We extend a result of Lewis and Reiner from finite Coxeter groups to all
Coxeter groups by showing that two reflection factorizations of a Coxeter
element lie in the same Hurwitz orbit if and only if they share the same
multiset of conjugacy classes.Comment: Comments welcom

Wegener, Patrick

Yahiatene, Sophiane

English

arXiv

Algebraic Combinatorics (ALCO)

ALGEBRAIC COMBINATORICSPatrick Wegener & Sophiane YahiateneA note on non-reduced reflection factorizations of Coxeter elementsVolume 3, issue 2 (2020), p. 465-469.<http://alco.centre-mersenne.org/item/ALCO_2020__3_2_465_0>© The journal and the authors, 2020.Some rights reserved.This article is licensed under theCREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.http://creativecommons.org/licenses/by/4.0/Access to articles published by the journal Algebraic Combinatorics onthe website http://alco.centre-mersenne.org/ implies agreement with theTerms of Use (http://alco.centre-mersenne.org/legal/).Algebraic Combinatorics is member of theCentre Mersenne for Open Scientific Publishingwww.centre-mersenne.orgAlgebraic CombinatoricsVolume 3, issue 2 (2020), p. 465–469https://doi.org/10.5802/alco.99A note on non-reduced reflectionfactorizations of Coxeter elementsPatrick Wegener & Sophiane YahiateneAbstract We extend a result of Lewis and Reiner from finite Coxeter groups to Coxeter groupsof finite rank by showing that two reflection factorizations of a Coxeter element lie in the sameHurwitz orbit if and only if they share the same multiset of conjugacy classes.1. IntroductionGiven a Coxeter system (W,S) with set of reflections T , the braid group (e.g. see [1]for a definition) acts on reflection factorizations of a given element w ∈ W , that isit acts on tuples (t1, . . . , tm) ∈ Tm of reflections such that w = t1 · · · tm. This actionis called Hurwitz action. A standard braid group generator σi (resp. its inverse σ−1i )acts by a Hurwitz move on a reflection factorization:σi(t1, . . . , ti−1, ti, ti+1, ti+2, . . . , tn) = (t1, . . . , ti−1, ttii+1, ti, ti+2, . . . , tn),σ−1i (t1, . . . , ti−1, ti, ti+1, ti+2, . . . , tn) = (t1, . . . , ti−1, ti+1, tti+1i , ti+2, . . . , tn),where we use the notation gh := hgh−1 for conjugation.We call a reflection factorization (t1, . . . , tm) of an element w ∈ W reduced if wcannot be written as a product of less than m reflections. It has been first observedby Deligne [3] that this action is transitive on reduced reflection factorizations of aCoxeter element if W is finite. The first published proof is due to Bessis [2, Propo-sition 1.6.1]. Igusa and Schiffler showed that this statement is true for every Coxetergroup [7, Theorem 1.4].The question of how these results extend to non-reduced reflection factorizationshas been first addressed by Lewis and Reiner.Theorem (Lewis–Reiner, [8, Theorem 1.1]). In a finite real reflection group, two re-flection factorizations of a Coxeter element lie in the same Hurwitz orbit if and onlyif they share the same multiset of conjugacy classes.Their proof makes heavy use of the following remarkable result for finite Coxetergroups.Manuscript received 21st January 2019, revised 25th June 2019, accepted 24th September 2019.Keywords. Coxeter groups, Hurwitz action, reflection factorizations, Coxeter element.ISSN: 2589-5486 http://algebraic-combinatorics.org/Patrick Wegener & Sophiane YahiateneLemma 1.1 (Lewis–Reiner, [8, Corollary 1.4]). Let (W,S) be a finite Coxeter systemand w ∈ W . Then every reflection factorization of w lies in the same Hurwitz orbitof some reflection factorization (t1, . . . , tm) of w such that (t1, . . . , t`) is a reducedreflection factorization of w for some ` 6 m andt`+1 = t`+2, t`+3 = t`+4, . . . , tm−1 = tm.This result is proved by a case-by-case analysis and seems not to extend to infiniteCoxeter groups in general. We give a case-free proof of a similar (but weaker) resultfor all Coxeter groups of finite rank (see Lemma 2.3). In this way, we obtain that theresult of Lewis and Reiner extends to all Coxeter groups of finite rank which providesa positive answer to a question of Lewis and Reiner [8, Question 6.2].Theorem 1.2. Let (W,S) be a Coxeter system of finite rank. Then two reflectionfactorizations of a Coxeter element in W lie in the same Hurwitz orbit if and only ifthey share the same multiset of conjugacy classes.2. The proofThroughout this note let (W,S) be a Coxeter system of finite rank n ∈ N with set ofreflections T = {wsw−1 | w ∈ W, s ∈ S}. All necessary definitions and facts aboutCoxeter groups we will use are covered by [1, 4, 5, 6].A subgroupW ′ ofW is called a reflection subgroup ifW ′ = 〈W ′∩T 〉. Each reflectionsubgroup W ′ admits a canonical set of generators χ(W ′) such that (W ′, χ(W ′)) is aCoxeter system and the set of reflections for (W ′, χ(W ′)) is given by W ′ ∩ T =⋃w∈W ′ wχ(W ′)w−1 (see [4, (3.3) Theorem]). A reflection subgroup of the form 〈I〉for some I ⊆ S, is called parabolic subgroup.Let S = {s1, . . . , sn}. For each permutation π of the numbers {1, . . . , n}, the ele-ment c = sπ(1) · · · sπ(n) is called a Coxeter element. A Coxeter element of a parabolicsubgroup is called parabolic Coxeter element.We denote by `S (resp. `T ) the length function onW with respect to the generatingset S (resp. T ).Definition 2.1. We define the Bruhat graph of (W,S) to be the directed graph Ω(W,S)on vertex set W with a directed edge from x to y if there exists t ∈ T such that y = xtand `S(x) < `S(y).Moreover, we denote by Ω(W,S) the corresponding undirected graph and for a subsetX ⊆W we denote by Ω(W,S)(X) the full subgraph of Ω(W,S) on the vertex set X.Note that Ω(W,S) is exactly the Cayley graph of W with generating set T .We use the notation (t1, . . . , tm) ∼ (r1, . . . , rm) to indicate that both tuples lie inthe same orbit under the Hurwitz action.The following fact is already part of the proof of [1, Proposition 2.2]. For sake ofcompleteness we include a proof (which can also be found in the first author’s Ph.D.thesis [9, Proposition 2.3.6]).Proposition 2.2. Let w ∈W and t1, t2 ∈ T with t1 6= t2 such thatw wt1 wt1t2in Ω(W,S). Then there exist t′1, t′2 ∈ 〈t1, t2〉 ∩ T with (t1, t2) ∼ (t′1, t′2) such that one ofthe following cases hold:(1) w wt′1 wt′1t′2 = wt1t2;(2) w wt′1 wt′1t′2 = wt1t2;(3) w wt′1 wt′1t′2 = wt1t2;Furthermore, in each of these cases we have `S(wt′1) < `S(wt1).Algebraic Combinatorics, Vol. 3 #2 (2020) 466Non-reduced reflection factorizations of Coxeter elementsProof. Consider the dihedral reflection subgroup W ′ := 〈t1, t2〉 and let S′ := χ(W ′).We have w,wt1, wt1t2 ∈ wW ′. Therefore we just have to consider the coset wW ′ toprove the claim. By [5, (1.4) Theorem] we haveΩ(W,S)(W ′) ∼= Ω(W,S)(wW ′) ∼= Ω(W ′,S′),where (W ′, S′) is dihedral and we can check the claim there directly. Any reflection(element of odd S′-length) of W ′ is joined by an edge to a rotation (element of evenS′-length) which in Ω(W ′,S′) is oriented towards the element of greater S′-length. Forx ∈W ′ there are three possible situations:• `S′(x) < `S′(xt1t2)• `S′(x) > `S′(xt1t2)• `S′(x) = `S′(xt1t2) (in particular x 6= e since t1 6= t2).We can choose t′1, t′2 ∈ W ′ ∩ T with t′1t′2 = t1t2 in the three situations such that wehave one of the following situations:• x xt′1 xt′1t′2• x xt′1 xt′1t′2• x xt′1 xt′1t′2To see this, note that x and xt1t2 are both either reflections or rotations. Thereforeboth are either of odd or even S′-length. Thus `S′(x) < `S′(xt1t2) implies `S′(x)+2 6`S′(xt1t2) and we find t′1 with `S′(x) < `S′(xt′1) < `S′(xt1t2). By setting t′2 := t′1t1t2we obtain x xt′1 xt′1t′2 and t′1t′2 = t1t2. The remaining cases are similar.It is easy to see that the Hurwitz orbit of (t1, t2) is the set of all pairs (r1, r2) ofreflections of W ′ such that t1t2 = r1r2 (see also [1]). Hence we have (t1, t2) ∼ (t′1, t′2).It remains to show that `S(wt′1) < `S(wt1) in each of the cases (1)-(3). Since theinitial path is of the form w wt1 wt1t2, we have:(i) `S(w) < `S(wt1), and(ii) `S(wt1t2) < `S(wt1).In case (1) we have an edge wt′1 wt′1t′2 = wt1t2, thus`S(wt′1) < `S(wt′1t′2) = `S(wt1t2)(ii)< `S(wt1).In cases (2) and (3) we have an edge w wt′1, thus`S(wt′1) < `S(w)(i)< `S(wt1). Lemma 2.3. Let w ∈ W with `S(w) = m and w = t1 · · · tm+2k with ti ∈ T for1 6 i 6 m+ 2k and some k ∈ Z>0. Then there exists a braid σ ∈ Bm+2k such thatσ(t1, . . . , tm+2k) = (r1, . . . , rm, ri1 , ri1 , . . . , rik , rik ).Proof. We proceed by induction on k. The case k = 0 is trivially satisfied. Thereforelet k > 1 and assume that all factorizations in Bm+2k(t1, . . . , tm+2k) consist of pairwisedifferent factors. Consider the path of Ω(W,S) starting in e and ending in w inducedby (t1, . . . , tm+2k). Then Proposition 2.2 allows us to replace successively the parts ofthe path of shape ? ? ? by? ? ?, ? ? ?, or ? ? ?only using the Hurwitz action. The latter is possible since the reflections of the fac-torizations in the Hurwitz orbit are pairwise different. Since by Proposition 2.2 eachreplacement reduces the sum of the length of the vertices, eventually we get afterfinitely many replacements a path of the forme t′1 t′1t′2 . . . t′1t′2 · · · t′p t′1t′2 · · · t′pt′p+1 . . . t′1 · · · t′m+2k = wAlgebraic Combinatorics, Vol. 3 #2 (2020) 467Patrick Wegener & Sophiane Yahiatenewith t′i ∈ T for 1 6 i 6 m+ 2k, that is, the path is first decreasing, then increasing.Since the path starts with e, it holds p = 0 and therefore it has no decreasing part.Altogether the initial path can be transformed toe t′1 t′1t′2 . . . t′1 · · · t′m+2k = wby using the Hurwitz action. Since the length of w is m and k > 1, there cannot be anincreasing path of length m+ 2k, so we arrive at a contradiction. Thus there exists afactorization (t′1, . . . , t′m+2(k−1), rik , rik ) in Bm+2k(t1, . . . , tm+2k). From the inductionhypothesis follows(t′1, . . . , t′m+2(k−1)) ∼ (r1, . . . , rm, ri1 , ri1 , . . . , rik−1 , rik−1)and the latter yields the assumption. Remark 2.4. Note that `T (w) 6 `S(w) for all w ∈W . Therefore the reflection factor-ization w = r1 · · · rm obtained by Lemma 2.3 does not have to be a reduced reflectionfactorization. This is the main difference compared with the key argument Lemma 1.1in the proof of Lewis and Reiner. However, by [1, Lemma 2.1] we have `S(w) = `T (w)for an element w ∈W if and only if w is a parabolic Coxeter element. Therefore, if wis a parabolic Coxeter element, then Lemma 2.3 generalizes Lemma 1.1. In particular,a reflection factorization of a parabolic Coxeter element can be reduced by just usingHurwitz moves and deleting matching neighbors.A proof of the following fact already implicitly appears in the proof of [8, Theo-rem 1.1].Lemma 2.5. Let t1, . . . , tn, t ∈ T . Then (t1, . . . , tn, t, t) ∼ (t1, . . . , tn, tw, tw) for allw ∈ 〈t1, . . . , tn〉.Proof. Let i ∈ {1, . . . , n} be arbitrary and assume w = ti (the general assertion followsby induction). Denoting an omitted entry by t̂i, we obtain(t1, . . . , tn, t, t) ∼ (t1, . . . , ti−1, t̂i, ttii+1 . . . , ttin , tti , tti , ti)∼ (t1, . . . , ti−1, t̂i, ttii+1 . . . , ttin , ti, tti , tti)∼ (t1, . . . , ti−1, ti, ti+1, . . . , tn, tti , tti). Proof of Theorem 1.2. Let c ∈W be a Coxeter element andc = t′1 · · · t′n+2k = r′1 · · · r′n+2ktwo reflection factorizations of c for some k ∈ Z>0 such that they share the samemultiset of conjugacy classes. By Lemma 2.3 we have(t′1, . . . , t′n+2k) ∼ (t1, . . . , tn, ti1 , ti1 , . . . , tik , tik )and (r′1, . . . , r′n+2k) ∼ (r1, . . . , rn, ri1 , ri1 , . . . , rik , rik ).Since c = t1 · · · tn = r1 · · · rn and `S(c) = `T (c) = n by [1, Lemma 2.1], (t1, . . . , tn)and (r1, . . . , rn) are reduced reflection factorizations of c. Hence we have (t1, . . . , tn) ∼(r1, . . . , rn) by [1, Theorem 1.3]. In particular (t1, . . . , tn) and (r1, . . . , rn) share thesame multiset of conjugacy classes. Hence ti1 , . . . , tik and ri1 , . . . , rik have to sharethe same multiset of conjugacy classes. Since (t, t, r, r) ∼ (r, r, t, t) for all r, t ∈ T , wecan assume after a possible renumbering that there exists wj ∈W such that twjij= rijfor all j ∈ {1, . . . , k}. We proceed by induction on k. As we have seen above, the casek = 0 is precisely [1, Theorem 1.3]. Therefore let k > 0. By induction we have(t1, . . . , tn, ti1 , ti1 , . . . , tik−1 , tik−1 , tik , tik )∼ (r1, . . . , rn, ri1 , ri1 , . . . , rik−1 , rik−1 , tik , tik ).Algebraic Combinatorics, Vol. 3 #2 (2020) 468Non-reduced reflection factorizations of Coxeter elementsAs a consequence of [1, Theorem 1.3], we have W = 〈r1, . . . , rn〉. By what we havepointed out before, there exists wk ∈ 〈r1, . . . , rn〉 such that twkik = rik . We conclude(t′1, . . . , t′n+2k) ∼ (r1, . . . , rn, ri1 , ri1 , . . . , rik−1 , rik−1 , tik , tik )∼ (r1, . . . , rn, tik , tik , ri1 , ri1 , . . . , rik−1 , rik−1)2.5∼ (r1, . . . , rn, twkik , twkik, ri1 , ri1 , . . . , rik−1 , rik−1)= (r1, . . . , rn, rik , rik , ri1 , ri1 , . . . , rik−1 , rik−1)∼ (r1, . . . , rn, ri1 , ri1 , . . . , rik−1 , rik−1 , rik , rik )∼ (r′1, . . . , r′n+2k). Corollary 2.6. If the Coxeter graph of (W,S) is connected and has a spanning treewith odd labels on all its edges, then two reflection factorizations of the same lengthof a Coxeter element in W lie in the same Hurwitz orbit.Proof. Since the Coxeter graph of (W,S) contains a spanning tree, all elements of Sare conjugate. Therefore all reflections in T are conjugate. Thus the assertion followsby Theorem 1.2. Acknowledgements. The authors thank two anonymous referees for helpful comments.References[1] Barbara Baumeister, Matthew Dyer, Christian Stump, and Patrick Wegener, A note on thetransitive Hurwitz action on decompositions of parabolic Coxeter elements, Proc. Am. Math.Soc., Ser. B 1 (2014), 149–154.[2] David Bessis, The dual braid monoid, Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), no. 5, 647–683.[3] Pierre Deligne, Letter to E. Looijenga, 1974, http://homepage.rub.de/christian.stump/Deligne_Looijenga_Letter_09-03-1974.pdf (accessed 30th September 2019).[4] Matthew Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), no. 1, 57–73.[5] , On the “Bruhat graph” of a Coxeter system, Compos. Math. 78 (1991), no. 2, 185–191.[6] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in AdvancedMathematics, vol. 29, Cambridge University Press, Cambridge, 1990.[7] Kiyoshi Igusa and Ralf Schiffler, Exceptional sequences and clusters, J. Algebra 323 (2010), no. 8,2183–2202.[8] Joel Brewster Lewis and Victor Reiner, Circuits and Hurwitz action in finite root systems, NewYork J. Math. 22 (2016), 1457–1486.[9] Patrick Wegener, Hurwitz action in Coxeter groups and elliptic Weyl groups, Ph.D. thesis, Uni-versität Bielefeld (Germany), 2017, https://pub.uni-bielefeld.de/record/2913106.Patrick Wegener, Technische Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049,67653 Kaiserslautern, GermanyE-mail : wegener@mathematik.uni-kl.deSophiane Yahiatene, Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, 33501Bielefeld, GermanyE-mail : syahiate@math.uni-bielefeld.deAlgebraic Combinatorics, Vol. 3 #2 (2020) 469

A note on non-reduced reflection factorizations of Coxeter elements

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A note on non-reduced reflection factorizations of Coxeter elements

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