Minimal length expressions

We make a precise definition of length on the words inW(sln(C)).The strong exchange condition tells us totally what the words can be under a certain condition (Corol- lary3.1.18) whereas the deletion condition tells us any expression can be reduced. Definition 3.1.15. Given a Coxeter system (W, S), every element w ∈ W can be expressed as a product of generators:

w=si1si2. . . sin where sij ∈S.

If n is the minimal among all such expressions for w, we write

`(w) =n

andnis theBruhat lengthofw. Consequently,si1si2. . . sin is areduced expression

for w.

Remark 3.1.16. The length function on the word is always non-negative. If e is the identity element in W, we may define `(e) := 0.

Theorem 3.1.17(Strong Exchange Condition). Supposew=si1si2. . . sin withsij ∈S

and t∈T. If`(tw)< `(w), then

tw=si1. . .scij. . . sin

Proof. Note that the condition

tw=si1. . .scij. . . sin is equivalent to show that

η(w, t) =−1.

If this is the case, then

t=si1si2. . . sij. . . si2si1 from some j. It follows thattw=si1. . .scij. . . sin.

First, suppose that`(tw)< `(w). We want to show that the formula from Lemma3.1.12,

η(w, t) = (−1)n(si₁si₂...sin,t)_=−1. In order to achieve a contradiction, assume η(w, t) = 1. Then,

π(tw)−1(t, ) =π_w−1πt(t, ) by Lemma3.1.12(1), =πw−1(t,−) by Lemma3.1.12(2), = (w−1tw,−η(w, t))

= (w−1tw,−) by assumption.

This implies that η(tw, t) =−1 which meansn(tw, t) is odd. Consequently, we obtain

`(ttw)< `(tw)⇐⇒`(w)< `(tw) contradicting our assumption.

QED Corollary 3.1.18. If w = si1si2. . . sin is reduced and t ∈ T, then the following are

equivalent:

(1) `(tw)< `(w),

(2) tw=si1. . .scij. . . sin, for some 1≤j≤n,

(3) t=si1si2. . . sij. . . si2si1,for some 1≤j≤n.

Furthermore, the index “ij” appearing in (ii) and (iii) is uniquely determined. Proof. (1)⇒ (2) It follows from the strong exchange property as proven before.

(2)⇒ (1) This is obvious. (2)⇔ (3) It is also easy to see.

The uniqueness of the indexij follows immediately from Proposition 3.1.11. QED

Inspired by these theorems, we establish the following definition. Definition 3.1.19. We call the set of left associated reflectionsto w,

TL(w) :={t∈T | `(tw)< `(w)} and the set of right associated reflectionsto w,

TR(w) :={t∈T |`(wt)< `(w)}. Remark 3.1.20. Note that TR(w) =TL(w−1).

Corollary 3.1.21. Let w=si1si2. . . sin be reduced. Then,

TL(w) ={si1si2. . . sij. . . si2si1 |1≤j ≤n} with |TL(w)|=`(w).

Proof. Write w=si1si2. . . sin with`(w) =n. By Corollary 3.1.18,

TL(w) ={si1si2. . . sij. . . si2si1 |1≤j≤n}.

Moreover, by Proposition3.1.11, the elements inTL(w) are all different from each other. QED Corollary 3.1.22. For all s∈S and w∈W, the following hold:

(1) s∈TL(w) if and only if some reduced expression for w begins with the letter s. (2) s∈TR(w) if and only if some reduced expression for w ends with the letter s. Proof. (1) (⇐=) If some reduced expression forwbegins with the letters, thenssurely satisfies the definition of TL(w).

(=⇒) Suppose `(sw) < `(w) with w = si1si2. . . sin. Then, by Corol- lary3.1.18,

sw=si1. . .scij. . . sin for someij. Since sis a Coxetor generator,

w=ssi1. . .scij. . . sin, as desired.

(2) (⇐⇒) Apply the proof of (1) on the word w−1.

QED Definition 3.1.23. Given a Coxeter syetem (W, S). For w ∈ W,the left descent set DL(w) is defined as the intersection TL(w)∩S. Similarly, the right descent set

Proposition 3.1.24. (Deletion Condition) Ifw=si1si2. . . sin and `(w)< k, then

w=si1. . .scij. . .scik. . . sin

for some 1≤j < k≤n.

Proof. Pickj maximal so that sijsij+1. . . sin is not reduced. Obviously,

`(sijsij+1. . . sin)< `(sij+1. . . sin). So, by Theorem3.1.17,

sijsij+1. . . sin =sij+1. . .scik. . . sin

for somej < k≤n. Now, multiplysi1si2. . . sij−1 on both side, we get

si1si2. . . sin =si1. . .scij. . .scik. . . sin, as desired.

QED Definition 3.1.25. Asubexpressionof an expressionsi1si2. . . sin in a Coxeter group

W is an expression of the form sj1sj2· · ·sjk where i1 ≤j1<· · ·< jk≤in.

Corollary 3.1.26. (i) Any expressionsi1si2. . . sin contains a reduced expression for

w as a subexpression, obtainable by deleting an even number of alphabets.

(ii) Supposew=si1si2. . . sin =s 0 i1s 0 i2. . . s 0

in are two reduced expression. Then, the set

of letters appearing in the word si1si2. . . sin equals the set of alphabets appearing

in s0_i1s0_i2. . . s0_in.

(iii) S is a minimal generating set for W; no Coxeter generator can be expressed in terms of the others.

Proof. (i) This is an easy consequence from the Proposition3.1.24.

(ii) Define S:={si1, . . . , sin}. Choose j minimal such thatsij is not in S. Then,

si1si2. . . sij. . . si2si1 =s 0 i1s 0 i2. . . s 0 ij· · ·s 0 i2s 0 i1

for somej by Corollary 3.1.18. So,

sij =sij−1. . . si1s 0 i1s 0 i2. . . s 0 ij. . . s 0 i2s 0 i1si1. . . sij−1

which reduced to an alphabets inS since every alphabets on the right is inS. However,

sij ∈S contradicts the assumption that sij ∈/S.

(iii) This follows immediately from (ii) whenw is a alphabet.

Definition 3.1.27. Given a Coxeter system (W, S), denote by αs,s0 the alternating

expression ss0ss0s . . . of finite length m(s, s0). Then, the deletion of a factor of the formss is called a nil-movewhile the replacement of a factor αs,s0 by α_s0_,s is called a

braid-move.

Theorem 3.1.28. (Expression Property) Let (W, S) be a Coxeter group and w∈W. (i) Any expression si1si2. . . sin for w can be transformed into a reduced expression

for w by a sequence of nil-moves and braid-moves.

(ii) Every two reduced expressions forwcan be connected via a sequence of braid-moves. Proof. Please refer to the proof of Theorem 3.3.1 in [BB05].

QED Remark 3.1.29. This theorem will be our powerful tool to manipulate expressions in our Coxeter system. In later chapter, it is extremely useful in proofs.