all i with 1 i n, on e can sho w tha t
q[BwSiB] + {q- l)[BwB], i f £(wsi) £(w),
[BwSiB], i f £{ws
) £(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 counting argumen t show s
that7 r is an isomorphism .
As ther e ar e a n infinit e numbe r o f primes , th e isomorphism s H
= Jf?R
also be used to show that J^R^ i s free of 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 that th e base ring R
is the field of complex numbers . The n H
i s semisimple and , therefore , s o is J#b,
Let e
= 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
= esCGeB- Thus , H
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]), the 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 C3n,
so this show s that th e irreducibl e constituent s o f Ind B(l) ar e indexe d b y th e ordi -
nary irreducible representations o f ©n. Thus , for every irreducible representation \
of &
ther e exist s a family o f representations { \q I Q a prim e powe r } such that Xq
is a n irreducibl e CGL n(g)-module whic h i s a direc t summan d o f Ind
(l). More -
over, wit h a little 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 r an Y Q (and 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, for an y complex reflection grou p [12]. I n addition to 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 broad 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 their Iwahori-Heck e algebra s in disguise because everything we
do including al l but on e of the proof s extends t o thi s mor e genera l situation .
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