Graded representations of Khovanov-Lauda-Rouquier algebras [PhD thesis]

265 Pages·2017·1.612 MB·English

by  Louise Sutton

Graded representations
of
Khovanov-Lauda-Rouquier algebras
Louise Sutton
Submitted in partial fulfilment of the requirements
of the degree of Doctor of Philosophy
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Abstract
The Khovanov–Lauda–Rouquier algebras R are a relatively new family of Z-graded
n
algebras. Their cyclotomic quotients RΛ are intimately connected to a smaller family
n
of algebras, the cyclotomic Hecke algebras H Λ of type A, via Brundan and Kleshchev’s
n
Graded Isomorphism Theorem. The study of representation theory of H Λ is well devel-
n
oped,partlyinspiredbytheremainingopenquestionsaboutthemodularrepresentations
of the symmetric group S .
n
Thereisa profound interplaybetween therepresentationsforS andcombinatorics,
n
wherebyeachirreduciblerepresentationincharacteristiczerocanberealisedasaSpecht
modulewhosebasisisconstructedfromcombinatorialobjects. ForRΛ, wecansimilarly
n
construct their representations as analogous Specht modules in a combinatorial fashion.
ManyresultscanbeliftedthroughtheGradedIsomorphismTheoremfromthesymmet-
ricgroupalgebras,andmoresofromH Λ,tothecyclotomicKhovanov–Lauda–Rouquier
n
algebras, providing a foundation for the representation theory of RΛ.
n
Following the introduction of RΛ, Brundan, Kleshchev and Wang discovered that
n
Specht modules over RΛ have Z-graded bases, giving rise to the study of graded Specht
n
modules. In this thesis we solely study graded Specht modules and their irreducible
quotients for RΛ. One of the main problems in graded representation theory of RΛ, the
n n
Graded Decomposition Number Problem, is to determine the graded multiplicities of
graded irreducible RΛ-modules arising as graded composition factors of graded Specht
n
modules.
WefirstconsiderRΛ inlevelone,whichisisomorphictotheIwahori–Heckealgebraof
n
type A, and research graded Specht modules labelled by hook partitions in this context.
In quantum characteristic two, we extend to RΛ a result of Murphy for the symmetric
n
groups, determining graded filtrations of Specht modules labelled by hook partitions,
whose factors appear as Specht modules labelled by two-part partitions. In quantum
characteristic at least three, we determine an analogous RΛ-version of Peel’s Theorem
n
for the symmetric groups, providing an alternative approach to Chuang, Miyachi and
Tan.
We then study graded Specht modules labelled by hook bipartitions for RΛ in level
n
two, which is isomorphic to the Iwahori–Hecke algebra of type B. In quantum character-
isitic at least three, we completely determine the composition factors of Specht modules
labelled by hook bipartitions for RΛ, together with their graded analogues.
n
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Declaration
I, Louise Sutton, confirm that the research included within this thesis is my own work
or that where it has been carried out in collaboration with, or supported by others, that
this is duly acknowledged below and my contribution indicated. Previously published
material is also acknowledged below.
I attest that I have exercised reasonable care to ensure that the work is original,
and does not to the best of my knowledge break any UK law, infringe any third partys
copyright or other Intellectual Property Right, or contain any confidential material.
I accept that the College has the right to use plagiarism detection software to check
the electronic version of the thesis.
Iconfirmthatthisthesishasnotbeenpreviouslysubmittedfortheawardofadegree
by this or any other university.
The copyright of this thesis rests with the author and no quotation from it or in-
formation derived from it may be published without the prior written consent of the
author.
Signature:
Date:
Details of collaboration and publications:
The contents in Part III Chapters 5 to 7 will form the paper Graded Specht modules
labelled by hook bipartitions I, arXiv:1707.01851, 2017.
The contents in Part III Chapters 8 to 13 will form the paper Graded Specht modules
labelled by hook bipartitions II, currently in preparation.
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Acknowledgements
Above all else, I am thankful to my supervisor, Matthew Fayers. This thesis would not
have been possible without his invaluable guidance and tireless patience.
ThisworkwassupportedbytheEngineeringandPhysicalSciencesResearchCouncil,
whichIammostthankfulfor,aswellasthesupportprovidedbyQueenMaryUniversity
of London.
On a personal note, I would like to especially thank my mum. She has always
encouraged me to achieve my goals with her limitless support and understanding; I
most certainly would not be where I am today without her love and many sacrifices.
I must also thank Andreas for his unfailing love and support throughout this time,
particularly, for humouring my erratic sleep schedule. Finally, a special thanks goes to
the following friends:
(cid:5) my oldest and dearest friends, Becca and Ruxandra, who have done their utmost
to keep me sane throughout my PhD studies with the help of guinea pigs;
(cid:5) my old university friends, Amy, Emma, Graeme and Sophie, my fellow foodies and
drinking companions;
(cid:5) my academic brother, Liron, for his helpful comments and late night discussions;
(cid:5) my new university friends, Andr´e, Ben, Chris, Diego, Emilio, Imen, Jack, Nils,
Paul, Sol, Sune, Tom, Trevor, and many others, for making my studies most
enjoyable;
(cid:5) my many flatmates that I have had the pleasure to meet and live with whilst in
London; Eion, Izzi, Jess, Katarina, Katie, Laura, Mahdi, Marc, Ollie, Rodrigo,
Sandra, Stewey, and in particular, Saffie cat, who has kept me company during
the early hours of the day.
Thank you all for your understanding and for being a great source of support.
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Contents
Abstract 3
Declaration 4
Acknowledgements 6
I Introductory materials 14
Introduction 15
1 Background 21
1.1 Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1.1 Graded algebras and graded modules . . . . . . . . . . . . . . . . 21
1.1.2 Permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.1.3 Lie-theoretic notation . . . . . . . . . . . . . . . . . . . . . . . . 23
1.1.4 Iwahori–Hecke alebras . . . . . . . . . . . . . . . . . . . . . . . . 24
1.1.5 Ariki–Koike algebras . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.1.6 Khovanov–Lauda–Rouquier algebras . . . . . . . . . . . . . . . . 26
1.1.7 Cyclotomic Khovanov–Lauda–Rouquier algebras . . . . . . . . . 27
1.1.8 Graded duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.1.9 The sign representation . . . . . . . . . . . . . . . . . . . . . . . 28
1.2 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.2.1 Young diagrams and partitions . . . . . . . . . . . . . . . . . . . 29
1.2.2 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.2.3 Ladders and e-regular partitions . . . . . . . . . . . . . . . . . . 31
1.2.4 Regular multipartitions . . . . . . . . . . . . . . . . . . . . . . . 32
1.2.5 Tableaux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3 Graded Specht modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Garnir tableaux and Garnir elements . . . . . . . . . . . . . . . . 37
1.3.2 Homogeneous presentation of Specht modules . . . . . . . . . . . 43
1.3.3 A standard homogeneous basis of Specht modules . . . . . . . . 43
1.3.4 Graded dimensions of Specht modules . . . . . . . . . . . . . . . 44
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CONTENTS CONTENTS
1.3.5 Weights of multipartitions . . . . . . . . . . . . . . . . . . . . . . 47
1.3.6 Graded duality of Specht modules . . . . . . . . . . . . . . . . . 48
1.4 Graded irreducible RΛ-modules . . . . . . . . . . . . . . . . . . . . . . . 49
n
1.5 Graded decomposition numbers for RΛ . . . . . . . . . . . . . . . . . . 50
n
1.6 Induction and restriction of RΛ-modules . . . . . . . . . . . . . . . . . . 51
n
1.7 Modular branching rules for RΛ-modules . . . . . . . . . . . . . . . . . 54
n
II Specht modules labelled by hook partitions 59
2 Gradings on S and S 60
(n−m,m) (n−m,1m)
2.1 The graded dimension of S when e = 2 . . . . . . . . . . . . . . 60
(n−m,m)
2.2 Degree of a standard (n−m,1m)-tableau . . . . . . . . . . . . . . . . . 64
3 Hook representations in quantum characteristic 2 67
3.1 Homogeneous basis elements of S . . . . . . . . . . . . . . . . . 68
(n−m,1m)
3.2 A Specht filtration of S . . . . . . . . . . . . . . . . . . . . . . 69
(n−2k,12k)
3.3 A Specht filtration of S . . . . . . . . . . . . . . . . . . . . 89
(n−2k−1,12k+1)
3.4 Graded decomposition numbers . . . . . . . . . . . . . . . . . . . . . . . 91
4 Hook representations in quantum characteristic at least 3 101
4.1 Ungraded decomposition numbers . . . . . . . . . . . . . . . . . . . . . 101
4.2 Graded Specht modules S with e (cid:45) n . . . . . . . . . . . . . . . 102
(n−m,1m)
4.3 Graded Specht modules S with e | n . . . . . . . . . . . . . . . 103
(n−m,1m)
III Specht modules labelled by hook bipartitions 112
5 Introducing Specht modules labelled by hook bipartitions 113
5.1 Homogeneous basis elements of S . . . . . . . . . . . . . . . . 114
((n−m),(1m))
5.2 Presentation of Specht modules labelled by hook bipartitions . . . . . . 114
5.3 The action of RΛ on S I . . . . . . . . . . . . . . . . . . . . . 115
n ((n−m),(1m))
6 Homomorphisms between Specht modules in level two of RΛ 128
n
6.1 Specht modules labelled by hook partitions in level two of RΛ . . . . . . 128
n
6.2 Specht module homomorphisms . . . . . . . . . . . . . . . . . . . . . . . 129
6.3 Exact sequences of Specht modules . . . . . . . . . . . . . . . . . . . . . 145
7 Composition series of S 150
((n−m),(1m))
7.1 The action of RΛ on S II . . . . . . . . . . . . . . . . . . . . 150
n ((n−m),(1m))
7.2 Linear combinations of basis vectors . . . . . . . . . . . . . . . . . . . . 164
7.3 Case I: κ (cid:54)≡ κ −1 (mod e) and n (cid:54)≡ κ −κ +1 (mod e) . . . . . . . . 167
2 1 2 1
7.4 Case II: κ (cid:54)≡ κ −1 (mod e) and n ≡ 0 (mod e) . . . . . . . . . . . . . 168
2 1
7.5 Case III: κ ≡ κ −1 (mod e) and n (cid:54)≡ κ −κ +1 (mod e) . . . . . . . 170
2 1 2 1
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