Parabolic Subgroups and Cosets

Coxeter Groups

2.3 Consequences of the Deletion Condition

2.3.2 Parabolic Subgroups and Cosets

Assume throughout this subsection that (W, S) satisfies the equivalent condi- tions (D), (E), and (F). We will derive some important consequences, mostly involving standard parabolic subgroups and cosets. These are defined exactly as in the case of finite reflection groups:

Definition 2.12.For any subset J ⊆S, we denote byW_J the subgroupJ generated byJ. We callWJastandard parabolic subgroup, or simply astandard subgroup. Any cosetwWJ will be called a standard coset.

Proposition 2.13.The function J → WJ is a poset isomorphism from the set of subsets of S to the set of standard subgroups ofW, where both sets are ordered by inclusion. The inverse is given by W→W∩S.

Proof. Consider the map from standard subgroups to subsets of S given by W →W∩Sfor any standard subgroupW. It is clear thatW =W∩Sif W is a standard subgroup. It is also clear thatJ ⊆WJ∩Sfor anyJ ⊆S. To prove the opposite inclusion, suppose s∈WJ∩S. Then we can expresssas a J-word and repeatedly apply the deletion condition until the word’s length has been reduced to 1; thus s ∈ J. Hence J =WJ ∩S, and our two maps are inverses of one another. Finally, both maps clearly preserve inclusions, so

they are poset isomorphisms.

Our next observation is that when dealing with elements w of standard subgroups, we can writel(w) without ambiguity.

Proposition 2.14.Let WJ be a standard subgroup, where J ⊆ S. For any w∈WJ,

lJ(w) =lS(w). Proof. Suppose we have aJ-reduced decomposition

w=s₁· · ·s_d. (2.2) Thussi∈J for eachiand there is no shorterJ-word representingw. We must show that there is no shorter S-word representingw. If there were a shorter S-word representingw, then we could get one by deleting two letters in (2.2).

But this would contradict the assumption that the decomposition in (2.2) is

J-reduced.

Here is another easy, but very useful, consequence of the deletion condition:

Lemma 2.15.Given J ⊆ S, w ∈ WJ, and s ∈ S J, we have l(sw) = l(w) + 1.

Proof. Choose a reduced decomposition w = s₁· · ·s_l with s_i ∈ J for all i.

Suppose l(sw) < l(w). Then w =ss₁· · ·ˆs_i· · ·s_l for somei by the exchange condition. This implies thats∈WJ∩S=J, where the equality follows from Proposition 2.13. But this contradicts our hypothesis thats /∈J, so we must have l(sw)≥l(w) and hencel(sw) =l(w) + 1.

This leads to the following useful result:

Proposition 2.16.For any w∈W there is a subset S(w)⊆S such that all reduced decompositions of w involve precisely the letters in S(w). Moreover, S(w)is the smallest subsetJ ⊆S with w∈WJ.

Proof. Both assertions will follow if we prove the following: Given two decom- positions w = s1· · ·sl = t1· · ·tr with the one on the left reduced, each si

is equal to some tj. We argue by induction on l = l(w), which may be as- sumed>0. LetJ ={t1, . . . , tr}. Sincew∈WJ andl(s1w)< l(w), the lemma implies thats1∈J. Nows2· · ·sl=s1w∈WJ, so we also haves2, . . . , sl∈J

by the induction hypothesis.

Proposition 2.17.Fix w ∈ W and let J := {s∈S |l(sw)< l(w)}. Then every reduced J-word can occur as an initial subword of a reduced decomposi- tion of w. Hence

l(ww) =l(w)−l(w) (2.3)

for every w∈W_J. In particular, the length function is bounded onW_J. Proof. Lett1· · ·tl be a reduced J-word. Arguing by induction onl, we may assume that we have a reduced decomposition

w=t₂· · ·t_ls₁· · ·s_r.

Sincel(t₁w)< l(w), we can exchange one of the letters in this decomposition for a t₁ in front. The exchanged letter cannot be a t_i, since that would con- tradict the assumption that the wordt₁· · ·t_lis reduced. So it must be ans_i; hence

w=t1· · ·tls1· · ·ˆsi· · ·sr.

This proves the first assertion of the proposition. Applying it to aJ-reduced decomposition of w⁻¹, we obtain (2.3). Finally, (2.3) shows that the length

function on W_J is bounded byl(w).

Note that the last assertion of the proposition implies thatWJ is finite if J is finite. Of courseJ is automatically finite ifS is finite, which it is in most applications of the theory. But Proposition 2.16 implies thatJ is finite even if Sis infinite, sinceJ ⊆S(w), and the latter is obviously finite. Proposition 2.17 therefore has the following consequence:

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Corollary 2.18.WithJ as in Proposition 2.17, the groupWJ is finite.

Here is an important special case:

Corollary 2.19.W is finite if and only if it has an element w0 such that l(sw0)≤l(w0) for all s∈S. In this case w0 has maximal length and is the unique element of maximal length, and it has order2. Moreover,

l(ww0) =l(w0)−l(w) (2.4)

for all w∈W.

Proof. IfW is finite, it obviously has an elementw0 of maximal length, and then necessarilyl(sw0)≤l(w0) for alls∈S. Conversely, ifw0 is an element such that l(sw0)≤l(w0) for all s, then in factl(sw0) < l(w0) for all s(see Section 2.3.1), so W is finite by Corollary 2.18, and equation (2.4) follows from (2.3). Takingw=w0 in (2.4), we see thatw²₀= 1. And taking w=w0

in (2.4), we see thatl(w)< l(w0), sow0has maximal length and is the unique

element of maximal length.

We have already encounteredw₀ several times in Chapter 1, starting in Section 1.5.2. See Exercise 1.59 for a geometric explanation of the fact that w₀ is characterized by the inequalityl(sw₀)< l(w₀) for alls. [Note that this can also be written as l(w0s)< l(w0) for all s.]

Next, we show how the deletion condition leads to an interesting result about standard cosets.

Proposition 2.20.Let WJ be a standard subgroup (J ⊆S). Then every left cosetwWJ has a unique representativew1 of minimal length. It is character- ized by the property

l(w₁s) =l(w₁) + 1 (2.5)

for all s∈J. Moreover,

l(w₁w_J) =l(w₁) +l(w_J) (2.6) for all wJ ∈WJ.

Proof. Choosew₁of minimal length in the coset. Thenl(w₁s)≥l(w₁) for all s∈J; hence (2.5) holds. To prove (2.6) (which implies the uniqueness ofw₁), choose a reduced decomposition w₁ = s₁· · ·s_l, and consider an arbitrary element wJ ∈ WJ and an arbitrary reduced J-decomposition wJ = t1· · ·tr

(ti ∈J). We must show that the words1· · ·slt1· · ·tr is reduced.

If s1· · ·slt1· · ·tr is not reduced, then we can delete two letters. Neither deleted letter can be ansi, since that would yield an element ofw1WJ shorter thanw1. On the other hand, the deleted letters cannot both betj’s, since that would yield a shorter decomposition of wJ. So we have a contradiction, and the decompositionw1wJ=s1· · ·slt1· · ·tr is indeed reduced.

Finally, ifwJ = 1, then r >0 andl(w1wJtr)< l(w1wJ). Thus w1 is the unique element of the coset satisfying (2.5) for alls∈J.

Definition 2.21.Given s ∈ S, we say that an element w ∈ W is (right) s-reduced ifl(ws) =l(w)+1. GivenJ⊆S, we say thatwis (right)J-reduced if it iss-reduced for alls∈J. One defines “lefts-reduced” and “leftJ-reduced”

elements similarly. Finally, given two subsetsJ, K ⊆S, we say that an element w∈W is (J, K)-reduced if it is leftJ-reduced and rightK-reduced.

Thus the proposition says that the minimal-length representative of a left W_J-coset is the unique right J-reduced element of that coset. Left-reduced elements are similarly related to right cosetsW_Jw, and, as we will see, (J, K)- reduced elements are related to double cosets W_JwW_K.

Remark 2.22.To get some geometric intuition for Proposition 2.20, sup- poseW is a finite reflection group. Then standard cosetswW_J correspond to simplices A, and coset representatives correspond to chambers D ≥A. The representative of minimal length corresponds to the chamber D₁ ≥ A that is closest to the fundamental chamber C, i.e., D₁ =AC (see Section 1.4.6).

Moreover, equation (2.6) is a restatement of the gate property of Exercise 1.42;

see Exercise 2.27 below.

It turns out that general productsAB(withB not necessarily a chamber) are related to double cosets. We will explain this in the next chapter in a more general setting where the groups are not necessarily finite. Our treatment will make use of the following generalization of Proposition 2.20 to double cosets:

Proposition 2.23.LetW_J andW_K be standard subgroups (J, K ⊆S). Then every(WJ, WK)-double cosetWJwWK has a unique representativew1of min- imal length. It is (J, K)-reduced and is the unique (J, K)-reduced element of the double coset. Moreover, every elementwof the double cosetWJw1WK can be written as

w=wJw1wK (2.7)

with wJ ∈WJ, wK∈WK, and

l(w) =l(w_J) +l(w₁) +l(w_K). (2.8) The proof will use the following consequence of (D):

Lemma 2.24.Let J and K be subsets of S, and let w1 be an element of minimal length in its double coset WJw1WK. Suppose u∈WJ and v ∈WK

are elements such that l(uw1v)< l(u) +l(w1) +l(v). Given reduced decom- positions u = s1· · ·sl and v = t1· · ·tr, we have uw1v = uw1v, where u = s1· · ·sˆi· · ·sl and v = t1· · ·ˆtj· · ·tr for some indices 1 ≤ i ≤ l and 1≤j≤r.

Proof. Consider the decomposition ofuw1v obtained by combining the given decompositions ofuandvwith a reduced decomposition ofw1. By hypothesis this is not reduced, so we can delete two letters. The assumption onw1implies that neither of the deleted letters can involve the w1-part of the word. Since we used reduced decompositions of u andv, the only possibility is that one

deleted letter is ansi and the other is a tj.

84 2 Coxeter Groups

Proof of the proposition. Choose w1 of minimal length in the double coset.

Then w₁ is trivially (J, K)-reduced. Next, repeated applications of Lemma 2.24 show that any element w ∈ W_Jw₁W_K can be written as in (2.7) in such a way that (2.8) holds; this implies that w₁ is the unique element of minimal length. Finally, ifw_Jin (2.7) is nontrivial thenwis not leftJ-reduced, and if w_K is nontrivial thenw is not rightK-reduced. Hencew₁ is the only

(J, K)-reduced element of the double coset.

We close with a technical result that will be needed in Chapter 5. It will actually fall out of our treatment of products in Section 3.6.4 (see Exer- cise 3.114), but we present here a purely group-theoretic proof.

Lemma 2.25.With the notation of Proposition 2.23, WJ∩w1WKw⁻₁¹=WJ1 , whereJ₁:=J∩w₁Kw⁻₁¹.

Proof. Givenu∈WJ∩w1WKw⁻¹₁ , we must show thatu∈WJ1. Equivalently, givenu∈WJ andv∈WK such thatuw1v=w1, we must show thatu∈WJ1. Note first that by repeated applications of Lemma 2.24, we have l(u) =l(v).

Write u = s1· · ·sn with s1, ..., sn ∈ J and n = l(u) = l(v). We show by induction onnthat si∈w1WKw⁻₁¹ for alli.

We havel(snw1v)< l(sn) +l(w1) +l(v) = 1 +l(w1) +n, since otherwise uw₁v= (s₁· · ·s_n−1)(s_nuv) would have length≥l(s_nw₁v)−(n−1) =l(w₁)+2.

We can now apply Lemma 2.24 again to get s_nw₁v = w₁v for some v ∈ W_K and hence s_n ∈ w₁W_Kw⁻₁¹. But we also have w₁ = uw₁v = (s₁· · ·s_n−1)s_nw₁v = (s₁· · ·s_n−1)w₁v. So we can apply the induction hy- pothesis to deduce that also s₁, . . . , s_n−1 ∈ w₁W_Kw⁻¹₁ . Finally, we observe that s=w₁xw⁻¹₁ with s∈J and x∈W_K impliesl(x) = 1 by the argument at the beginning of the proof (sw1x⁻¹ =w1). So indeed we havesi ∈J1 for

alli; henceu∈WJ1.

Exercises

Assume throughout these exercises that (W, S) satisfies the deletion condition.

2.26.Given two standard subgroups WJ, WK (J, K ⊆ S), show that their intersection WJ∩WK is the standard subgroupW_J∩K. Generalize to an ar- bitrary family of standard subgroups.

2.27.IfW is a finite reflection group, rewrite equation (2.6) to get the gate property of Exercise 1.42.

2.28.Let w =s₁· · ·s_n, where the s_i are distinct elements of S. Show that l(w) =n.

2.29.Assume that the Coxeter diagram of (W, S) has no isolated nodes. Show that every proper standard subgroup ofW has index at least 3.

2.30.Suppose that the elements ofScan be enumerated ass1, . . . , s_nso that m(s_i, s_i+1)>2 fori= 1, . . . , n−1. Thus the Coxeter diagram contains a path of lengthn−1. Set w_i:=s₁s₂· · ·s_i fori= 0, . . . , n.

(a) Show that the word definingw_i is reduced, i.e.,l(w_i) =i.

(b) Ifj=i, show thatl(w_is_j) =i+ 1, i.e.,w_i is rights_j-reduced in the sense of Definition 2.21.

2.31.LetW be the finite reflection group of type A_n−1(symmetric group on nletters) with its standard generatorss₁, . . . , s_n−1, and setJ :=S{s₁}.

(a) Show that the right J-reduced elements are the n elements w_i :=

s_i· · ·s₂s₁ (i= 0, . . . , n−1).

(b) List the leftJ-reduced elements ofW and the (J, J)-reduced elements.

(c) Generalize to the casen=∞.