4 1. 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^ .. . Sik 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 5n, write (ii,..., ik) ~ 6 (ji,. . , jk) i f one ex-
pression ca n be transformed int o the other usin g only the braid relations . Thus , ^ 5
is the equivalenc e relatio n o n fc-tuples generate d b y (... , z , j , . . .) ~ 6 (... , j , z,...) ,
when \i j \ 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 they correspon d t o reduce d
expressions o f the sam e elemen t o f S
n
; surprisingly , th e convers e i s also true .
1.8 Theore m (Matsumoto ) Suppose that s^ ,..., Si
k
and Sj ±,..., Sj
k
are ele-
ments 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 the remarks above , we need to show 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
il
Sj1.. . Sj
k
= Sj
1
...2j
a
.. . sJfe fo r som e in -
teger a by th e exchang e condition . Therefore , (i ) s;
2
.. . s
ik
= 5
JX
... 3ja ... Sj
fc
an d
(ii) Si 1Sj1 .. . Sj a_1 = Sj
1
... Sj a; further , al l four o f these expression s ar e reduced .
If a ^ fc then th e reduce d expression s i n (i ) an d (ii ) al l have length strictl y les s
thanA: . 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 by induction 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, ji, •, 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 ji| 1. The n (ii , ji) ~ 6 (ji»^i ) becaus e s^s ^ = Sj
1
Si1
is a brai d relatio n o f length 2 . Furthermore , b y (ii ) abov e an d it s mirro r image ,
SiiSjiSJ2 ' ' ' 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
.. . Sjk_1 ar e reduced expressions for the same element
in 6
n
. Consequently , (z
2
,... ,ik-i) ~ 6 0*2 ? •»Jfc-i) b y inductio n o n A: ; therefore,
(ii,ji,j
2
...,j/e-i)~ & (ii,ji,z2 ,...,2/e-i) ~b (ji , H, ^2, •. •, ik-i) a s required .
Finally, conside r th e cas e where |i i ji| = 1. Initially , we wanted t o prove tha t
(ii,..., ik) ~ & (ji,..., jk) an d we argued to 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 the last fc 1 entries in each k-
tuple with the 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 1ilSj1i s 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.
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