1. TH E SYMMETRI C GROU P 3
Next, suppos e tha t t eT. The n t = s
ix
. .. s
il
_1ilSil_1... s s
i:L
, for som e s
io
G 5,
where thi s expressio n i s reduced. Therefore , b y the las t paragraph ,
iV [t) = ^S{
1
, Si
1
Si
2
Si1, . . . , Sj1 . . . Si
l
_1Si
t
S{
1
_1. . . Si1 j .
In particular , thi s show s that t G N(t) fo r al l t eT.
Finally we prove (ii) . Le t N(w) = {t\, ..., tk] a s above. Then , fo r a = 1,..., /c,
£aw = Si
±
... si
a
.. . s*fc; so £(taw) £(w) . Therefore ,
(ii)/ iV(w ) C { t T | £(to ) ^(w ) } .
To prove th e converse , le t t G T an d suppos e tha t t £ N(w). Now , b y Lemm a 1.2,
iV(tty) = N(t)+tN(w)t an d w e have just see n tha t t G iV(£); therefore, t G iV(tty)
since t £ tN(w)t. Consequently , £(t tw) £{tw) b y (ii) ' applie d t o N(tw); tha t
is, £(w) £{tw). Thi s complete s th e proo f o f (ii ) an d henc e th e Proposition .
Prom th e definition s an d par t (ii ) o f the Propositio n w e deduce th e following .
1.4 Corollar y Suppose that w G (5n and that Si G S. Then
0(
x J ^ M + 1, if(i)w(i + l)w,
v }
\l(w)-l, if(i)w ( i + l)w .
There i s a similar descriptio n o f £{wsi).
1.5 Theore m (Th e stron g exchang e condition ) Le t s^,..., Si
k
be elements of
S and suppose that t eT and that £{tsi
r
.. . Si k) £(si1 .. . Si k). Then
tSi1 . . . Sik Si
1
. . . Sia . . . Si
k
for some a. Furthermore, t = S*, ... si ,Si Si . . . . 5 ^ .
Proof. Le t w = s
ix
.. . s
ik
and , fo r 1 a fc, let £
a
= s
i]L
... s
ia
__1siasia_1 .. . s
ix
as in the proof o f the Proposition 1.3. The n N(w) {ti}-j \-{tk} b y Lemma 1.2
and t G iV(w) b y Propositio n 1.3(h). Therefore , t = t
a
fo r som e a an d everythin g
follows.
Notice tha t i n th e stron g exchang e conditio n w e do no t assum e tha t s^ .. . Sik
is reduced . Th e stron g exchang e conditio n ha s a n equivalen t righ t han d versio n
which w e leave a s a n exercise . I f we assume i n th e statemen t o f Theore m 1.5 tha t
t G S the n th e correspondin g resul t i s known a s the exchang e condition .
1.6 Corollar y (Th e deletio n condition ) Suppose that w G 6n , £{w) k and
that w = s
tl
.. . Sik for some Si
j
G S.
(i) There exist integers 1 a b k such that w = s ^ .. . si
a
.. . s^
b
... Si k.
(ii) A reduced expression for w can be obtained from 5 ^ .. . Si
k
by deleting an even
number of the simple reflections Si
j
.
Proof. Le t a be maximal such that £(si
a
. .. S{
k
) £(si
a+1
.. . Si
fc
); such an a exists
with 1 a k becaus e £(w) k. The n Si
a
... Sik = s^ a+1.. . sib ... s
ik
fo r som e
b with a b k b y takin g £ = s
a
i n th e stron g exchang e condition . Therefore ,
w = Si
1
... sia .. . ?ib ... s^
fc
, provin g (i) . Par t (ii ) i s now immediate .
1.7 Corollar y Suppose that w G 6n an d s e S. Then
(i) £(sw) £(w) if and only if w has a reduced expression beginning with s;
(ii) £(ws) £(w) if and only if w has a reduced expression ending with s.
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