Selected Title s i n Thi s Serie s

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Probeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 S tephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 D u s a McDuf F an d Dietma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/015

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Iwahori-Hecke Algebra s and Schu r Algebra s

of th e Symmetri c Grou p

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University

LECTURE Series

Volume 1 5

Iwahori-Hecke Algebra s and Schu r Algebra s

of th e Symmetri c Grou p Andrew Matha s

American Mathematica l Societ y Providence, Rhod e Islan d

Editorial Boar d

Jerry L . Bon a (Chair ) Nicola i Reshetikhi n Jean-Luc Brylinsk i Leonar d L . Scot t

T h e au tho r gratefull y acknowledge s th e suppor t o f th e Sonderforschungsbereich 34 3 a t t h e Universi ta t Bielefel d and o f a U200 0 Fellowshi p a t t h e Universit y o f Sydney .

1991 Mathematics Subject Classification. P r imar y 20C30 , 16G99 ; Secondary 05E10 , 20G05 , 20C20 .

ABSTRACT. Thi s boo k give s a full y self-containe d introductio n t o th e modula r representatio n the -ory o f the Iwahori-Heck e algebra s o f the symmetri c group s an d o f the associate d g-Schu r algebras . The mai n landmark s tha t w e reac h ar e th e classificatio n o f th e simpl e module s an d th e block s of thes e algebras . Alon g th e wa y th e theor y o f cellula r algebra s i s develope d an d a n analogu e of Jantzen' s su m formul a i s proved . Combinatoria l motif s pervad e th e text , wit h standar d an d semistandard tableau x bein g use d t o inde x explici t (cellular ) bases ; thes e base s ar e particularl y well adapte d t o th e representatio n theory . Thi s result s i n clea n an d elegan t proof s o f mos t o f th e basic result s abou t thes e algebras . Th e final chapte r give s a surve y o f som e recen t an d excitin g developments i n th e field an d discusse s ope n problems .

The boo k shoul d b e accessibl e t o advance d graduat e student s an d als o usefu l t o researcher s i n the field.

Library o f Congres s Cataloging-in-Publicatio n Dat a

Mat has, Andrew , 1966 -Iwahori-Hecke algebra s an d Schu r algebra s o f th e symmetri c grou p / Andre w Mathas .

p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 15 ) Includes bibliographica l reference s an d indexes . ISBN 0-8218-1926- 7 (alk . paper ) 1. Symmetr y groups . 2 . Representation s o f algebras . I . Title . II . Series : Universit y lectur e

series (Providence , R.I. ) ; 15. QA 174.2.M3 8 199 9 512'.2-dc21 99-2931 0

CIP

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

Republication, systemati c copying , o r multipl e reproductio n o f any materia l i n thi s publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Assistan t t o the Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permission@ams. org .

© 199 9 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

@ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability .

Visit th e AM S hom e pag e a t URL : http: / /www.ams.org /

10 9 8 7 6 5 4 3 2 1 0 4 0 3 0 2 0 1 0 0 9 9

Contents

Introduction i x

Chapter 1 . Th e Iwahori-Hecke algebr a o f the symmetri c group 1 1. Th e Symmetric group 1 2. Th e Iwahori-Hecke algebr a 5 3. Th e O-Hecke algebra 1 0

Chapter 2 . Cellula r algebra s 1 5 1. Cellula r base s 1 5 2. Simpl e modules in a cellular algebr a 1 9

Chapter 3 . Th e modular representatio n theor y o f 3riP 2 7 1. Th e combinatorics o f tableaux 2 7 2. Th e Murphy basi s 3 2 3. Spech t module s and Jucys-Murph y element s 3 9 4. Irreducibl e ^-modules 4 5

Chapter 4 . Th e g-Schur algebr a 5 5 1. Semistandar d tableau x 5 5 2. A Specht filtratio n o f M» 5 9 3. Th e semistandard basi s theorem 6 1

Chapter 5 . Th e Jantzen su m formula an d the blocks of Jf? 6 9 1. Gra m determinant s o f Weyl modules 6 9 2. Th e Jantzen su m formula 8 1 3. Th e blocks of y(n) an d 3ff 8 4 4. Irreducibl e Weyl modules and Spech t module s 8 7

Chapter 6 . Branchin g rules , canonical bases and decompositio n matrice s 9 5 1. Th e LLT algorithm 9 5 2. Decompositio n map s an d adjustmen t matrice s 11 5 3. Th e Kleshchev-Brundan modula r branchin g rule s 11 8 4. Rule s for computin g decomposition matrice s 12 2 5. Th e g-Schur algebra s and GL n(q) 12 9 6. Th e Ariki-Koike algebra s and cyclotomi c g-Schur algebra s 13 1

Appendix A . Finit e dimensiona l algebra s over a field 13 7 1. Filtration s an d composition serie s 13 7 2. Idempotent s an d indecomposabl e module s 13 8 3. Th e blocks of A 14 4 4. Semisimpl e symmetric algebra s 14 6

viii C O N T E N T S

Appendix B . Decompositio n matrice s 14 9 1. Decompositio n matrice s whe n e = 2 14 9 2. Decompositio n matrice s whe n e = 3 15 6 3. Decompositio n matrice s whe n e — 4 16 1

Appendix C . Elementar y divisor s o f integra l Spech t module s 16 5

Index o f notatio n 17 7

References 18 1

Index 187

What is now proved was once only imagined. William Blak e

Introduction

The symmetri c grou p 6 n i s th e grou p o f permutation s o n 1,2 , . . . , n . Th e ordinary irreducible representations of &n ar e very well understood, wit h everythin g from thei r degree s an d characte r formulae , t o explici t matri x representation s bein g known fo r man y years . I n contrast , ver y littl e detaile d informatio n i s known abou t the modula r representation s o f th e symmetri c groups ; i n fact , i n spit e o f a grea t deal o f effort , no t tha t muc h progres s ha s bee n mad e sinc e Jame s [84 ] classifie d and constructe d th e irreducibl e modula r representation s o f th e symmetri c group s in 1976 .

The ai m o f thi s boo k i s t o giv e a self-containe d introductio n t o th e modula r representation theor y o f the Iwahori-Heck e algebra s o f the symmetri c groups ; thi s includes the modula r representatio n theor y o f 6 n a s a special case . I n studying th e Iwahori-Hecke algebra s i t i s profitabl e t o wide n th e scop e o f ou r investigation s t o include the g-Schu r algebras . Th e study o f these algebra s wa s pioneered b y Dippe r and Jame s i n a serie s o f landmar k paper s [37,39-41] . Her e w e recas t thi s theory , taking accoun t o f recen t advances , wit h a primar y goa l o f classifyin g th e block s and th e simpl e module s o f both algebras . W e have written thes e note s s o a s t o b e accessible t o th e advance d graduat e studen t an d als o to b e usefu l t o researcher s i n the field .

Apart fro m bein g interestin g i n an d o f themselves , th e mai n motivatio n fo r studying the Iwahori-Heck e algebra s i s that the y provide a bridge between th e rep -resentation theor y o f th e symmetri c an d genera l linea r groups ; thes e connection s are eve n mor e transparen t wit h th e g-Schu r algebras . I n th e classica l cas e (tha t is, q = 1) , th e Schu r algebra s wer e introduce d b y Schu r [158 ] wh o use d them , together wit h the representation theor y o f the symmetri c groups , t o classify th e or -dinary irreducibl e polynomia l representation s o f GLn(C); se e also [74] . Dippe r an d James' motivatio n fo r introducin g th e g-Schu r algebr a wa s t o stud y th e modula r representation theor y o f GL n(q) ove r fields o f characteristic no t dividin g q (that is , in non-definin g characteristic) ; the y showe d tha t th e g-Schu r algebra s completel y determine th e decompositio n matri x o f GL n{q) i n thi s cas e [40] .

Our motivatio n fo r studyin g th e g-Schu r algebr a i s mor e modes t i n tha t w e view th e g-Schu r algebr a a s a too l fo r studyin g th e Iwahori-Heck e algebra . A s we will see, the g-Schur algebra s have a rich and beautiful combinatoria l representatio n theory whic h i s closely allie d wit h tha t o f the Iwahori-Heck e algebras . Indeed , ful l knowledge o f th e representatio n theor y o f on e clas s o f thes e algebra s i s equivalen t to ful l knowledg e o f the other . Further , i t i s often th e cas e tha t th e easies t wa y t o prove a resul t abou t on e o f thes e algebra s i s to firs t prov e a n analogou s resul t fo r the other . Throughou t th e 'classical ' theor y fo r th e symmetri c an d genera l linea r groups ca n b e obtaine d b y settin g q = 1 .

ix

x INTRODUCTIO N

These notes adopt th e view that th e Iwahori-Hecke algebra s — rather tha n the g-Schur algebra s — are the objects o f central importance . Thi s i s partly a matte r of persona l tast e an d partl y expedience ; othe r authors , suc h a s Donki n [48 ] and Green [74] , travel i n the reverse direction . On e consequence o f our perspective i s that ou r definition o f the Iwahori-Hecke algebr a ma y strike som e reader s a s being contrived; t o remedy thi s w e now provide additiona l motivation .

First recal l that , a s an abstract group , the symmetric grou p has a presentatio n with generator s s±,.. ., s n - i an d relation s

si = 1 , fo r i = 1 , 2 , . .. , n - 1 , SiSj = SjSi, fo r 1 < i < j — 1 < n — 2,

SiSi+iSi = Si+iSiSi +i, fo r i = 1 , 2 , . .. , n - 2 .

Identifying Si with th e transposition (z, i + 1 ) shows tha t & n i s a quotien t o f the group W wit h th e presentation above ; a little mor e wor k verifie s tha t W = & n.

Now fix a ring R and let q be an element of R. Th e Iwahori-Hecke algebra J$? = J%R,q(&n) is the associative algebr a wit h generator s Xi , . . . , Tn_i an d relation s

(T2-g)(T2 + l ) = 0 , fo r i = 1,2,.. . , n - l , TtT3 = TjTi, fo r 1 < i < j - 1 < n - 2 ,

T.T^T, = T i + i T i r i + i , fo r 2 = 1,2,.. . , n - 2 .

In particular , J^ 7 and i ^6n ar e isomorphic whe n q = 1 . Thus , J f i s a deformation of the group rin g i26 n o f the symmetric group ; tha t is , the Iwahori-Hecke algebra s of 6 n ar e a famil y o f algebra s whic h 'loo k like ' th e grou p rin g o f the symmetri c group excep t tha t th e multiplication i s 'deformed' b y q.

Prom th e presentatio n o f ffl i t seem s likel y tha t M* will collaps e fo r som e choices o f the parameter q\ in fact , w e will sho w i n chapter 1 that, independentl y of q, the Iwahori-Hecke algebr a i s always a free i^-modul e o f rank n\ = \& n\. Th e best explanatio n o f why the ran k o f Jif i s independen t o f q is tha t th e Iwahori -Hecke algebra s appea r naturall y i n the representation theor y o f the general linea r groups; we now describe ho w this come s about .

Assume tha t q i s a prim e powe r an d le t G = GL n(q) b e th e genera l linea r group ove r th e field o f q elements . Le t B = B(q) b e a Bore l subgrou p o f G ; thus, B i s conjugate to the subgroup of upper triangula r matrice s in G. Le t Ind B(l) be th e induce d it!G-representatio n o n th e righ t coset s o f B i n G an d le t H q = End^G ( lnd B ( l ) ) b e the endomorphis m algebr a o f thi s module . Amazingly , th e algebras H q an d J4fR,q(<Sn) ar e canonically isomorphic ; so , Jif i s also a deformatio n of the endomorphism algebr a H q\

Here i s a roug h proo f o f the isomorphism J4?R, q(<£>n) — H q. Fo r any subset X of G le t [X] — J2xexx' a n e l e m e n t o f RG. The n Ind B( l) i s fre e a s a n RG-module wit h basi s [Bg], where g run s ove r a se t o f righ t cose t representative s of B i n G. Therefore , H q ha s basis [BgB], wher e g run s ove r th e (B,B)-doub\e coset representatives . No w 6n i s the Weyl grou p o f G , s o G = \Jwee BwB b y the Bruha t decomposition ; her e we identify (£> n wit h th e subgroup o f permutatio n matrices i n G. Therefore , H q i s R-free wit h basi s { [BwB] \ w G 6n }.

As above , le t s i , . . . , s n _ i b e generator s o f 6 n an d writ e £(w) = k i f k i s minimal suc h tha t w = Si 1 ... Si k fo r som e 1 < i 3 < n. The n fo r al l w G (5n and

INTRODUCTION

all i wit h 1 < i < n, on e ca n sho w tha t

[BwB][BslB] q[BwSiB] + {q- l)[BwB], i f £(wsi) < £(w),

[BwSiB], i f £{ws z) >£(w).

This implie s tha t th e itHinea r ma p IT : J^R^q(&n) —» Hq give n b y 7r(Ti) — [BsiB], for 1 < i < n , i s a surjectiv e algebr a homomorphism . A countin g argumen t show s that 7 r is an isomorphism .

As ther e ar e a n infinit e numbe r o f primes , th e isomorphism s H q = Jf?R iq can also be used to show that J^R^ i s free o f rank n ! for an y q. Se e Note 1. 7 on page 13.

To get a little more mileage out o f this discussion, suppos e tha t th e base ring R is the field o f complex numbers . The n H q i s semisimple and , therefore , s o i s J#b, q. Let e B = I^ I [B] G CG; the n e% — e# , s o e # i s idempotent . Furthermore , i t i s no t

hard t o chec k tha t Ind B(l) = esCG an d tha t H q = esCGeB- Thus , H q i s what i s known a s a Hecke algebra. (Mor e generally , a Heck e algebr a i s an y subalgebr a o f an algebr a A o f the for m eAe fo r som e idempoten t e £ A.)

Using th e elementar y theor y o f Heck e algebra s (se e [31 , §12]) , th e irreducibl e constituents o f Ind B(l) ar e i n canonica l one-to-on e correspondenc e wit h th e irre -ducible representations of Hq. A s we are in the semisimple case, Hq = 3%c,q — C<3n, so this show s tha t th e irreducibl e constituent s o f Ind B( l) ar e indexe d b y th e ordi -nary irreducible representation s o f ©n . Thus , fo r every irreducible representatio n \ of & n ther e exist s a family o f representations { \q I Q a prim e powe r } such tha t Xq is a n irreducibl e CGL n(g)-module whic h i s a direc t summan d o f Ind B( l ) . More -over, wit h a littl e mor e work , i t i s possible t o prov e th e astoundin g fac t tha t ther e exists a polynomia l D x(x), whic h depend s onl y o n x ? suc h tha t D x(q) i s th e di -mension o f Xq f° r an Y Q (a nd D x(l) i s th e dimensio n o f x). Mor e generally , th e characters o f th e representation s Xq a r e a l s o polynomial s i n q(\). Th e represen -tations Xq a re th e unipotent principal series representation s o f G. Proof s o f thes e results ca n b e foun d i n [22,31,68] .

The theor y w e hav e jus t sketche d connectin g th e Iwahori-Heck e algebra s o f the symmetri c group s wit h th e genera l linea r group s applie s mor e generall y t o th e Iwahori-Hecke algebra s o f arbitrar y Wey l group s an d th e correspondin g group s o f Lie type . Further , Iwahori-Heck e algebra s ma y b e define d fo r an y Coxete r grou p and, mor e recently, fo r an y complex reflection grou p [12] . I n addition t o these link s with the group s of Lie type, the Iwahori-Hecke algebra s play a role in the represen -tation theor y o f quantu m group s an d affin e Heck e algebra s an d hav e application s to kno t theor y an d statistica l mechanics . A t best , w e touc h onl y briefl y o n thes e matters here ; th e intereste d reade r i s referre d t o [29 , 64,102,103,131,138,168 ] and th e reference s therein .

The representatio n theor y o f th e Iwahori-Heck e algebra s an d th e g-Schu r al -gebras i s a very ric h an d beautifu l subject . Ou r approac h i s largely combinatorial , involving generalization s o f well-known concept s suc h a s tableau x fro m th e repre -sentation theor y o f the symmetri c group . Her e i s a broa d outlin e o f the book .

Chapter 1 begins by establishing som e basic properties o f the symmetri c grou p and it s Iwahori-Heck e algebr a Jif. I n fact , thi s chapte r i s reall y a chapte r abou t Coxeter group s and thei r Iwahori-Heck e algebra s in disguise because everything we do — including al l bu t on e of the proof s — extends t o thi s mor e genera l situation .

xii I N T R O D U C T I O N

The secon d chapte r develop s Graha m an d Lehrer' s [72 ] theory o f cellular alge-bras. Thes e ar e a clas s o f algebras whic h com e equipped wit h a distinguished basi s which i s particularly well-adapte d t o th e representatio n theor y o f the algebra . Fo r example, given a cellular basi s one can immediately writ e down a collection of mod-ules whic h contain s al l o f the simpl e module s o f the algebra . Cellula r algebra s ar e one of the unifyin g thread s runnin g throug h thes e note s a s we construct th e simpl e modules o f th e Iwahori-Heck e algebra s an d th e g-Schu r algebra s b y firs t showin g that thes e algebra s ar e cellular .

The thir d chapte r embark s upo n th e stud y o f the representatio n theor y o f JF. Following Murph y [150 ] w e sho w tha t J(? ha s a natura l basi s indexe d b y pair s o f standard tableaux ; importantly , Murphy' s basi s i s cellular . A s a consequence , fo r each partition A we obtain a Spech t modul e S x; thi s modul e i s a ^-analogue o f th e usual Spech t modul e o f @ n. B y th e theor y o f chapte r 2 , ther e i s a n intrinsicall y defined bilinea r form o n each Specht module ; furthermore , S x modul o the radical of its form i s either zer o or absolutely irreducible . Th e last tw o sections of this chapte r are a detailed stud y o f the Spech t modules , culminatin g i n th e classificatio n o f th e simple ^ -modu les . Al l of this theor y closel y parallel s th e modula r representatio n of the symmetri c group .

Apart fro m bein g cellular , Murphy' s basi s ha s anothe r marvelou s propert y i n that i t i s possible t o 'lift ' thi s basi s t o giv e a cellula r basi s fo r th e g-Schu r algebra ; this basi s i s indexe d b y pair s o f semistandar d tableaux , an d eac h basi s elemen t is essentiall y a su m o f Murph y basi s element s ( a semistandar d tableau x ca n b e thought o f a s a n orbi t o f standar d tableau x an d th e sum s ar e ove r thes e orbits) . In thi s wa y w e obtai n a ver y clea n an d ver y elegan t constructio n o f th e simpl e modules o f th e g-Schu r algebras . Fo r free , w e discove r tha t th e g-Schu r algebra s are quasi-hereditary . Al l o f these result s ar e prove d i n chapte r 4 .

Chapter 5 i s devote d t o classifyin g th e block s o f th e g-Schu r algebras ; a s a corollary thi s yield s th e block s o f Jif. I n orde r t o classif y th e block s w e first prov e an analogu e o f the Jantze n su m formul a [98 ] fo r th e Wey l module s o f the g-Schu r algebras. Th e Jantze n su m formul a i s a stron g resul t whic h give s informatio n about th e compositio n factor s o f Wey l module s an d Spech t modules ; i t i s prove d by computing the determinant s o f the Gra m matrice s o f the Weyl modules (tha t w e can comput e thes e determinant s i n genera l i s in itsel f surprising) . Thi s calculatio n requires a heavy dos e o f combinatorics; i t i s by fa r th e mos t technica l par t o f thes e notes.

The fina l chapte r i s a survey o f some recen t an d importan t result s an d conjec -tures i n the field ; her e we abandon ou r claim s o f being self-contained . Th e chapte r begins wit h a reasonabl y thoroug h accoun t o f th e LL T algorith m whic h compute s the decompositio n matrice s o f th e Iwahori-Heck e algebra s define d ove r th e com -plex field. W e then discuss adjustment matrices , the Kleshchev-Brundan branchin g rules, th e theor y o f Dippe r an d Jame s [40 ] connectin g th e g-Schu r algebra s an d the finit e genera l linea r groups , rule s fo r computin g decompositio n matrice s an d the Ariki-Koik e algebra s an d cyclotomi c g-Schu r algebras .

In addition , th e boo k contain s thre e appendices . Th e firs t o f thes e provide s a quick treatment o f the assume d representatio n theor y o f finite dimensiona l algebra s over a field . Thi s appendi x i s intende d a s a prime r fo r thos e ne w t o th e subject ; although no t necessary , previou s exposur e t o th e ordinar y representatio n theor y of finit e group s woul d b e advantageous . Th e secon d appendi x contain s table s o f the crystallize d decompositio n matrice s an d adjustmen t matrice s fo r n < 1 0 (se e

INTRODUCTION x m

chapter 6) , and th e third appendi x contain s tables o f the elementary divisor s of th e Gram matrice s o f the integra l Spech t module s fo r n < 12.

There ar e few ne w results i n this book ; however , man y o f the argument s eithe r do no t appea r i n th e literatur e or , whe n the y do , thei r simplicit y i s obscured b y a more genera l framework . Throughou t w e hav e trie d t o attribut e th e mai n result s to thei r rightfu l owners ; w e hop e tha t w e hav e succeede d — a t th e ver y leas t w e have provide d a n extensiv e bibliography . Eac h chapter , excep t fo r th e last , end s with a serie s o f exercise s (coverin g materia l w e have no t ha d tim e fo r i n th e text) , together wit h som e historica l note s an d references .

These notes are based upon a series of lectures I gave at the Universitat Bielefel d in 199 7 and a t th e University of Sydney in 1998 . I would like to thank th e member s of bot h audience s fo r thei r encouragement ; especially , Steffe n Koni g an d Clau s Ringel without who m thi s boo k would no t hav e been written . I am als o grateful t o the participant s o f the representatio n theor y worksho p hel d i n Blaubeure n i n Ma y 1999. Thank s ar e als o du e t o th e productio n staf f o f the AMS .

Finally, specia l thanks g o to Susumu Ariki , Jon Brundan , Richar d Dipper , Bo b Howlett, Gordo n James , Steffe n Konig , Bernar d Leclerc , Fran k Liibec k an d th e referees fo r thei r man y comments , correction s an d suggestions .

Andrew Matha s Sydney, 199 9

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Index o f notatio n

We us e standar d symbol s N , Z an d C fo r th e set s o f non-negativ e integers, integer s an d comple x number s respectively ; i n addition , 6ij is the Kronecke r delta .

Chapter 1 . 1.1 & n Th e symmetri c grou p o n {1 , 2 , . . ., n} 1

Si Th e simpl e transpositio n (z, i +1) 1 S Th e Coxete r generator s { s i , . . . , sn_i} o f & n 1 £{w) Th e lengt h o f w G 6n 2

1.2 Jl? Th e Iwahori-Heck e algebr a o f 6 n ; als o ^ ) g ( 6 n ) 5 T,, Ts Th e generator s r x , . . . , T n_ 1 o f JT ' 5 Tw A basis elemen t o f Jff, t o G 6 n 5 Z Z[g,g _1], wher e q is an indeterminat e ove r Z 8 ^ 2 Th e generi c Iwahori-Heck e algebr a J%?z,q(&n) 8

Chapter 2 . 2.1 (A , >) Th e pose t indexin g th e cellula r basi s 1 6

T(A) A n indexin g se t fo r th e cellula r basi s 1 6 Ax Th e spa n o f the c^ d wher e \i > A 1 6 * Th e automorphis m o f A give n b y c A

t = c A5 1 6

,4A Th e spa n o f the c^ wher e / i > A 1 7 CA A (right ) cel l modul e 1 7 C*A A lef t cel l module 1 7 Dx C A / r adC A 1 8

2.2 dx^ Th e compositio n multiplicit y o f D^ i n C A 2 1

Chapter 3 . 3-1 \n\ EiM i 2 7

A N n A is a compositio n o f n 2 8 /i h n ̂ x is a partition o f n 2 8 [/i] Th e diagra m { (z, j) \ 1 < j < fa an d z > 1 } of \i 2 8 Shape(t) Th e shap e o f the tablea u t 2 8 s, t , . .. (Standard ) tableau x (o f type a; ) 2 8 Std(/i) Th e se t o f standard /i-tableau x 2 8 P Th e standar d /i-tablea u wit h 1 , 2 , . . ., n entere d 2 8

in orde r alon g it s rows .

177

178 INDEX O F NOTATIO N

S M Th e Young subgrou p S M l x 6 M 2 x • • • 2 8 <#?(&„) Th e Iwahori-Heck e algebr a o f 6 M 2 8

M^ Th e "permutatio n module " m ^ 2 8 ^M Distinguishe d cose t representative s o f 6 M 2 9 d(t) Th e elemen t o f 6 n suc h tha t t = t^d(t ) 2 9 >, > Th e dominanc e orde r 3 0 t j ra Th e subtablea u o f t containin g 1 , 2 , . . ., m 3 0

3.2 * Th e ^-linea r antiautomorphis m o f J4? 3 2 determined by T^ = Tw-i fo r al l it ; G S n

rd%)mMTd(t) 3 2 M Th e Murph y basi s {ra st} o f J f 3 4 A+ Th e se t o f partitions o f n ordere d b y dominanc e 3 5 J^x Th e idea l o f Jff wit h basi s th e se t o f ra UD wher e u 3 7

and t ) are standar d /i-tableau x wit h \x \> A JfA Th e idea l o f Jf " wit h basi s th e se t o f ra UD wher e u 3 7

and t ) are standar d //-tableau x wit h \x > A SA Th e Specht modul e indexe d b y A 3 8 rat A standar d basi s elemen t o f a Spech t modul e 3 8 Dx A simple JT-module ; £> A = S x/radSx 3 8

3.3 L fc g - 1 T ( f c _ 1 ^ ) +g- 2 L ( f c _ 2 5 f c ) + . . . + 91- / cT ( 1 , f c ) 3 9

[m]q Th e 'quantu m integer ' 1 -f- g -j - • • • - f q 171"1 4 1 [ra]g Th e 'quantu m factorial ' [l]g[2] q . .. [ra] g 4 1 e Th e smalles t positiv e intege r suc h tha t [e] q =0 4 1 res(x) Th e e-residu e o f x 4 1 ft A n orthogona l basi s elemen t o f S x 4 2

3.4 l e{hj) = j — z + e( z — 1) , the ladder numbe r o f the node (i,j) 4 6 Mg [ /^ l ]g [M2]g . . . [ ^ 4 6 A' Th e partition conjugat e t o A 4 9

Chapter 4 . 4.1 A(d,n ) {/iN= n | / i = ( /x i , . . . , / x d )} 5 5

y(d,n) Th e g-Schu r algebra ; als o S^R iq{d,n) 5 5 J^(n) «^(n,n ) ' 5 5 UJ Th e partitio n ( l n) 5 6 S, T, . .. (Semistandard ) tableau x o f typ e /j, 5 6 7o(//, ^) Th e se t o f semistandard /i-tableau x o f type v 5 6 T M Th e uniqu e semistandar d /i-tablea u o f type / i 5 6 z/(t) Th e tablea u o f type i / obtained fro m t b y replacin g 5 6

each entr y i n t b y it s ro w inde x i n i v

fidDis Th e compositio n suc h tha t &^dnv = d _ 1 6 M d D 6^ 5 6 5 ^ Th e se t o f distinguished (6 M , 6^)-double cose t 5 7

representatives

INDEX O F NOTATIO N 179

first(T) last(T)

Mu*

mst mST

A+(d ,n)

</?ST

^o(A)

yx(d,n)

yx(d,n)

*

wx

raT

(pT

Lx

first (T) > t wheneve r u(t) = T t > last(T ) wheneve r u(t) = T

(MUY =3tfm v

YJB m s t wher e /x(s ) = S S s t

m5 t wher e /i(s ) = S an d i/(t ) = T The se t o f parti t ion s i n A(d , n ) A semistandar d basi s elemen t

U M G A ( d , n ) T 0 ( A ^ ) The idea l o f y{d,n) wit h basi s th e <^ ST

where S, T ar e i n %(a) an d a > A The idea l o f 5?{d,n) wit h basi s th e <^ ST

where S , T ar e i n To (a) an d a > A The automorphis m o f y(d,n) wit h <£>g T = ip TS

The Wey l modul e indexe d b y A

J2t m t wher e /i(t ) = T A semistandar d basi s elemen t o f W x

The simpl e ^ - m o d u l e W x/mdWx

57 58 59 59 60

61 61 61

61

61

61 62 62 62 63

Chapter 5 .

wx

4 K [resT(fc)]g

7 T ( ? )

< 9XAQ) a,/? dT(a)

PC

n hx

nab JX

lab Vp

w¥a

o¥

w*{i) vv

"e,p{k) (5A)°

The /x-weigh t spac e o f W x

The Gra m determinan t o f W x

A Jucys-Murph y elemen t o f E n d ^ ( M M )

Ex:T(x) = fc[ re8Wl? A rationa l functio n Another rationa l functio n

IIT7T(<Z) Sequences o f integer s (bet a numbers ) ± di m W" wher e v i s determine d b y a A sequenc e o f be t a number s { 1 , 2 , . . . , n } The lengt h o f th e (a , b) th hoo k i n [A ] The le g lengt h o f th e (a , b) th hoo k i n A The p-adi c valuatio n map , p a prim e The virtua l Wey l modul e correspondin g t o a The virtua l Spech t modul e correspondin g t o a The i th modul e i n th e Jantze n filtratio n o f W ¥

The p-adi c valuation , fo r p a prim e intege r vp(k) i f e|fc ; — 1 otherwise . The dua l o f S x

69 69 69 70

71 73 75 76 76 76 76 79 79 81 82 82 83 88 89 89

Chapter 6 . 6.1 j€n Jt?(& n) 9 6

Res Th e restrictio n functo r fro m Jff n to c ^ _ i 9 6

180 INDE X O F NOTATIO N

Ind v—• A

Qn \E\ i-Res i-Ind

i / - i » A

Ql \Pf J n

[Sv] uv Ai Ei,Fi,Kh

L(A) B0(A) B(A) J y

B, pn

m(/i) D n = (d Xfl)

Cn = (cxfj)

The inductio n functo r fro m J4? n to J4? n+i [i/] C [A ] and |i/ | = |A | - 1 The complexifie d Grothendiec k grou p o f J$?n

The equivalenc e clas s o f E i n Q n

The i-restrictio n functo r fro m Mn t o J ^ _ i The z-inductio n functo r fro m J ^ t o J ^ + i

z/ —» A , where th e adde d nod e ha s e-residu e i The projectiv e Grothendiec k grou p o f J^ n

The equivalenc e clas s o f P i n C/ £ The Grothendiec k grou p o f a semisimpl e

Iwahori-Hecke algebr a A basi s elemen t o f T n

U v ( s t e ) , th e quantize d affin e specia l linea r grou p The fundamenta l weight s Generators o f U v

An integra l highes t weigh t modul e The crysta l grap h o f JL(A ) The canonica l basi s o f L(A ) The Foc k spac e 0 n 0 A C{v)X B^ = ^2 X b\y\, a canonica l basi s elemen t i n T v

A principa l indecomposabl e The Mullineu x ma p Decomposition matr i x Inverse decompositio n matr i x

96 96 97 97 97 97

97 98 98 99

99

100 100 100 101 101 102 103 108 111 120 122 122

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Index

This i s mainl y a n inde x o f definitions ; occasion -ally mor e genera l reference s ar e given .

abacus, 8 5 absolutely irreducible , 14 3 addable node , 4 2 adjustment matrix , 11 7 affme quantu m group , 10 0 Ariki-Koike algebra , 13 2 Art in brai d group , 5

beta numbers , 76 , 12 4 blocks, 86 , 14 4 braid relations , 1 branching rule , 9 6

modular, 12 0 Bruhat-Chevalley order , 3 0

canonical basis , 102 , 10 8 Cart an matrix , 2 1 Carter criterion , 12 5 cell linked , 2 3 cell module , 1 7 cellular algebra , 1 6 cellular basis , 1 6 character, 9 completely reducible , 14 2 composition, 2 7 composition factor , 13 7 composition series , 13 7 Coxeter group , 1 3 crystal graph , 10 2 crystallized decompositio n matrix , 11 0 cyclotomic g-Schu r algebra , 13 4

decomposition matrix , 21 , 48, 63 , 8 3 diagram, 2 8 distinguished cose t representatives , 2 9

double cose t representatives , 5 7 dominance orde r

partitions, 3 0 tableau, 30 , 7 0

double centralize r property , 6 5 dual, 2 4

e-content, 4 9

e-core, 8 5 e-quotient, 12 4 e-regular, 5 4 e-residue, 41 , 70 e-weight, 8 5 elementary divisors , 8 1 exact sequence , 14 3 exchange condition , 3

filtration, 13 7 Fock space , 10 3 fundamental weights , 10 0

Garnir tableau , 3 3 generic degree , 9 generic Iwahori-Heck e algebra , 8 good i-node , 10 5 Gram determinant , 6 9 Grothendieck group , 9 7

highest weigh t module , 10 1 hook, 7 9 hook length , 7 9 hook lengt h formula , 5 3

indecomposable, 13 8 induced representation , 2 9 Iwahori-Hecke algebra , 5

Jacobson radical , 14 1 Jantzen filtration , 81 , 8 3 Jantzen su m formula , 8 3 Jordan-Holder theorem , 13 8 Jucys-Murphy elements , 39 , 41 , 69, 13 2

Krull-Schmidt theorem , 13 9

ladder number , 4 6 left cel l module , 1 7 leg length , 7 9 length, 2 lexicographic order , 3 3 Littlewood-Richardson rule , 12 3 LLT algorithm , 11 0

187

188

LLT-conjecture, 10 2

Maschke's theorem , 14 2 modular system , 11 6 Mullineux map , 12 0 Murphy basis , 35 , 59 , 6 7

Nakayama conjecture , 8 6 nilpotent ideal , 14 0 node, 2 8

above, below , 10 4 i-node, 10 4

normal i-node , 10 5

partition, 2 8 conjugate, 4 9 diagram, 2 8 e-content, 4 9 e-core, 8 5 e-restricted, 4 5 e-weight, 8 5 parts, 2 8

Poincare polynomial , 9 principal indecomposabl e modules , 14 0 projective module , 14 2

g-Schur algebra , 5 5 quantum group , 10 0 quantum hyperalgebra , 11 9

reduced expression , 2 removable node , 10 4 rim hook , 7 9 Robinson-Schensted correspondence , 5 2 row standard , 2 8 row standar d basis , 3 0

Schur algebra , see g-Schu r algebr a Schur functor , 64 , 83 , 8 6 Schur's lemma , 14 3 Scopes equivalence , 12 7 semisimple, 14 2 semistandard basi s theorem , 6 1 sign representation , 2 , 8 Specht filtration , 6 0 Specht module , 38 , 5 4

dual Spech t module , 3 8 specialization, 8 splitting field , 14 3 standard basis , 35 , 3 8 standard basi s theorem , 3 7 Steinberg tenso r produc t theorem , 12 3 subexpression, 3 0 symmetric algebra , 14 6

tableau, 2 8 row semistandard , 5 6 e-restricted, 4 5 row standard , 2 8 semistandard, 5 6

INDEX

shape, 2 8 standard, 2 8 type, 5 6

trivial representation , 8

Wedderburn decomposition , 14 5 weight space , 6 9 Weyl module , 6 2

Young subgroup , 2 8 Young's seminorma l form , 43 , 5 4