondence p-regular m-partitions with the p-regular conjugacy classes of Snm; see [13].Finally, considering the generic case wherep and m may not be coprime, write m=pαk, where α∈N0 and p does not divide k. Suppose that K contains the kth roots of the unity. Let us consider ψ~:Cmn→Ckn, the canonical projection, given by the decomposition(101)Cmn=Cpαn⊕Ckn.This morphism induces the projection ψ:Snm→Snk given by(102)πd,τ=πd,τd,τ∈Snm.By the previous theorem, for each p-regulark-partition Λ of n we have an irreducible representation of Snk(103)ρΛ:Snk⟶AutKNΛand thus ρΛ∘ψ is an irreducible representation of Snm.Keeping the previous notation, we have following theorem which establishes the irreducible modular representations ofSnm in the general case.Theorem 25.
IfK contains the kth roots of the unity, the following statements hold:(i)
IfΛ is a p-regular k-partition of n, then ρΛ∘π is an irreducible representation of Snm.(ii)
IfΛ and Υ are p-regular k-partitions of n and Λ≠Υ, then ρΛ∘π and ρΥ∘π are nonequivalent.(iii)
Every irreducible representation ofSnm is equivalent to ρΛ∘π for some p-regular k-partition Λ of n.Proof.
(i) and (ii) follow from Theorem24.
(iii) By (i) and (ii) we have as many nonequivalent irreducible representations ofSnm as p-regular k-partitions of n. On the other hand, it follows from [13] that every p-regular conjugacy class of Snm has a representative Snk; in consequence Snm and Snk have the same number of p-regular classes. Since the number of p-regular classes of Snm agrees with the number of p-regular k-partitions of n it follows that the representations ρΛ∘π are all irreducible presentations of Snm.



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