CHAPTER 1
The Iwahori-Heck e algebr a o f th e
symmetric grou p
This chapte r introduce s th e mai n characte r o f ou r story : th e Iwahori-Heck e
algebra ¥? o f th e symmetri c group . A s i n mos t o f Shakespeare' s plays , w e mus t
first mee t a secondar y player ; namely , th e symmetri c grou p itself . Literar y aspi -
rations aside , thi s i s necessar y becaus e w e hav e littl e hop e o f understandin g th e
Iwahori-Hecke algebra s unti l w e hav e firs t establishe d th e necessar y fact s abou t
the symmetri c groups .
These note s begi n the n wit h a quic k stud y o f th e symmetri c grou p 5 n. Th e
properties o f 6
n
tha t w e nee d al l deriv e fro m a particula r presentatio n whic h
shows tha t &
n
i s a Coxeter group, tha t is , a grou p generate d b y reflections . Thi s
presentation wa s discusse d i n th e introductio n an d appear s agai n i n (1.1 ) below ;
it arise s whe n w e thin k o f &
n
a s bein g generate d b y th e n 1 transposition s
(1,2), (2,3),..., (n 1, n). Th e ke y result o f the firs t sectio n i s Matsumoto's The -
orem (1.8), which describe s ho w certai n minima l lengt h word s i n thes e generator s
are related .
The secon d sectio n introduce s th e Iwahori-Heck e algebr a Jif, definin g i t a s a n
associative it!-algebr a wit h certai n generator s an d relations . Th e relation s i n Jf?
are a deformatio n o f th e relation s fo r th e symmetri c grou p an d the y depen d o n a
parameter q G R. Ou r firs t ste p i s to prov e that M^ is free a s an /^-module . T o d o
this w e show tha t J ^ ha s a n i?-fre e basi s o f th e for m { Tw \ w G &n } ; the result s
of sectio n 1 tell u s tha t th e element s T
w
ar e well-defined . Th e remainde r o f th e
chapter look s at ho w Jf 7 behave s when we change the ring R (specialization) , som e
'generic' propertie s o f J^, an d th e representatio n theor y o f Jlf whe n q = 0 .
We remark tha t thi s chapter i s really a chapter abou t Coxete r group s an d thei r
Iwahori-Hecke algebras . Al l of the results that w e prove (an d all of their proofs, ba r
Lemma 1.2, whic h ca n b e modifie d b y usin g roo t systems) , ar e vali d fo r arbitrar y
Coxeter groups . Thi s them e i s taken u p i n the exercise s a t th e en d o f the chapter .
1. Th e Symmetri c grou p
Throughout thes e note s w e fix a n intege r n\ an d le t 6
n
b e th e symmetri c
group actin g o n 1, 2,..., n fro m th e right Fo r i = 1, 2,..., n 1 let S{ be the basi c
transposition (i,i + 1) an d le t S {si,..., s
n
-i}- Then , a s a Coxete r group , &
n
is generated b y si , S2? sn-i, subjec t t o th e relation s
sf = 1, fo r i 1, 2,..., n 1,
1.1 SiSj SjSi, fo r 1 i j 1 n 2,
SiS^iSi = s
i+1
SiSi+i, fo r i = 1,2,... ,n - 2 .
The reade r i s invited t o prove that thi s doe s indeed giv e a presentation o f & n. Th e
second an d thir d relation s ar e calle d th e brai d relation s of 6
n
.
l
http://dx.doi.org/10.1090/ulect/015/01
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