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).
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