Modular Branching Rules for Projective Representations of Symmetric Groups and Lowering Operators for the Supergroup Q (n)

148 Pages·2012·1.345 MB·English

by  Alexander Kleshchev, Vladimir Shchigolev

MEMOIRS
of the
American Mathematical Society
Number 1034
Modular Branching Rules for
Projective Representations of
Symmetric Groups and Lowering
Operators for the Supergroup Q(n)
Alexander Kleshchev
Vladimir Shchigolev
November 2012 • Volume 220 • Number 1034 (second of 4 numbers) • ISSN 0065-9266
American Mathematical Society
Number 1034
Modular Branching Rules for
Projective Representations of
Symmetric Groups and Lowering
Operators for the Supergroup Q(n)
Alexander Kleshchev
Vladimir Shchigolev
November2012 • Volume220 • Number1034(secondof4numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Kleshchev,A.S.(AleksandrSergeevich)
Modular branching rules for projective representationsofsymmetricgroups andlowering op-
eratorsforthesupergroupQ(n)/AlexanderKleshchev,VladimirShchigolev.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 1034)
“November2012,volume220,number1034(secondof4numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-7431-8(alk. paper)
1.Symmetrygroups. 2.Modules(Algebra). 3.Operatortheory. I.Shchigolev,Vladimir.
II.Title
QA174.7.S96K54 2012
515(cid:2).724—dc23 2012025981
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Contents
Introduction vii
Set up vii
Projective representations and Sergeev algebra vii
Crystal graph approach ix
Schur functor approach xi
Modular branching rules xiii
Connecting the two approaches xiii
Some tensor products over Q(n) xiv
Strategy of the proof and organization of the paper xv
Chapter 1. Preliminaries 1
1.1. General Notation 1
1.2. The supergroup Q(n) and its hyperalgebra 2
1.3. Highest weight theory 5
Chapter 2. Lowering operators 11
2.1. Definitions 11
2.2. Properties of Sε ({j¯}) and Sε ({j}) 12
i,j i,j
2.3. Supercommutator [Hδ,Sε (M)] 15
k i,j
2.4. Supercommutator [Eδ,Sε (M)] 17
j i,j
2.5. More on EδSε (M) 18
l i,j
2.6. Some coefficients 26
Chapter 3. Some polynomials 29
3.1. Operators σk 29
i,j
3.2. Polynomials fD,l(S) 30
i,j
3.3. Polynomials g(1)(S) 33
i,j
3.4. Polynomials g(2) (S) 36
i,k,q,j
Chapter 4. Raising coefficients 51
4.1. Inductive formulas 51
4.2. The case of signed sets with only even elements 53
4.3. The case of signed sets with one odd element 55
Chapter 5. Combinatorics of signature sequences 61
5.1. Marked signature sequences 61
5.2. Normal and good indices 66
5.3. Tensor conormal and tensor cogood indices 68
5.4. Removable and addable nodes for dominant p-strict weights 70
iii
iv CONTENTS
Chapter 6. Constructing U(cid:2)((cid:3)n−1)-primitiv(cid:4)e vectors 75
6.1. Construction: case r (λ) =−m 75
(cid:2)(cid:3)i<k(cid:2)n β k(cid:4)
6.2. Construction: case r (λ) = −m and λ , λ are not both
i<k<n β k i n
divisible by p (cid:2)(cid:3) (cid:4) 78
6.3. Construction: case λ ≡1 (mod p) and r (λ) =+−m 81
i i<k(cid:2)n 0(cid:2)(cid:3) k (cid:4)
6.4. Extension: case λ ≡0 (mod p), λ ≡1 (mod p), r (λ) =
h i h<k(cid:2)i 0 k
−m (cid:2)(cid:3) (cid:4) 85
6.5. Extension: case λ (cid:3)≡0 (mod p), r (λ) =−m,
i h<k(cid:2)i β k
and λ ≡0 (mod p), λ ≡1 (mod p) do not both hold 86
h i
6.6.(cid:3)Extension: case λh ≡ 1 (mod p), λi ≡ 0 (mod p), and
[ r (λ) ]=+−m 89
h<k(cid:2)i 0 k
Chapter 7. Main results on U(n) 93
7.1. Normal indices and primitive vectors 93
7.2. Criterion for existence of nonzero U(n−1)-primitive vectors 95
7.3. The socle of the first level 98
7.4. Complement pairs 99
7.5. Primitive vectors in L(λ)⊗V∗ 106
7.6. Primitive vectors in L(λ)⊗V 110
Chapter 8. Main results on projective representations of symmetric groups 113
8.1. Representations of Sergeev superalgebras 113
8.2. Proof of Theorem A 115
8.3. Proof of Theorem B 116
8.4. Projective representations of symmetric groups 118
Bibliography 121
Abstract
Therearetwoapproachestoprojectiverepresentationtheoryofsymmetricand
alternatinggroups,whicharepowerfulenoughtoworkformodularrepresentations.
OneisbasedonSergeevduality,whichconnectsprojectiverepresentationtheoryof
the symmetric group and representation theory of the algebraic supergroup Q(n)
via appropriate Schur (super)algebras and Schur functors. The second approach
follows the work of Grojnowski for classical affine and cyclotomic Hecke algebras
andconnectsprojectiverepresentationtheoryofsymmetricgroupsincharacteristic
p tothe crystal graph of the basic module of the twisted affine Kac-Moody algebra
of type A(2) .
p−1
The goal of this work is to connect the two approaches mentioned above and
toobtainnewbranchingresultsforprojectiverepresentationsofsymmetricgroups.
This is achieved by developing the theory of lowering operators for the supergroup
Q(n) which is parallel to (although much more intricate than) the similar theory
for GL(n) developed by the first author. The theory of lowering operators for
GL(n) is a non-trivial generalization of Carter’s work in characteristic zero, and it
has received a lot of attention. So this part of our work might be of independent
interest.
Oneoftheapplicationsofloweringoperatorsistotensorproductsofirreducible
Q(n)-modules with natural and dual natural modules, which leads to important
special translation functors. We describe the socles and primitive vectors in such
tensor products.
ReceivedbytheeditorNovember8,2010,and,inrevisedform,August8,2011.
ArticleelectronicallypublishedonMarch7,2012;S0065-9266(2012)00657-5.
2010 MathematicsSubjectClassification. Primary20C30;Secondary20C25,20C20,17B10.
Key wordsand phrases. projectiverepresentations,symmetricgroups,Liesuperalgebras.
ThiscollaborationbeganwhenthesecondauthorvisitedUniversityofOregonin2007. The
secondauthorisgratefultotheDepartmentofMathematics,UniversityofOregon,forhospitality.
Thesecondauthorwouldlike toacknowledgethe financialsupport fromthe RussianFederation
President Grant MK-2304.2007.1 which made the visit possible. The first author supported in
part by the NSF grant no. DMS-0654147 and the Alexander von Humboldt Foundation. Both
authors supported by the Isaac Newton Institute in Cambridge, U.K. Both authors are grateful
toJonBrundanformanyusefulcommentsandsuggestions.
Alexander Kleshchev, Department of Mathematics, University of Oregon, Eugene, Oregon
97403;email: [email protected].
VladimirShchigolev,email: shchigolev [email protected].
(cid:2)c2012 American Mathematical Society
v

Introduction
Set up
Therearetwoapproachestoprojective representationtheoryofsymmetricand
alternatinggroups,whicharepowerfulenoughtoworkformodularrepresentations.
OneisbasedonSergeevduality,whichconnectsprojectiverepresentationtheoryof
the symmetric group and representation theory of the algebraic supergroup Q(n)
viaappropriate Schur (super)algebras and Schur functors. This approach has been
developed in [14].
The second approach follows the work of Grojnowski for the classical affine
and cyclotomic Hecke algebras and connects projective representation theory of
symmetricgroupsincharacteristicptothecrystalgraphofthebasicmoduleofthe
twistedaffineKac-MoodyalgebraoftypeA(2) . Thisapproachhasbeendeveloped
p−1
in [13] and [35].
The goal of this work is to connect the two approaches described above and to
obtain new branching results for projective representations of symmetric groups.
This is achieved by developing the theory of lowering operators for the supergroup
Q(n) which is parallel to (although much more intricate than) the similar theory
for GL(n) developed in [31].
The theory of lowering operators for GL(n) is a non-trivial generalization of
Carter’swork[20]incharacteristiczero, andithasreceivedquitealotofattention
recently, see for example [33, 34, 6, 7, 11, 38, 47, 48, 49, 50, 42]. So this
part of our work might be of independent interest, since it should be a useful tool
for studying representation theory of Q(n) and for obtaining further results on
projective representations of symmetric groups.
Oneoftheapplicationsofloweringoperatorsistotensorproductsofirreducible
Q(n)-modules with natural and dual natural modules, which leads to important
special translation functors. In this paper we describe the socles and primitive
vectors in such tensor products.
Projective representations and Sergeev algebra
We now describe the contents of this work more carefully. Let F be an alge-
braically closed field of characteristic p (cid:3)= 2. Let S be the symmetric group on n
n
letters, and A be the alternating group on n letters.
n
Studyingprojective representationsofS overFisequivalenttostudyinglinear
n
representations of the twisted group algebra T of S , see for example [35, Section
n n
13.1]. Explicitly, Tn is theF-algebra generatedby theelementst1,...,tn−1 subject
vii
viii INTRODUCTION
only to the relations
t2 =1 (1≤i<n),
i
t t t =t t t (1≤i≤n−2),
i i+1 i i+1 i i+1
t t =−t t (1≤i,j <n, |i−j|>1).
i j j i
Inside the algebra T we have the subalgebra
n
U :=span{t |g ∈A }.
n g n
This is a twisted group algebra of the alternating group A , and its representation
n
theory is equivalent to the projective representation theory of A over F.
n
We consider T as a superalgebra with respect to the following grading:
n
(T ) =U , (T ) =span{t |g ∈S \A }.
n 0 n n 1 g n n
(When Z/2Z is used for grading, its zero and identity elements are be denoted 0
and 1). To understand the usual irreducible modules over T and U , it suffices to
n n
understandtheirreduciblesupermodulesoverT . Thisisexplainedpreciselyin[35,
n
Proposition 12.2.11]. So from now on, we are mainly interested in supermodules
over the superalgebra T . Note that understanding irreducible supermodules over
n
T , among other things, now entails understanding their type, which can be M or Q,
n
see [35, Section 12.2].
LetC betheClifford(super)algebragivenbyoddgeneratorsc ,...,c subject
n 1 n
only to the relations
c2 =1 (1≤i≤n),
i
c c =−c c (1≤i(cid:3)=j ≤n).
i j j i
The superalgebra T is ‘Morita superequivalent’ to the Sergeev superalgebra
n
Y :=T ⊗C ,
n n n
where the tensor product is the tensor product of superalgebras. To be more
precise, recall that the Clifford superalgebra C is simple as a superalgebra, so
n
it has only one irreducible supermodule denoted by U , see [35, Example 12.2.14].
n
The supermodule U is of type M if n is even and of type Q if n is odd.
n
We have functors
?(cid:2)U =F :T −smod→Y −smod
n n n n
and
HomCn(Un,?)=Gn :Yn−smod→Tn−smod.
Here and throughout the paper, if A is a superalgebra, we denote by A−smod
the category of all finite dimensional A-supermodules, and by Hom (U,V) we un-
A
derstand all (not just even) A-homomorphisms from an A-supermodule U to an
A-supermodule V. Thus
Hom (U,V)=Hom (U,V) ⊕Hom (U,V) ,
A A 0 A 1
a direct sum of the spaces of even and odd homomorphisms.
Main properties of the functors F and G are described in [35, Proposition
n n
13.2.2]. In particular, F and G are exact, left and right adjoint to each other,
n n
and behave nicely with respect to restriction and induction.
CRYSTAL GRAPH APPROACH ix
If n is even, then F and G are quasi-inverse equivalences of categories which
n n
induce a type-preserving bijection between the set of isomorphism classes of T -
n
supermodules and the set of isomorphism classes of Y -supermodules.
n
If n is odd, F sends irreducible T -supermodules of type M to irreducible Y -
n n n
supermodules of type Q, while G sends irreducible Y -supermodules of type M
n n
to irreducible T -supermodules of type Q; in this way we get a type-reversing bi-
n
jection between the set of isomorphism classes of T -supermodules and the set of
n
isomorphism classes of Y -supermodules. This is all explained in detail in [35,
n
Section 13.2].
In particular, the information about irreducible supermodules, including their
type,iseasilytransferablebetweenT andY . ItturnsoutthatY isalittleeasier
n n n
to work with than T , so from now on, let us concentrate on Y (we return to T
n n n
in the final Section 8.4).
Crystal graph approach
This approach has been realized in [13], see also [35], following the work of
Grojnowski [22] for the usual symmetric groups (and cyclotomic Hecke algebras).
The original idea here is due to Leclerc and Thibon [39].
Set (cid:5)
∞ if p=0,
(cid:5):=
(p−1)/2 if p>0;
and (cid:5)
I := Z≥0 if p=0,
{0,1,...,(cid:5)} if p>0.
Y
The block components of the restriction res n are naturally labeled by the ele-
Y
ments of I. For any irreducible Y -supermodun−le1L, this gives us a natural decom-
n
position into block components [35, Section 19.1]:
(cid:6)
Y
res n L= res L.
Y i
n−1
i∈I
Ifres L(cid:3)=0,thenuptoa(notnecessarilyeven)isomorphism, thereisonlyone
i
irreducible Yn−1-supermodule, denoted e˜iL, such that
HomYn−1(e˜iL,resiL)(cid:3)=0.
If res L = 0 in the formula above, we set e˜L = 0. Now, let B be the set of
i i n
the isomorphism classes of irreducible Y -supermodules, and
n
(cid:7)
B := B .
n
n≥0
WemakeB intoanI-coloredgraphasfollows: [L ]→i [L ]ifandonlyifL ∼=e˜L .
1 2 1 i 2
One of the main result of [13] is that the colored graph B is the crystal graph
B(Λ ) of the basic representation of the twisted affine Kac-Moody Lie algebra of
0
type A(p2−)1 (interpreted as type B∞ if p=0).
Kang[24]hasgivenaconvenientcombinatorialdescriptionofthecrystalgraph
B(Λ )intermsofYoungdiagrams,whichwenowexplain. Thefollowingnotionsof
0
p-strict and p-restricted partitions were first suggested in [39]. These notions arise
naturally in [14] and [17] from completely different Lie theoretic considerations.