Involution products in Coxeter groups

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Hart, Sarah and Rowley, P.J. (2011) Involution products in Coxeter groups.

Journal of Group Theory 14 (2), pp. 251-259. ISSN 1435-4446.

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Involution products in Coxeter groups

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Hart, S.B.; Rowley, P.J. (2011)

Involution products in Coxeter groups

J. Group Theory14(2011), 251–259 DOI 10.1515/JGT.2010.053

Journal of Group Theory (de Gruyter 2011

Involution products in Coxeter groups

S. B. Hart and P. J. Rowley

(Communicated by C. W. Parker)

Abstract.ForW a Coxeter group, let

W¼ fwAWjw¼xywherex;yAW andx2¼1¼y2g:

It is well known that ifW is finite thenW ¼W. Suppose that wA_W. Then the minimum value oflðxÞ þlðyÞ lðwÞ, wherex;yAW withw¼xyandx2_¼1¼y2_{, is called theexcess}

ofw(lis the length function ofW). The main result established here is thatwis alwaysW -conjugate to an element with excess equal to zero.

1 Introduction

The study of a Coxeter group W frequently weaves together features of its root system F and properties of its length function l. The delicate interplay between

lðw1w2Þ and lðw1Þ þlðw2Þ for variousw1;w2AW is often to be seen in investiga-tions into the structure ofW. Instances of the additivity of the length function, that islðw1w2Þ ¼lðw1Þ þlðw2Þ, are of particular interest. For example, ifWJ is a

stan-dard parabolic subgroup ofW, then there is a setXJ of so-called distinguished right

coset representatives for WJ in W with the property that lðwxÞ ¼lðwÞ þlðxÞ for

all wAWJ, xAXJ ([6, Proposition 1.10]). There is a parallel statement to this for

double cosets of two standard parabolic subgroups of W ([5, Proposition 2.1.7]). Also, whenWis finite it possesses an elementw0, the longest element ofW, for which

lðw0Þ ¼lðwÞ þlðww0Þfor allwAW ([5, Lemma 1.5.3]).

Of the involutions (elements of order 2) inW, the reflections, and particularly the fundamental reflections, more often than not play a major role in investigating W. This is due to there being a correspondence between the reflections in W and the roots inF. In general, involutions occupy a special position in a group and it is some-times possible to say more about them than it is about other elements of the group.

This is true in the case of Coxeter groups. For example, Richardson [7] gives an e¤ective algorithm for parameterizing the involution conjugacy classes of a Coxeter group. In something of the same vein we have the fact that an involution can be ex-pressed as a canonical product of reflections (see Deodhar [4] and Springer [8]).

Suppose thatW is a Coxeter group (not necessarily finite or even of finite rank), and put

W¼ fwA_{Wjw¼xywhere x};yA_{W andx}2_{¼1¼ y}2_g:

That is, W is the set of strongly real elements of W. For wA_W we define the excessofw,eðwÞ, by

eðwÞ ¼minflðxÞ þlðyÞ lðwÞ jw¼xy;x2¼y2¼1g:

ThuseðwÞ ¼0 is equivalent to there existing x;yA_{W with x}2_¼1¼y2_{, w¼xy} andlðwÞ ¼lðxÞ þlðyÞ. We shall call ðx;yÞ, where x, y are involutions, a spartan pair for w if w¼xy with lðxÞ þlðyÞ lðwÞ ¼eðwÞ. As a small example take w¼ ð1234Þin Symð4ÞGWðA3Þ(the Coxeter group of typeA3). ThenlðwÞ ¼3 and wcan be written in four ways as a product of involutions (see Table 1).

ThuseðwÞ ¼2 withðð24Þ;ð12Þð34ÞÞandðð12Þð34Þ;ð13ÞÞbeing spartan pairs forw. To give some idea of the distribution of excesses we briefly mention two other exam-ples. The number of elements with excess 0, 2, 4, 6, 8 in Symð6ÞGWðA5Þis, respec-tively, 489, 173, 46, 10, 2, the maximum excess being 8. For Symð7ÞGWðA6Þ, the number of elements with excess 0, 2, 4, 6, 8, 10, 12 is, respectively, 2659, 1519, 574, 228, 50, 8, 2 and here the maximum excess is 12.

In the case whenW is finite we have W¼W, and so excess is defined for every element ofW. (SinceW is a direct product of irreducible Coxeter groups, it su‰ces to check this forW an irreducible Coxeter group. IfW is a Weyl group see Carter [3]. The case when W is a dihedral group is straightforward to verify while types H3 orH4 may be checked using [2].) However ifW is infinite we can haveW0W. This can be seen when W is of type AAf22. Then W¼HN, the semidirect product of N¼ fðl1;l2;l3Þ jliAZ;l1þl2þl3¼0gGZZ and HGWðA2ÞGSymð3Þ withH acting onN by permuting the co-ordinates ofðl1;l2;l3Þ. Let g¼ ð12ÞAH

Table 1. w¼ ð1234Þ ¼xy

x y lðxÞ þlðyÞ

(13) (14)(23) 3þ6¼9 (14)(23) (24) 6þ3¼9 (24) (12)(34) 3þ2¼5 (12)(34) (13) 2þ3¼5 252 S. B. Hart and P. J. Rowley

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and 00_lA_{Z, and set w¼gðl};l;2lÞ. Clearly g and ðl;l;2lÞ commute and

ðl;l;2lÞhas infinite order. Thereforewhas infinite order. Ifwcan be written as a product of two involutions, then there existhm;knA_{W withh};kA_{H, m};nA_{N and}

ðhmÞðknÞ ¼w. Thereforeh,kare self-inverse elements ofHwithhk¼ ð12Þ. So one of horkisð12Þand the other is the identity. ButNhas no elements of order 2, so either hmorknis the identity, contradicting the fact thatwis not an involution. So we con-clude thatwcannot be expressed as a product of two involutions and henceW0_W. As we have observed, a Coxeter group may have many elements with non-zero ex-cess. Nevertheless our main theorem shows the zero excess elements are ubiquitous from a conjugacy class viewpoint.

Theorem 1.1.Suppose that W is a Coxeter group,and let wA_W. Let X denote the W -conjugacy class of w. Then there exists wAX such that eðwÞ ¼0.

We prove Theorem 1.1 in Section 3, after gathering together a number of prepara-tory results about Coxeter groups in Section 2. Also some easy properties of excess are noted and, in Proposition 2.7, we demonstrate that there are Coxeter groups in which elements can have arbitrarily large excess.

2 Background results and notation

Assume, for this section, thatW is a finite rank Coxeter group. So, by its very defini-tion,W has a presentation of the form

W ¼hRj ðrsÞmrs_¼1;r;sARi

where Ris finite,mrr¼1,mrs¼msrAZþUfyg andmrsd2 forr;sARwithr0s.

The elements ofRare called the fundamental reflections ofW and the rank ofW is the cardinality ofR. The length of an elementwofW, denoted bylðwÞ, is defined to be

lðwÞ ¼ minfljw¼r1r2. . .rl:riARg if w01;

0 if w¼1:

Now letV be a real vector space with basisP¼ farjrARg. Define a symmetric

bilinear formh;i_onVby

h_ar;asi¼ cos

p mrs

;

where r;sARand themrsare as in the above presentation ofW. Lettingr;sARwe

define

ras¼as2har;asiar:

This then extends to an action ofW on V which is both faithful and respects the bilinear form h;i _{(see [6, (5.6)]) and the elements of R act as reflections uponV.} The moduleV is called a reflection module forW and the subset

F¼ fwarjrAR;wAWg

ofV is the all important root system ofW. Setting

Fþ¼ X

rAR

lrarAFjlrd0 for allr

and F¼ Fþ

we have the basic fact that F is the disjoint union FþUU_F (see [6, (5.4)–(5.6)]). Elements ofFþ andF are referred to, respectively, as positive and negative roots ofF.

ForwA_{W we define}

NðwÞ ¼ faA_Fþ_jwaA_Fg:

The connection between lðwÞ and the root system of W is contained in our next lemma.

Lemma 2.1.Let wA_{W and r}A_R.

(i) IflðwrÞ>lðwÞthen warAFþand iflðwrÞ<lðwÞthen warAF. In

particu-lar,lðwrÞ<lðwÞif and only ifarANðwÞ.

(ii) lðwÞ ¼ jNðwÞj.

Proof.See [6, §§5.4, 5.6]. r

Lemma 2.2.Let g;hA_{W . Then}

NðghÞ ¼NðhÞnðh1NðgÞÞU_h1_{ðNðgÞnNðh}1_ÞÞ:

HencelðghÞ ¼lðgÞ þlðhÞ 2jNðgÞVNðh1_Þj.

Proof.Let aA_Fþ_{. Suppose that a}A_{NðhÞ. Thengha¼g ðhaÞis negative if and} only ifhaB_{NðgÞ. That is,a}B_h1_{NðgÞ. Thus}

NðghÞVNðhÞ ¼NðhÞnh1NðgÞ:

If on the other hand aB_{NðhÞ, then gha}A_{F if and only if ha}A_{NðgÞ. That is,} aA_FþV_h1_{NðgÞ. Thus}

NðghÞnNðhÞ ¼h1½ðNðgÞnNðh1ÞÞ:

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Combining the two equations gives the expression for NðghÞstated in Lemma 2.2. Since NðghÞV_{NðhÞ and NðghÞnNðhÞ are clearly disjoint, using Lemma 2.1 (ii) we} deduce that

lðghÞ ¼lðgÞ þlðhÞ 2jNðgÞV_Nðh1_Þj;

which completes the proof of the lemma. r

Proposition 2.3.Let wA_{W and r}A_{R. IflðrwÞ}<lðwÞandlðwrÞ<lðwÞ,then either rwr¼w orlðrwrÞ ¼lðwÞ 2.

Proof.See [6, §5.8, Exercise 3]. r

ForJJRdefineWJto be the subgroup ofW generated byJ. Such a subgroup of

W is referred to as a standard parabolic subgroup. Standard parabolic subgroups are Coxeter groups in their own right with root system

FJ ¼ fwarjrAJ;wAWJg

(see [6, §5.5] for more on this). A conjugate of a standard parabolic subgroup is called a parabolic subgroup ofW. Finally, acuspidalelement ofW is an element which is not contained in any proper parabolic subgroup of W. Equivalently, an element is cuspidal if itsW-conjugacy class has empty intersection with all the proper standard parabolic subgroups ofW.

Theorem 2.4. Let00vAV . Then the stabilizer of v in W is a parabolic subgroup of W . Furthermore,if vAF,then the stabilizer of v in W is a proper parabolic subgroup of W.

Proof.The fact that the stabilizer ofvis a parabolic subgroup is proved in [1, Chapter V, §3.3]. IfvA_{F, thenv¼warfor somer}A_R,wA_{W and henceðwrw}1_{Þ v¼ v.} Thus the stabilizer ofvcannot be the whole ofW, so is a proper parabolic subgroup ofW. r

Theorem 2.5.Suppose that w is an involution in W . Then there exists JJR such that w is W -conjugate to wJ,an element of WJ which acts as1uponFJ.

Proof.See Richardson [7]. r

Next we give some easy properties of excess.

Lemma 2.6.Let wA_W. Then the following hold.

(i) If w is an involution or the identity element,then eðwÞ ¼0.

(ii) eðwÞis non-negative and even.

(iii) If w¼xy where x and y are involutions and2jNðxÞV_{NðyÞj ¼eðwÞ,thenðx};yÞis a spartan pair for w.

Proof. If w is an involution or w¼1, then w¼w1 with w2_¼1¼12_{, whence} eðwÞ ¼0. For (ii), suppose thatx2 _¼y2_{¼1 and xy¼w. Then, using Lemma 2.2,} we havelðwÞ ¼lðxÞ þlðyÞ 2jNðxÞV_{NðyÞjand hencelðxÞ þlðyÞ lðwÞis even} and (ii) follows. Part (iii) is also immediate from Lemma 2.2. r

We now have the tools needed for the proof of Theorem 1.1, but before continuing with this we calculate the excess of the elementð12. . .nÞof SymðnÞ. The aim of this is to show that there are Coxeter groups in which elements may have arbitrarily large excess. Before stating our next result we require some notation. Forqa rational num-ber,bqcdenotes the floor ofq(that is, the largest integer less than or equal toq), and

dqedenotes the ceiling ofq(that is, the smallest integer greater than or equal toq). Letnd2. Then SymðnÞis isomorphic to the Coxeter groupWðAn1Þof typeA. If W¼WðAn1ÞGSymðnÞ, then we setR¼ fð12Þ;ð23Þ;. . .;ððn1ÞnÞgand the set of positive roots isFþ¼ feiejj1ci< jcng. An alternative description oflðwÞfor

wA_{W in this case is}

lðwÞ ¼ jfði;jÞ j1ci< jcn;wðiÞ>wðjÞgj:

For 0ckcn1, letskbe the longest element of Symðf1;2;. . .;kgÞand lettkbe the

longest element of Symðfkþ1;kþ2;. . .;ngÞ. A straightforward calculation shows that

sk ¼ ð1 kÞð2 k1Þ. . .

k 2

k 2 þ1

;

and

tk ¼ ðkþ1 nÞðkþ2 n1Þ. . .

nk 2

þk nk

2

þkþ1

:

Note thats0¼s1¼tn1¼1. Finally, for 0ckcn1, set yk ¼sktk.

Proposition 2.7.Let w¼ ð12. . .nÞA_W¼WðAn1ÞGSymðnÞ. Put

Iw¼ fxAWjx2¼1;wx¼w1g:

Then

(i) Iw¼ fykj0ckcn1g.

(ii) If n is odd,thenðwyðn1Þ=2;yðn1Þ=2Þis a spartan pair for w.

(iii) If n is even,thenðwyn=2;yn=2Þandðwyn=21;yn=21Þare both spartan pairs for w. (iv) eðwÞ ¼ bðn2Þ2=2c.

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Proof. It is easy to check that yk AIw whenever 0ckcn1. Since

jIwjcjCWðwÞj ¼n, part (i) follows immediately.

Suppose that yA_{Iw. Write}

eyðwÞ ¼lðwyÞ þlðyÞ lðwÞ; so that eðwÞ ¼minfeyðwÞ jyAIwg:

We have

eyðwÞ ¼lðwyÞ þlðyÞ lðwÞ

¼lðwÞ þlðyÞ 2jNðyÞV_{NðwÞj þlðyÞ lðwÞ}

¼2ðlðyÞ jNðyÞV_NðwÞjÞ:

NowNðwÞ ¼ feienj1ci<ng. Therefore

jNðyÞV_{NðwÞj ¼ jfijyðiÞ}>yðnÞgj ¼nyðnÞ:

Let y¼yk for some 0ckcn1. We have

lðykÞ ¼lðskÞ þlðtkÞ ¼

kðk1Þ

2 þ

ðnkÞðnk1Þ

2 :

MoreoverjNðykÞVNðwÞj ¼nykðnÞ ¼n ðkþ1Þ. Therefore

eykðwÞ ¼2ðlðykÞ jNðykÞVNðwÞjÞ

¼kðk1Þ þ ðnkÞðnk1Þ 2ðnk1Þ

¼2k2_{2kðn1Þ þn}2_3nþ2

¼2 k1

2ðn1Þ

2

þ1

2ðn

2_4nþ3Þ

¼2 k1

2ðn1Þ

2

þ1

2ðn2Þ 2

1

2:

Ifnis odd, then this quantity is minimal whenk¼1

2ðn1Þ. Hence part (ii) holds. In this case,

eðwÞ ¼eyðn1Þ=2ðwÞ ¼ 1 2ðn2Þ

21 2¼

1 2ðn2Þ

2

:

Ifnis even, theneyk is minimal whenk¼n=2 ork¼n=21. Hence we obtain part

(iii), and in either case, eðwÞ ¼1 2ðn2Þ

2

. Combining the odd and even cases we see thateðwÞ ¼ bðn2Þ2=2c. r

3 Zero excess in conjugacy classes

Proof of Theorem1.1.Suppose thatW is a Coxeter group,wA_{WandX is theW} -conjugacy class ofw. Noww¼r1. . .rlfor certainriARand some finitel¼lðwÞ. So

wA h_r1;. . .;rli. Thus it su‰ces to establish the theorem forW of finite rank.

Ac-cordingly we argue by induction onjRj. Suppose thatKY_{R. IfX}V_WK0 q_{, then} by induction there exists w0A_XV_{WK with eW}

Kðw

0_{Þ ¼0, whence eðw}0_{Þ ¼0 and we}

are done. So we may suppose thatXV_WK¼q_{for allK}Y_{R. That is,X is a}

cuspi-dal class ofW.

Choose wAX. If w¼1 or w is an involution, then eðwÞ ¼0 by Lemma 2.6 (i). Thus we may suppose thatw¼xywhere xand yare involutions. By Theorem 2.5 we may conjugate w so that yAWJ for some JJR, with y acting as 1 onFJ.

Thus yAZðWJÞ. Now choosezto have minimal length infg1xgjgAWJg. So we

havez¼g1_{xgfor somegAW}

J.

Suppose for a contradiction that there existsrA_{J withlðzrÞ}<lðzÞ. Sincezandr are involutions,

lðrzÞ ¼lððrzÞ1Þ ¼lðzrÞ<lðzÞ:

Applying Proposition 2.3 yields that eitherrzr¼zorlðrzrÞ ¼lðzÞ 2<lðzÞ. Since rAWJ, the latter possibility would contradict the minimal choice ofz. Hencerzr¼z.

So r ðzarÞ ¼z ðrarÞ ¼ zar. It is well known that the only roots b for

whichrb¼ b arear andar. Thuszar¼Gar. By assumptionlðzrÞ<lðzÞand

therefore zarAF by Lemma 2.1 (i), whence zar¼ ar. Combining this with

yar¼ ar we then deduce that zyar¼ar. Then Theorem 2.4 gives that zy is

in a proper parabolic subgroup ofW. Noting thatzy¼g1_xgy¼g1_xyg¼g1_wg, as gA_{WJ, we infer that X is not a cuspidal class, a contradiction. We conclude} therefore that lðzrÞ>lðzÞ for all rA_{J. Consequently NðzÞ}V_Fþ

J ¼q. Since

NðyÞ ¼Fþ_J we deduce that NðzÞV_{NðyÞ ¼}q _{and hence, using Lemma 2.2, that}

lðzyÞ ¼lðzÞ þlðyÞ. Setting w¼zy¼g1wg we have wAX and eðwÞ ¼0, so

completing the proof of Theorem 1.1. r

References

[1] N. Bourbaki.Lie Groups and Lie Algebras, chapters 4–6 (Springer-Verlag, 2002).

[2] J. J. Cannon and C. Playoust. An introduction to algebraic programming with Magma

[draft] (1997).

[3] R. W. Carter. Conjugacy classes in the Weyl Group.Compos. Math. 25(1972), 1–59. [4] V. V. Deodhar. On the root system of a Coxeter group. Comm. Algebra 10 (1982),

611–630.

[5] M. Geck and G. Pfei¤er.Characters of finite Coxeter groups and Iwahori–Hecke algebras. London Math. Soc. Monographs, New Series 21 (Oxford University Press, 2000).

[6] J. E. Humphreys.Reflection groups and Coxeter groups. Cambridge Studies in Adv. Math. 29 (Cambridge University Press, 1990).

[7] R. W. Richardson. Conjugacy classes of involutions in Coxeter groups. Bull. Austral. Math. Soc.26(1982), 1–15.

258 S. B. Hart and P. J. Rowley

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[8] T. A. Springer. Some remarks on involutions in Coxeter groups.Comm. Algebra10(1982), 631–636.

Received 4 May, 2010

S. B. Hart, Department of Economics, Mathematics and Statistics, Birkbeck, University of London, Malet Street, London WC1E 7HX, United Kingdom

E-mail: s.hart@bbk.ac.uk

P. J. Rowley, School of Mathematics, The University of Manchester, Oxford Road, Manches-ter M13 9PL, United Kingdom

E-mail: peter.j.rowley@manchester.ac.uk