CHAYAN KARMAKAR ABSTRACT . The aim of this paper is to give another proof of a theorem of D .Prasad

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arXiv:2210.03544v2 [math.RT] 13 Jun 2023

CHARACTER FACTORIZATIONS FOR REPRESENTATIONS OF GL(n, C)

CHAYAN KARMAKAR

ABSTRACT. The aim of this paper is to give another proof of a theorem of D.Prasad,

Theorem 2 of [DP], which calculates the character of an irreducible representation of

GL(mn, C)at the diagonal elements of the form t·cn, where t= (t1, t2,··· , tm)∈

(C∗)mand cn= (1, ωn, ω2

n,··· , ωn−1

n), where ωn=e

2πı

n, and expresses it as a product

of certain characters for GL(m, C)at (tn).

June 14, 2023

CONTENTS

1. Introduction 1

2. Calculation of the Weyl Denominator Aρ2

3. Calculation of the Weyl numerator 3

References 9

1. INTRODUCTION

In the work [DP] of Dipendra Prasad, he discovered a certain factorization theorem for

characters of GL(mn, C)at certain special elements of the diagonal torus, those of the

form

t·cn=



...

ωn−1

nt

,

where t= (t1, t2,··· , tm)and ωnis a primitive nth root of unity. D. Prasad proved that

the character of a ﬁnite dimensional highest weight representation πλof GL(mn, C)of

highest weight λat such elements t·cnis the product of characters of certain highest

weight representations of GL(m, C)at the element tn= (tn

1, tn

2,··· , tn

m). This work

of D.Prasad was recently generalized for all classical groups in [AK]. Both the works

[DP] and [AK] are achieved via direct manipulation with the Weyl character formula

expressed as a determinantal identity. The present work aims at giving another proof of

Prasad’s factorisation theorem in which we manipulate directly with the Weyl numerator

which is an alternating sum over the Weyl group. Assume that for each k,0≤k≤

n−1, there are exactly mintegers in λ+ρmn that are congruent to kmodulo n; this is

a necessary condition for the character of the associated representation of GL(mn, C)to

be nonzero at some element of the form t·cn(see Proposition 1below for a proof). If

2 CHAYAN KARMAKAR

this necessary condition on λ+ρmn is satisﬁed, we can replace λ+ρmn by its conjugate

w0(λ+ρmn) = µ,w0∈Smn, which affects the numerator in the Weyl character formula

only by a sign, such that the ﬁrst mentries in µare a set of (distinct) integers that are

congruent to 0modulo n, the next mentries are congruent to 1modulo nand so on. Then

we sum the Weyl numerator over various left cosets of a particularly chosen subgroup

Rwhich is isomorphic to Sn

m. We ﬁnd that there is a subgroup Csuch that those cosets

of Rwhich have a representative from Ccontribute a nonzero term whereas sum over

cosets not contained in CR contribute zero. Adding the summations obtained for each

of these distinct cosets, we get the Weyl numerator of a representation of GL(m, C)nat

the diagonal element tnin each of GL(m, C)in GL(m, C)n. This gives a proof of the

factorization theorem of Prasad, hopefully a more conceptual one; this is Theorem 4.1 of

this paper which we refer to for a more precise statement of the Theorem.

The paper is written in the hope that manipulations with the Weyl group may be available

in many more situations, yielding character identities of the kind obtained in [DP] as well

as [AK].

2. CALCULATION OF THE WEYL DENOMINATOR Aρ

Our work involves factorizing the Weyl numerator and the denominator. Since the Weyl

denominator is a special case of the Weyl numerator (for λ= 0), in particular if we

prove a factorization theorem for the numerator, it proves a factorization thorem for the

Weyl denominator. However, the Weyl denominator is much a simpler expression, and its

factorization is easy enough, so we begin with the factorization of the Weyl denominator

Aρmn (t·cn)in this section.

Aρmn (t·cn)(1)

{k<l,i≥j}

(ωk

nti−ωl

ntj)×Y

{i<j,k≤l}

(ωk

nti−ωl

ntj),

(2)

k<l,i=j

(ωk

n−ωl

n)ti×Y

{k<l,i>j}

(ωk

nti−ωl

ntj)×Y

{i<j,k≤l}

(ωk

nti−ωl

ntj),

=A×B×C.

We have

A=Y

i<j

(ωi

n−ωj

n)m·(

s=1

ts)n(n−1)

2,(1)

B×C=Y







k < l,

i > j







(ωk

nti−ωl

ntj)×Y







i < j,

k≤l







(ωk

nti−ωl

ntj),(2)

n−1

k=0

Pk,

where Pk=Qi<j Ql≥k(ωk

nti−ωl

ntj)×Qi>j Q{0≤s<k}(ωs

nti−ωk

ntj).

Let us evaluate Pk. Now

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摘要：

arXiv:2210.03544v2[math.RT]13Jun2023CHARACTERFACTORIZATIONSFORREPRESENTATIONSOFGL(n,C)CHAYANKARMAKARABSTRACT.TheaimofthispaperistogiveanotherproofofatheoremofD.Prasad,Theorem2of[DP],whichcalculatesthecharacterofanirreduciblerepresentationofGL(mn,C)atthediagonalelementsoftheformt·cn,wheret=(t1,t2,···...

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CHAYAN KARMAKAR ABSTRACT . The aim of this paper is to give another proof of a theorem of D .Prasad

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