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 reduced 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 see 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 will 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 sets 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 for

some s e S an d u G 6n an d wit h £(v) = £(u) + 1. Therefore , b y induction o n the

length of v,

N(vw) = N(s(uw)) = N(s)+sN(uw)s = iV(5)+sAr(w)s4-5'uiV(ii;)u~

1s

= N^+vN^v'

1

as required . D

For convenience 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 1.17).

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 that s ^ ... 5^

f c

is a reduced expressio n for w and, for a = 1,..., fc,

let t

a

= Si

x

... Si

a

_1SiaSia_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 for a ^ b. B y wa y

of contradiction, suppos e tha t t

a

= tb for some a b. The n

^ = = t

a

tbW = S ^ . . . 52a_1 5fa Sj

a

_

1

. . . Si1Si1 . . . Sib_1SibSib_1 . . . Si-^W

= Si

l

. . . S{a . . . Sib . . . Si

k

,

where Sj indicates tha t th e transpositio n Sj is omitted. However , thi s contradict s

the assumption that w has length k so t\,..., tk must al l be distinct afte r all . Hence ,

N(w) = {ti,... ,tk} an d £{w) = k = \N(w)\ provin g (i).