Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group (University Lecture Series) [UK ed.] 0821819267, 9780821819265

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Selected Title s i n Thi s Serie s 15 A n d r e w M a t h a s , Iwahori-Heck e algebra s an d Schu r algebra s o f th e symmetri c group , 1 99 9 14 Lar s Kadison , Ne w example s o f Probeniu s extensions , 1 99 9 13 Yako v M . Eliashber g an d W i l l i a m P . T h u r s t o n , Confoliations , 1 99 8 12 I . G . M a c d o n a l d , Symmetri c function s an d orthogona l polynomials , 1 99 8 11 Lar s Garding , Som e point s o f analysi s an d thei r history , 1 99 7 10 V i c t o r Kac , Verte x algebra s fo r beginners , Secon d Edition , 1 99 8 9 S t e p h e n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 1 99 6 8 B e r n d Sturmfels , Grobne r base s an d conve x polytopes , 1 99 6 7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 1 99 4 6 D u s a McDuf F an d D i e t m a 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 , 1 99 4 4 D a v i 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 , 1 99 2 2 Frit z J o h n , Nonlinea r wav e equations , formatio n o f singularities , 1 99 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 , 1 98 9

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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 J e r r y L . B o n a (Chair ) Nicola J e a n - L u c Brylinsk i Leonar

i Reshetikhi n d L . Scot t

T h e a u t h o r gratefull y acknowledge s t h e s u p p o r t o f t h e Sonderforschungsbereich 34 3 a t t h e Universita t Bielefel d a n d o f a U200 0 Fellowshi p a t t h e Universit y o f Sydney .

1991 Mathematics Subject Classification. P r i m a r y 20C30 , 1 6G99 ; Secondary 05E1 0 , 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 th e Iwahori-Heck e algebra s o f th e symmetri c group s an d o f th e 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 D a t a Mat has, Andrew , 1 966 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 1 047-399 8 ; v. 1 5 ) Includes bibliographica l reference s an d indexes . ISBN 0-821 8-1 926- 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 1 74.2.M3 81 99 9 512'.2-dc21 99-293 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 an y 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 addresse d t o th e Assistan t t o th e 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 . © 1 99 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 03 02 01 00 9 9

Contents Introduction i

x

Chapter 1 . Th e Iwahori-Hecke algebr a o f the symmetri 1 c grou p 1. Th e Symmetri c grou p 2. Th e Iwahori-Hecke algebr a 5 3. Th e O-Heck e algebr a

0

Chapter 2 . Cellula r algebra s 1. Cellula r base s 2. Simpl 1 e modules i n a cellular algebr a

5 5 9

Chapter 3 . Th e modula r representatio n theor y o f 3riP 2 1. Th e combinatoric s o f tableaux 2 2. Th e Murph y basi s 3 3. Spech t module s an d Jucys-Murph y element s 3 4. Irreducibl e ^-module s 4

7 7 2 9 5

Chapter 4 . Th e g-Schu r 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 Jantze n su m formula an d th e block s of Jf? 6 9 1. Gra m determinant s o f Weyl modules 6 9 2. Th e Jantze n su m formul a 8 1 3. Th e blocks of y(n) an d 3ff 8 4 4. Irreducibl e Wey l modules an d Spech t module s 8 7 Chapter 6 . Branchin g rules , canonical base s and decompositio n matrice s 9 5 1. Th e LL T algorithm 9 5 2. Decompositio n map s an1 1 d adjustmen t matrice s 5 3. Th e Kleshchev-Brunda n modula 1 1 r branchin g rule s 8 4. Rule s for computin 1 g decompositio n matrice s 2 2 5.1 Th e g-Schu r algebra s an d GL n(q) 2 9 6. Th e Ariki-Koik e algebra s an d cyclotomi c g-Schu1 r algebra s 3 1 Appendix A . Finit e dimensiona l algebra s over a field 1 3 1.1 Filtration s an d compositio n serie s 3 2. Idempotent s an1 d indecomposabl e module s 3 3. Th e block s of A 4 1 4. Semisimpl e symmetri c algebra s 4

7 7 8 4 6

viii C O N T E N T

S

Appendix 1 B . Decompositio n matrice s 4 9 1. Decompositio 1 n matrice s whe n e = 2 4 9 2. Decompositio 1 n matrice s whe n e = 3 5 6 3. Decompositio 1 n matrice s whe n e — 4 6 1 Appendix C . Elementar y divisor s o f integra l Spech 1 t module s 6

5

Index o f notatio n 7

7

References 8 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 1 976 . 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 th e symmetri c groups ; thi s includes the modula r representatio n theor y o f 6 n a s a special case . I n studyin g 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 th e g-Schu r algebras . Th e stud y 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 bot h algebras . W e hav e writte n 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 t o 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 th e Iwahori-Heck e algebra s i s that the y provid e 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 [1 58 ] wh o use d them , together wit h th e representatio n theor y o f the symmetri c groups , t o classif y th e or dinary irreducibl e polynomia l representation s o f GL n (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 field s 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 w e 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 th e other . Further , i t i s ofte n 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 t o 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 note s adop t 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 perspectiv e i s that ou r definition o f the Iwahori-Hecke algebr a ma y strike som e reader s a s bein g 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 ha s a presentatio n with generator s s±,.. ., s n - i an d relation s si = 1 , fo SiSj = SjSi, fo SiSi+iSi = Si+iSiSi + i, fo

r i = 1 , 2 , . .. , n - 1 , r 1 < i < j — 1 < n — 2, r i = 1 , 2 , . .. , n - 2 .

Identifying Si with th e transpositio n (z, i + 1 ) show s 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 o f R. Th e Iwahori-Hecke algebr a J$? = J%R,q(&n) is the associative algebr a wit h generator s X i , . . . , Tn _i an d relation s (T 2 -g)(T 2 + l ) = 0 , fo TtT3 = TjTi, fo T.T^T, = T i + i T i r i + i , fo

r i = 1 ,2,.. . , n - l , r1 s n-i, subjec t t o th e relation s 1.1 SiSj

sf = 1 , fo — SjSi, fo SiS^iSi = s i + 1 SiSi + i, fo

r i — 1, 2 , . . ., n — 1 , r 1 < i < j — 1 < n — 2, r i = 1 ,2,.. . ,n - 2 .

The reade r i s invited t o prov e tha t thi s doe s indee d giv e a presentatio n o f & n. Th e second an d thir d relation s ar e calle d th e brai d relation s o f 6 n . l

2

1. TH E IWAHORI-HECK E ALGEBR A O F TH E SYMMETRI C GROU P

Suppose tha t w is an element o f (5n an d writ e w = sil . . . S{ k where s ix,..., Si k are element s o f S. I f k is minimal w e say tha t w has lengt h k and writ e £{w) = k; in thi s case , Si 1 . .. Si k i s called a reduce d expressio n fo r w. I n general, a n element of & n wil l hav e man y reduce d expression s (fo r example, th e braid relation s giv e two reduce d expression s fo r various element s o f @n ); nonetheless , bot h th e lengt h function an d reduce d expression s ar e usefu l asset s whe n studyin g & n. These note s ar e concerne d wit h representations . Pro m th e presentatio n o f &n given i n (1.1), w e se e tha t S n ha s a non-trivial on e dimensiona l representatio n e which i s determined b y e(s) = — 1 for all s G S. Thi s i s the sign representatio n of 6 n ; notice tha t e{w) = (— l)£(w) fo r all w G 6n . Ultimately , w e wil l construc t all of the irreducibl e representation s o f &n. By definition , i f s G S an d w G &n the n £(sw) = £(w) ± 1 . W e now giv e another descriptio n o f the lengt h functio n whic h wil l allo w u s to determine whe n £(sw) = £{w) + 1 and when £{sw) = £(w) — 1 . Dyer' s reflection cocycle is th e function give n by N(w) = { (j, k) G 6 n | 1 < j < k < n and jw > kw} . For example , N(l) = 0 an d N(s) = {5}, for all s G S. Shortly , w e will prov e tha t £{w) = \N(w)\ fo r all we & n. Given set s A and B let A+B = (AuB)\(Ar\B) b e their symmetri c difference . 1.2 Lemm a (Dyer ) Suppose that v,w G &n. Then N(vw) — N(v)+vN(w)v~

1

.

Proof. I f s G S and (j , k) G 6n then s(j, k)s = (js, ks) s o N(sw) = N(s)+sN(w)s for an y w G 6n ; so the Lemm a hold s whe n £{v) = 1 . If £(v) > 1 then v = su fo r some s e S an d u G 6n an d wit h £(v) = £(u) + 1 . Therefore , b y induction o n the length o f v, N(vw) = N(s(uw)) = N(s)+sN(uw)s =

= N^+vN^v'

iV(5)+sAr(w)s4-5'uiV(ii;)u~ 1 s

1

as required . D For convenienc e let T = {(i , j) G 6n } = [Jwee n wSw ~1 - T h e n ^V(^ ) Q T and , when w e consider 6 n a s a Coxete r group , T i s the set of reflections i n © n (se e Exercise 1 .1 7) . 1.3 Propositio n Suppose that w G 6n . T/ie n (i) l(w) = \N(w)\; and,

(ii) JV H = { * G T I £{tw) < £{w) }. Proof. Suppos e tha t s ^ . .. 5^ fc is a reduced expressio n fo r w and, for a = 1 , . . . , fc, let t a = Si x . . . Si a_1 SiaSia_1 ...Si 1 . The n t a £ T fo r all a and , b y Lemma 1 .2 , iV(w;) = Ar(5 ^ . . . Si k) = {ti}-\ \-{tk}. W e clai m tha t t a =£ tt fo r a ^ b. B y wa y of contradiction , suppos e tha t t a = tb for some a < b. The n ^

=

=

t atbW =

S ^ . . . 52a _ 1 5 fa S j a _ 1 . . . Si1Si1 . = Si

l

. . Sib_1SibSib_1 .

. . Si-^W

. . . S{a . . . Sib . . . Si k,

where Sj indicates tha t th e transpositio n Sj is omitted. However , thi s contradict s the assumption tha t w has length k so t\,..., tk mus t al l be distinct afte r all . Hence , N(w) = { t i , . .. ,t k} an d £{w) = k = \N(w)\ provin g (i).

3

1. T H E S Y M M E T R I C G R O U P

Next, suppos e tha t t eT. The n t = s ix . .. s il_1 silSil_1 ... s i:L, fo r som e s io G 5, where thi s expressio n i s reduced. Therefore , b y th e las t paragraph , iV [t) = ^S{ 1 , Si

Si 2Si1 ,

1

. . . , Sj >1 . . . Si l_1 Si

t

S{ 1 _1 .

. . Si 1 j .

In particular , thi s show s tha t t G N(t) fo r al l t eT. Finally w e prove (ii) . Le t N(w) = {t\, ..., tk] a s above. Then , fo r a = 1 , . . . , /c, £aw = Si ± . .. si a . . . s* fc; so £(t aw) < £(w) . Therefore , ) C { t € T | £(to ) < ^(w ) } .

(ii)/ iV(w

To prove th e converse , le t t G T an d suppos e tha t t £ N(w). Now , b y Lemm a 1 .2 , iV(tty) = N(t)+tN(w)t an d w e hav e just see n tha t t G iV(£); therefore , t G iV(tty) since t £ tN(w)t. Consequently , £(t • tw) < £{tw) b y (ii) ' applie d t o N(tw); tha t is, £(w) < £{tw). Thi s complete s th e proo f o f (ii ) an d henc e th e Proposition . • Prom th e definition s an d par t (ii ) o f th e Propositio n w e deduc e th e following . 1.4 Corollar y Suppose that w G (5n and that Si G S. Then 0( x v}

J ^ M + 1 , if(i)w ( i + l)w .

There i s a simila r descriptio n o f £{wsi). 1.5 Theore m (Th e stron g exchang e condition ) Le t s ^ , . . ., Si k be elements of S and suppose that t eT and that £{tsi r . . . Si k) < £(si 1 . . . Si k). Then tSi1 .

. . Si k —

Si 1 . . . Si a . . . Si

k

for some a. Furthermore, t = S*, ... si ,Si Si . . . . 5 ^ . Proof. Le t w = s ix . . . s ik and , fo r 1 < a < fc, let £ a = s i]L . . . s ia__1 siasia_1 . . . s ix as in the proof o f the Propositio n 1 .3 . The n N(w) — {ti}-j \-{tk} b y Lemma 1 . 2 and t G iV(w) b y Propositio n 1 .3(h) . Therefore , t = t a fo r som e a an d everythin g follows. • Notice tha t i n th e stron g exchang e conditio n w e d o no t assum e tha t s^ . . . Si k is reduced . Th e stron g exchang e conditio n ha s a n equivalen t righ t han d versio n which w e leave a s a n exercise . I f w e assum e i n th e statemen t o f Theore m 1 . 5 tha t t G S the n th e correspondin g resul t i s known a s th e exchang e condition . 1.6 Corollar y (Th e deletio n condition ) Suppose that w G 6 n , £{w) < k and that w = s tl . . . Si k for some Si j G S. (i) There exist integers 1 < a < b < k such that w = s ^ . . . si a . . . s^ b . .. Si k. (ii) A reduced expression for w can be obtained from 5 ^ . . . Si k by deleting an even number of the simple reflections Si j. Proof. Le t a be maximal such that £(si a . .. S{ k) < £(si a+1 . . . Si fc); suc h a n a exists with 1 < a < k becaus e £(w) < k. The n Si a . .. Si k = s^ a+1 . . . si b . . . s ik fo r som e b with a < b < k b y takin g £ = s a i n th e stron g exchang e condition . Therefore , w = Si 1 ... si a . . . ?i b . .. s^ fc, provin g (i) . Par t (ii ) i s no w immediate . • 1.7 Corollar y Suppose that w G 6n an d s e S. Then (i) £(sw) < £(w) if and only if w has a reduced expression beginning with s; (ii) £(ws) < £(w) if and only if w has a reduced expression ending with s.

1 4

. TH E IWAHORI-HECK E ALGEBR A O F TH E SYMMETRI C GROU P

Proof. Becaus e £(sw) = £(w~ ls), part s (i ) an d (ii ) ar e equivalen t an d i t suffice s to conside r (i) . Suppos e the n tha t £(sw) < £{w) an d le t s^ . . . Si k b e a reduce d expression fo r w. B y th e exchang e conditio n ther e exist s a n intege r a suc h tha t sw = Si ± . .. si a . . . Si k. Therefore , w = ssi 1 . . . si a . . . Si k; thi s expressio n i s reduce d since i t ha s lengt h fc. The convers e i s clear . • We no w prov e th e mai n resul t neede d fro m thi s section . Give n tw o reduce d expressions Si 1 . .. Si k an d s 3l . . . Sj k i n 1 , an d (... , i, i + 1 , i , . . .) ~ ^ (... , i + 1 , i, i + 1 ,...) . Observ e tha t two fc-tuples ar e i n th e sam e equivalenc e clas s onl y i f the y correspon d t o reduce d expressions o f th e sam e elemen t o f S n ; surprisingly , th e convers e i s als o true . 1.8 Theore m (Matsumoto ) Suppose that s^ ,..., Si k and Sj ±,..., Sj k are elements of S such that Si 1 . .. Si k and Sj 1 . . . Sj k are two reduced expressions in & n. Then (ii,...,ik) ~ 6 (ji , • • • ,j/c) if and only if s^ ...s ik = s jl ...s jk. Proof. B y th e remark s above , we need t o sho w that i f s^ . . . Si k — Sj 1 ... Sj k, an d these expression s ar e bot h reduced , the n (zi,..., ik) ~ 6 (ji ? • • • > jk)> We argu e b y inductio n o n fc. I f fc < 1 ther e i s nothin g t o prov e s o sup pose tha t k > 1 . Becaus e thes e tw o expression s ar e reduced , Si 1 Sj1 ... Sj k i s not a reduce d expression ; therefore , s ilSj1 . . . Sj k = Sj 1 . . . 2 j a . . . s Jfe fo r som e in teger a b y th e exchang e condition . Therefore , (i ) s; 2 . . . s ik = 5 JX . .. 3j a . .. Sj fc an d (ii) Si 1 Sj1 . . . Sj a_1 = Sj 1 . .. Sj a; further , al l fou r o f thes e expression s ar e reduced . If a ^ fc then th e reduce d expression s i n (i ) an d (ii ) al l hav e lengt h strictl y les s than A: . Therefore , usin g inductio n an d (i ) an d (ii ) i n turn , ( i l , i 2 , . . . , Z / c ) ~b (ilJl,".,ja-l.Ja+l,".Jk)

~b

Ul,'--JaJa+l,--Jk)

as required . If a = k onl y th e expression s i n (i ) hav e lengt h les s than fc, so b y inductio n w e can onl y deduc e tha t (z 2 ,... ,i^) ~t> (ji > • • • ,jk-i)', b y symmetr y w e ma y als o as sume tha t {J2,'.',jk) ~b (zi,...,ifc-i) . Hence , (ii,i 2 , • • • ,ik) ~b {h, j i , • • •, jfc-i) and (ji , j 2 , • - • ,jk) ~b (ji,h, • • • > ^ - i ); s o th e theore m wil l follo w i f w e ca n sho w that (ii , J ! , . .. ,.7'fe-i ) ~ 6 (ji,h,.' • ,*fe-i)Suppose first tha t |i i — j i| > 1 . The n (ii , j i) ~ 6 (ji»^i ) becaus e s ^ s ^ = Sj 1 Si1 is a brai d relatio n o f lengt h 2 . Furthermore , b y (ii ) abov e an d it s mirro r image , S S S

ii ji J2 '

' ' S jk-i ~

S

jiSJ2 •

' • Sjk =

S

iiSi2 •

• ' S ik ~

s

jisiisi2 '

* '

s

ik-i-

Therefore, Si 2... Si k_1 an d s j2 . . . Sj k_1 ar e reduced expression s for the same elemen t in 6 n . Consequently , (z 2 ,... ,ik-i) ~ 6 0*2 ? • • •»Jfc-i) b y inductio n o n A: ; therefore, ( i i , j i , j 2 . . . , j / e - i ) ~ & (ii,ji,z 2 ,...,2/e-i) ~b (ji , H, ^2, •. •, ik-i) a s required . Finally, conside r th e cas e where |i i — j i| = 1 . Initially , w e wanted t o prov e tha t (ii,..., ik) ~ & ( j i , . . ., jk) an d we argued t o show that i t was sufficient t o prove tha t (ii, jfi,..., jk-i) ~b 0'i ? ^1 , • • •, ^fc-i); s o w e replaced th e las t fc—1 entries in each ktuple with th e first fc — 1 entries in the other . Therefore , b y repeating this argumen t we ar e reduce d t o showin g tha t (i x , ju iu z 2 ,... r ik-2) ~b (ji,ii,ji,J2,---Jk-2)However, Si 1 Sj1 Si1 = Sj 1 silSj1 i s a brai d relatio n o f lengt h 3 ; so , a s before , th e result follow s b y inductio n o n fc. •

5

2. T H E I W A H O R I - H E C K E A L G E B R A

Let 9 5 n b e th e Ar t in brai d grou p o f 6 n ; tha t is , *B n i s th e abstrac t grou p generated b y elements si, S2,..., s n - i whic h satisfy onl y the braid relation s of (1.1). Notice tha t ther e i s a surjectiv e grou p homomorphis m n : *B n — > @ n whic h i s determined b y 7r(si ) — Si for i = 1 , 2 , . . ., n — 1 . Matsumoto' s Theore m i s reall y a result abou t th e element s o f Q3 n; specifically, i t say s tha t i f s^ . . . 5^ fc an d s ^ . . . Sj k are reduce d expression s i n 6 n the n s^ . . . s^ = s jx . . . Sj k i n 6 n i f an d onl y i f s

ii •

• • s ik =

S

ji •

' " S jk m

-**n -

The final resul t i n thi s sectio n i s a technica l resul t whic h i s neede d below . 1.9 Lemm a Suppose that £{w) = £(siWSj) and £{siw) = £(WSJ) for some w G & n and some integers i and j with 1 < i , j < n. Then SiW = WSJ. Proof. Firs t conside r th e cas e wher e l(siw) — N(w) =

N(WSJSJ)

—

£(WSJ)

N(wsj)+wSjN(sj)sjW~ 1 —

< £(w) — £(siWSj). The n N(WSJ)^-{WSJW~

1

},

by Lemm a 1 .2 , sinc e N(SJ) — {SJ}. However , s^ G N(w) an d Si £ N(WSJ) b y Proposition 1 .3(h) . Therefore , Si — WSJW~ 1 an d siw — WSJ a s required . On th e othe r hand , i f £(siw) = £{WSJ) > £(w) le t w 1 — SiW. The n £{siw') — r 1 £(W'SJ) < £{w') = £{siW Sj) an d SiW — W'SJ b y th e las t paragrap h applie d t o w'; whence SiW = WSJ, onc e w e unravel th e notation . •

2. Th e Iwahori—Heck e algebr a We ar e no w read y t o introduc e th e Iwahori-Heck e algebr a o f & n. Let R b e a commutative domai n wit h 1 and le t q be a n arbitrar y elemen t o f R. The Iwahori-Heck e algebra J$? — JffR, q(&n) o f S n i s the unital associativ e i£-algebr a with generator s 7\ , T 2 , . . ., T n _i an d relation s 1.10 T

(Ti - q)(Ti + 1 ) - 0 , fo lT3=T3Ti, fo TzTl+1T% = Ti+xTiTi+u fo

r i - 1 , 2 , . . ., n - 1 , r 1 < i < j- 1 < n - 2 , r i = 1 ,2,.. . ,n - 2 .

It i s sometimes usefu l t o thin k o f Jif a s belongin g t o a famil y o f jR-algebra s whic h depend upo n a paramete r q. I n particular , whe n q — 1 the first relatio n i n (1 .1 0 ) reduces t o Tf = 1 . Thus , b y (1 .1 ) , Jf? i s isomorphi c t o th e grou p rin g R n o f th e symmetric grou p whe n q = 1 ; for thi s reaso n w e cal l Jf? a deformation o f R& nBy the remarks in the introduction, £(w), + (q - l)T w, if ws

l{ws) < £(w).

Proof. Suppos e tha t £(ws) > £(w) an d le t s^ ... Si k b e a reduce d expressio n for w. The n Si 1 . .. s^s i s a reduce d expressio n fo r ws s o TWTS = T ws b y definition . On th e othe r hand , i f £(ws) < £(w) the n w ha s a reduced expressio n endin g i n s b y Corollary 1 .7(h) ; therefore , r~p rp

rj-i

rjiZ

rji

q + (q- l)T

a

=

qT ws + (q - 1 )T

W.

Notice tha t whe n q = 1 the multiplicatio n i n Jif reduce s t o tha t i n R, i \qew + (q- 1 ) \qe3iW + (q - l)e

f

K siw) > Z( w)i otherwise . w,

Therefore, Of — (q — 1 )6i + OIE fo r al l 1 < i < n; her e 1 ^ i s the identit y ma p o n E. Equivalents, (0 % - q){0i + l E) = 0 . To sho w tha t th e Oi als o satisf y th e brai d relation s i n Jrf? w e first tak e a smal l detour. Fo r i = 1 , 2 , . . ., n — 1 let di b e th e elemen t o f E n d ^ i ? ) give n b y ajx

_ f

e

wsx, \{£{wSi)>

£(w),

1 qe-wsi + {q — l) eu;? otherwise

.

(So Oi correspond s t o lef t multiplicatio n b y Ti an d $ ; correspond s t o righ t multi plication b y Ti.) Fix integer s i an d j wit h 1 < z , j < n. W e clai m tha t O^j = djOi. T o prov e the clai m i t suffice s t o sho w tha t Oidj{e w) — $jOi(ew) fo r al l w G &n. No w eithe r £(siWSj) = £(w) o r £(siWSj) = £(w) ± 2 ; so there ar e si x case s t o consider .

7

2. T H E I W A H O R I - H E C K E A L G E B R A

First, i f £(siWSj) = £(w) + 2 the n 9i$j(e w) = if £(siWSj) = £(w) — 2 the n Oi$j(ew) = q 2eSiws0 + g(< 7 - l)e^

Sj

e

SiWS.

= $j9i(e w)] secondly ,

+ g( g - l)e s.w + (q- l)

2

ew = $

J9l(ew)

by direc t calculation . Next, i f £(siw) = £{ws 3) < £{w) = £(siWSj) the n 9 i,dj(ew) i s equal t o