For W a Coxeter group, let



= {w ∈ W | w = xy where x, y ∈ W and x 2 = 1 = y 2}.



It is well known that if W is finite then W = . Suppose that w ∈ . Then the minimum value of ℓ(x) + ℓ(y) – ℓ(w), where x, y ∈ W with w = xy and x 2 = 1 = y 2, is called the excess of w (ℓ is the length function of W). The main result established here is that w is always W-conjugate to an element with excess equal to zero

Carter R. W.

P. J. Rowley

S. B. Hart

English

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Journal of Group Theory

Involution products in Coxeter groups

For W a Coxeter group, let = {w ∈ W | w = xy where x, y ∈ W and x2 = 1 = y2}. It is well known that if W is finite then W = . Suppose that w ∈ . Then the minimum value of ℓ(x) + ℓ(y) - ℓ(w), where x, y ∈ W with w = xy and x2 = 1 = y2, is called the excess of w (ℓ is the length function of W). The main result established here is that w is always W-conjugate to an element with excess equal to zero. © 2011 de Gruyter

Hart, S. B.

Rowley, P. J.

The University of Manchester - Institutional Repository

For W a Coxeter group, let\ud
\ud
= {w ∈ W | w = xy where x, y ∈ W and x 2 = 1 = y 2}.\ud
\ud
It is well known that if W is finite then W = . Suppose that w ∈ . Then the minimum value of ℓ(x) + ℓ(y) – ℓ(w), where x, y ∈ W with w = xy and x 2 = 1 = y 2, is called the excess of w (ℓ is the length function of W). The main result established here is that w is always W-conjugate to an element with excess equal to zero

Hart, Sarah B.

Rowley, P.J.

Name not available

Hart, Sarah

Birkbeck Institutional Research Online

   BIROn - Birkbeck Institutional Research Online  Enabling open access to Birkbeck’s published research output    Involution products in Coxeter groups  Journal Article   http://eprints.bbk.ac.uk/811    Version: Published (Refereed)  Citation:      © 2011 de Gruyter   Publisher version   ______________________________________________________________  All articles available through Birkbeck ePrints are protected by intellectual property law, including copyright law. Any use made of the contents should comply with the relevant law. ______________________________________________________________  Deposit Guide  Contact: lib-eprints@bbk.ac.uk Birkbeck ePrints Hart, S.B.; Rowley, P.J.  (2011) Involution products in Coxeter groups Journal of Group Theory 14(2), pp.291-299 J. Group Theory 14 (2011), 251–259DOI 10.1515/JGT.2010.053Journal of Group Theory( de Gruyter 2011Involution products in Coxeter groupsS. B. Hart and P. J. Rowley(Communicated by C. W. Parker)Abstract. For W a Coxeter group, letW ¼ fw AW jw ¼ xy where x; y AW and x2 ¼ 1 ¼ y2g:It is well known that if W is ﬁnite then W ¼W. Suppose that w AW. Then the minimumvalue of lðxÞ þ lðyÞ  lðwÞ, where x; y AW with w ¼ xy and x2 ¼ 1 ¼ y2, is called the excessof w (l is the length function of W ). The main result established here is that w is always W -conjugate to an element with excess equal to zero.1 IntroductionThe study of a Coxeter group W frequently weaves together features of its rootsystem F and properties of its length function l. The delicate interplay betweenlðw1w2Þ and lðw1Þ þ lðw2Þ for various w1;w2 AW is often to be seen in investiga-tions into the structure of W . Instances of the additivity of the length function, thatis lðw1w2Þ ¼ lðw1Þ þ lðw2Þ, are of particular interest. For example, if WJ is a stan-dard parabolic subgroup of W , then there is a set XJ of so-called distinguished rightcoset representatives for WJ in W with the property that lðwxÞ ¼ lðwÞ þ lðxÞ forall w AWJ , x A XJ ([6, Proposition 1.10]). There is a parallel statement to this fordouble cosets of two standard parabolic subgroups of W ([5, Proposition 2.1.7]).Also, whenW is ﬁnite it possesses an element w0, the longest element ofW , for whichlðw0Þ ¼ lðwÞ þ lðww0Þ for all w AW ([5, Lemma 1.5.3]).Of the involutions (elements of order 2) in W , the reﬂections, and particularly thefundamental 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 theroots in F. 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.The authors wish to acknowledge partial support for this work from the Birkbeck Faculty ofSocial Science and the Manchester Institute for Mathematical Sciences (MIMS).Brought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AMThis is true in the case of Coxeter groups. For example, Richardson [7] gives ane¤ective algorithm for parameterizing the involution conjugacy classes of a Coxetergroup. 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 that W is a Coxeter group (not necessarily ﬁnite or even of ﬁnite rank),and putW ¼ fw AW jw ¼ xy where x; y AW and x2 ¼ 1 ¼ y2g:That is, W is the set of strongly real elements of W . For w AW we deﬁne theexcess of w, eðwÞ, byeðwÞ ¼ minflðxÞ þ lðyÞ  lðwÞ jw ¼ xy; x2 ¼ y2 ¼ 1g:Thus eðwÞ ¼ 0 is equivalent to there existing x; y AW with x2 ¼ 1 ¼ y2, w ¼ xyand lðwÞ ¼ lðxÞ þ lðyÞ. We shall call ðx; yÞ, where x, y are involutions, a spartanpair for w if w ¼ xy with lðxÞ þ lðyÞ  lðwÞ ¼ eðwÞ. As a small example takew ¼ ð1234Þ in Symð4ÞGWðA3Þ (the Coxeter group of type A3). Then lðwÞ ¼ 3 andw can be written in four ways as a product of involutions (see Table 1).Thus eðwÞ ¼ 2 with ðð24Þ; ð12Þð34ÞÞ and ðð12Þð34Þ; ð13ÞÞ being spartan pairs for w.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 Symð6ÞGWðA5Þ is, respec-tively, 489, 173, 46, 10, 2, the maximum excess being 8. For Symð7ÞGWðA6Þ, thenumber 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 when W is ﬁnite we have W ¼W, and so excess is deﬁned for everyelement of W . (Since W is a direct product of irreducible Coxeter groups, it su‰cesto check this for W an irreducible Coxeter group. If W is a Weyl group see Carter[3]. The case when W is a dihedral group is straightforward to verify while typesH3 or H4 may be checked using [2].) However if W is inﬁnite we can have W0W.This can be seen when W is of type fA2. Then W ¼ HN, the semidirect productof N ¼ fðl1; l2; l3Þ j li A Z; l1 þ l2 þ l3 ¼ 0gGZ Z and HGWðA2ÞG Symð3Þwith H acting on N by permuting the co-ordinates of ðl1; l2; l3Þ. Let g ¼ ð12Þ A HTable 1. w ¼ ð1234Þ ¼ xyx y lðxÞ þ lðyÞ(13) (14)(23) 3þ 6 ¼ 9(14)(23) (24) 6þ 3 ¼ 9(24) (12)(34) 3þ 2 ¼ 5(12)(34) (13) 2þ 3 ¼ 5252 S. B. Hart and P. J. RowleyBrought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AMand 00 l A Z, and set w ¼ gðl; l;2lÞ. Clearly g and ðl; l;2lÞ commute andðl; l;2lÞ has inﬁnite order. Therefore w has inﬁnite order. If w can be written as aproduct of two involutions, then there exist hm; kn AW with h; k A H, m; n A N andðhmÞðknÞ ¼ w. Therefore h, k are self-inverse elements of H with hk ¼ ð12Þ. So one ofh or k is ð12Þ and the other is the identity. But N has no elements of order 2, so eitherhm or kn is the identity, contradicting the fact that w is not an involution. So we con-clude that w cannot be expressed as a product of two involutions and hence W0W.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 ubiquitousfrom a conjugacy class viewpoint.Theorem 1.1. Suppose that W is a Coxeter group, and let w AW. Let X denote the W-conjugacy class of w. Then there exists w A X such that eðwÞ ¼ 0.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 excessare noted and, in Proposition 2.7, we demonstrate that there are Coxeter groups inwhich elements can have arbitrarily large excess.2 Background results and notationAssume, for this section, that W is a ﬁnite rank Coxeter group. So, by its very deﬁni-tion, W has a presentation of the formW ¼ hR j ðrsÞmrs ¼ 1; r; s A Riwhere R is ﬁnite, mrr ¼ 1, mrs ¼ msr A Zþ U fyg and mrsd 2 for r; s A R with r0 s.The elements of R are called the fundamental reﬂections of W and the rank of W isthe cardinality of R. The length of an element w of W , denoted by lðwÞ, is deﬁned tobelðwÞ ¼ minfl jw ¼ r1r2 . . . rl : ri A Rg if w0 1;0 if w ¼ 1:Now let V be a real vector space with basis P ¼ far j r A Rg. Deﬁne a symmetricbilinear form h ; i on V byhar; asi ¼ cos pmrs ;where r; s A R and the mrs are as in the above presentation of W . Letting r; s A R wedeﬁner  as ¼ as  2har; asiar:Involution products in Coxeter groups 253Brought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AMThis then extends to an action of W on V which is both faithful and respects thebilinear form h ; i (see [6, (5.6)]) and the elements of R act as reﬂections upon V .The module V is called a reﬂection module for W and the subsetF ¼ fw  ar j r A R;w AWgof V is the all important root system of W . SettingFþ ¼Xr ARlrar A F j lrd 0 for all rand F ¼ Fþwe have the basic fact that F is the disjoint union Fþ _UF (see [6, (5.4)–(5.6)]).Elements of Fþ and F are referred to, respectively, as positive and negative rootsof F.For w AW we deﬁneNðwÞ ¼ fa A Fþ jw  a A Fg:The connection between lðwÞ and the root system of W is contained in our nextlemma.Lemma 2.1. Let w AW and r A R.(i) If lðwrÞ > lðwÞ then w  ar A Fþ and if lðwrÞ < lðwÞ then w  ar A F. In particu-lar, lðwrÞ < lðwÞ if and only if ar A NðwÞ.(ii) lðwÞ ¼ jNðwÞj.Proof. See [6, §§5.4, 5.6]. rLemma 2.2. Let g; h AW. ThenNðghÞ ¼ NðhÞnðh1NðgÞÞU h1ðNðgÞnNðh1ÞÞ:Hence lðghÞ ¼ lðgÞ þ lðhÞ  2jNðgÞVNðh1Þj.Proof. Let a A Fþ. Suppose that a A NðhÞ. Then gh  a ¼ g  ðh  aÞ is negative if andonly if h  a B NðgÞ. That is, a B h1NðgÞ. ThusNðghÞVNðhÞ ¼ NðhÞnh1NðgÞ:If on the other hand a B NðhÞ, then gh  a A F if and only if h  a A NðgÞ. That is,a A Fþ V h1NðgÞ. ThusNðghÞnNðhÞ ¼ h1½ðNðgÞnNðh1ÞÞ:254 S. B. Hart and P. J. RowleyBrought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AMCombining the two equations gives the expression for NðghÞ stated in Lemma 2.2.Since NðghÞVNðhÞ and NðghÞnNðhÞ are clearly disjoint, using Lemma 2.1 (ii) wededuce thatlðghÞ ¼ lðgÞ þ lðhÞ  2jNðgÞVNðh1Þj;which completes the proof of the lemma. rProposition 2.3. Let w AW and r A R. If lðrwÞ < lðwÞ and lðwrÞ < lðwÞ, then eitherrwr ¼ w or lðrwrÞ ¼ lðwÞ  2.Proof. See [6, §5.8, Exercise 3]. rFor JJR deﬁneWJ to be the subgroup of W generated by J. Such a subgroup ofW is referred to as a standard parabolic subgroup. Standard parabolic subgroups areCoxeter groups in their own right with root systemFJ ¼ fw  ar j r A J;w AWJg(see [6, §5.5] for more on this). A conjugate of a standard parabolic subgroup is calleda parabolic subgroup of W . Finally, a cuspidal element of W is an element which isnot contained in any proper parabolic subgroup of W . Equivalently, an element iscuspidal if its W -conjugacy class has empty intersection with all the proper standardparabolic subgroups of W .Theorem 2.4. Let 00 v A V. Then the stabilizer of v in W is a parabolic subgroup ofW. Furthermore, if v A F, then the stabilizer of v in W is a proper parabolic subgroupof W .Proof. The fact that the stabilizer of v is a parabolic subgroup is proved in [1, ChapterV, §3.3]. If v A F, then v ¼ w  ar for some r A R, w AW and hence ðwrw1Þ  v ¼ v.Thus the stabilizer of v cannot be the whole of W , so is a proper parabolic subgroupof W . rTheorem 2.5. Suppose that w is an involution in W. Then there exists JJR such thatw is W-conjugate to wJ , an element of WJ which acts as 1 upon FJ .Proof. See Richardson [7]. rNext we give some easy properties of excess.Lemma 2.6. Let w AW. Then the following hold.(i) If w is an involution or the identity element, then eðwÞ ¼ 0.(ii) eðwÞ is non-negative and even.Involution products in Coxeter groups 255Brought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AM(iii) If w ¼ xy where x and y are involutions and 2jNðxÞVNðyÞj ¼ eðwÞ, then ðx; yÞ isa spartan pair for w.Proof. If w is an involution or w ¼ 1, then w ¼ w1 with w2 ¼ 1 ¼ 12, whenceeðwÞ ¼ 0. For (ii), suppose that x2 ¼ y2 ¼ 1 and xy ¼ w. Then, using Lemma 2.2,we have lðwÞ ¼ lðxÞ þ lðyÞ  2jNðxÞVNðyÞj and hence lðxÞ þ lðyÞ  lðwÞ is evenand (ii) follows. Part (iii) is also immediate from Lemma 2.2. rWe now have the tools needed for the proof of Theorem 1.1, but before continuingwith this we calculate the excess of the element ð12 . . . nÞ of SymðnÞ. The aim of this isto show that there are Coxeter groups in which elements may have arbitrarily largeexcess. Before stating our next result we require some notation. For q a rational num-ber, bqc denotes the ﬂoor of q (that is, the largest integer less than or equal to q), anddqe denotes the ceiling of q (that is, the smallest integer greater than or equal to q).Let nd 2. Then SymðnÞ is isomorphic to the Coxeter group WðAn1Þ of type A. IfW ¼ WðAn1ÞG SymðnÞ, then we set R ¼ fð12Þ; ð23Þ; . . . ; ððn 1Þ nÞg and the set ofpositive roots is Fþ ¼ fei  ej j 1c i < jc ng. An alternative description of lðwÞ forw AW in this case islðwÞ ¼ jfði; jÞ j 1c i < jc n;wðiÞ > wð jÞgj:For 0c kc n 1, let sk be the longest element of Symðf1; 2; . . . ; kgÞ and let tk be thelongest element of Symðfk þ 1; k þ 2; . . . ; ngÞ. A straightforward calculation showsthatsk ¼ ð1 kÞð2 k  1Þ . . . k2 k2 þ 1 ;andtk ¼ ðk þ 1 nÞðk þ 2 n 1Þ . . . n k2 þ k n k2 þ k þ 1 :Note that s0 ¼ s1 ¼ tn1 ¼ 1. Finally, for 0c kc n 1, set yk ¼ sktk.Proposition 2.7. Let w ¼ ð12 . . . nÞ AW ¼ WðAn1ÞG SymðnÞ. PutIw ¼ fx AW j x2 ¼ 1;wx ¼ w1g:Then(i) Iw ¼ fyk j 0c kc n 1g.(ii) If n is odd, then ðwyðn1Þ=2; yðn1Þ=2Þ is a spartan pair for w.(iii) If n is even, then ðwyn=2; yn=2Þ and ðwyn=21; yn=21Þ are both spartan pairs for w.(iv) eðwÞ ¼ bðn 2Þ2=2c.256 S. B. Hart and P. J. RowleyBrought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AMProof. It is easy to check that yk A Iw whenever 0c kc n 1. SincejIwjc jCW ðwÞj ¼ n, part (i) follows immediately.Suppose that y A Iw. WriteeyðwÞ ¼ lðwyÞ þ lðyÞ  lðwÞ; so that eðwÞ ¼ minfeyðwÞ j y A Iwg:We haveeyðwÞ ¼ lðwyÞ þ lðyÞ  lðwÞ¼ lðwÞ þ lðyÞ  2jNðyÞVNðwÞj þ lðyÞ  lðwÞ¼ 2ðlðyÞ  jNðyÞVNðwÞjÞ:Now NðwÞ ¼ fei  en j 1c i < ng. ThereforejNðyÞVNðwÞj ¼ jfi j yðiÞ > yðnÞgj ¼ n yðnÞ:Let y ¼ yk for some 0c kc n 1. We havelðykÞ ¼ lðskÞ þ lðtkÞ ¼ kðk  1Þ2þ ðn kÞðn k  1Þ2:Moreover jNðykÞVNðwÞj ¼ n ykðnÞ ¼ n ðk þ 1Þ. Thereforeeyk ðwÞ ¼ 2ðlðykÞ  jNðykÞVNðwÞjÞ¼ kðk  1Þ þ ðn kÞðn k  1Þ  2ðn k  1Þ¼ 2k2  2kðn 1Þ þ n2  3nþ 2¼ 2 k  12ðn 1Þ 2þ 12ðn2  4nþ 3Þ¼ 2 k  12ðn 1Þ 2þ 12ðn 2Þ2  12:If n is odd, then this quantity is minimal when k ¼ 12 ðn 1Þ. Hence part (ii) holds. Inthis case,eðwÞ ¼ eyðn1Þ=2ðwÞ ¼12ðn 2Þ2  12¼ 12ðn 2Þ2 :If n is even, then eyk is minimal when k ¼ n=2 or k ¼ n=2 1. Hence we obtain part(iii), and in either case, eðwÞ ¼ 12 ðn 2Þ2. Combining the odd and even cases we seethat eðwÞ ¼ bðn 2Þ2=2c. rInvolution products in Coxeter groups 257Brought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AM3 Zero excess in conjugacy classesProof of Theorem 1.1. Suppose that W is a Coxeter group, w AW and X is the W -conjugacy class of w. Now w ¼ r1 . . . rl for certain ri A R and some ﬁnite l ¼ lðwÞ. Sow A hr1; . . . ; rli. Thus it su‰ces to establish the theorem for W of ﬁnite rank. Ac-cordingly we argue by induction on jRj. Suppose that KYR. If X VWK0q, thenby induction there exists w 0 A X VWK with eWK ðw 0Þ ¼ 0, whence eðw 0Þ ¼ 0 and weare done. So we may suppose that X VWK ¼q for all KYR. That is, X is a cuspi-dal class of W .Choose w A X . If w ¼ 1 or w is an involution, then eðwÞ ¼ 0 by Lemma 2.6 (i).Thus we may suppose that w ¼ xy where x and y are involutions. By Theorem 2.5we may conjugate w so that y AWJ for some JJR, with y acting as 1 on FJ .Thus y A ZðWJÞ. Now choose z to have minimal length in fg1xg j g AWJg. So wehave z ¼ g1xg for some g AWJ .Suppose for a contradiction that there exists r A J with lðzrÞ < lðzÞ. Since z and rare involutions,lðrzÞ ¼ lððrzÞ1Þ ¼ lðzrÞ < lðzÞ:Applying Proposition 2.3 yields that either rzr ¼ z or lðrzrÞ ¼ lðzÞ  2 < lðzÞ. Sincer AWJ , the latter possibility would contradict the minimal choice of z. Hence rzr ¼ z.So r  ðz  arÞ ¼ z  ðr  arÞ ¼ z  ar. It is well known that the only roots b forwhich r  b ¼ b are ar and ar. Thus z  ar ¼Gar. By assumption lðzrÞ < lðzÞ andtherefore z  ar A F by Lemma 2.1 (i), whence z  ar ¼ ar. Combining this withy  ar ¼ ar we then deduce that zy  ar ¼ ar. Then Theorem 2.4 gives that zy isin a proper parabolic subgroup of W . Noting that zy ¼ g1xgy ¼ g1xyg ¼ g1wg,as g AWJ , we infer that X is not a cuspidal class, a contradiction. We concludetherefore that lðzrÞ > lðzÞ for all r A J. Consequently NðzÞVFþJ ¼q. SinceNðyÞ ¼ FþJ we deduce that NðzÞVNðyÞ ¼q and hence, using Lemma 2.2, thatlðzyÞ ¼ lðzÞ þ lðyÞ. Setting w ¼ zy ¼ g1wg we have w A X and eðwÞ ¼ 0, socompleting the proof of Theorem 1.1. rReferences[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. RowleyBrought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AM[8] T. A. Springer. Some remarks on involutions in Coxeter groups. Comm. Algebra 10 (1982),631–636.Received 4 May, 2010S. B. Hart, Department of Economics, Mathematics and Statistics, Birkbeck, University ofLondon, Malet Street, London WC1E 7HX, United KingdomE-mail: s.hart@bbk.ac.ukP. J. Rowley, School of Mathematics, The University of Manchester, Oxford Road, Manches-ter M13 9PL, United KingdomE-mail: peter.j.rowley@manchester.ac.ukInvolution products in Coxeter groups 259Brought to you by | Birkbeck College Library (Library & Learning Res.)Authenticated | 172.16.1.226Download Date | 3/29/12 9:31 AM

Involution products in Coxeter groups

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