On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups´swiatosław

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. Introduction The central object of this paper is the growth series of a Coxeter group. Many geometric features of such a group (or any group in general) reflect in properties of the growth series. We describe in detail the normal closure of a standard parabolic in a Coxeter group W. It is again a Coxeter group and its Coxeter presentation is given explicitly. We give two formulae for the growth series of such a normal subgroup. The first one is given as a specialization of a multi-variable version of the growth series of W. The second formula works when the normal subgroup is right-angled (this can be easily checked by analysis of the Dynkin diagram of W). The formula is based on counting special subdiagrams of the Dynkin diagram of W. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. The author would like to thank Tadeusz Januszkiewicz for useful discussions, Paweł Goldstein for his extensive help with the final version of this manuscript and Mike Davis for a preliminary version of [D2]. 1. Preliminaries on Coxeter Groups Definition. A Coxeter system (W,S) is a group W together with a set of generators S, and presentation W = 〈S|(st)st = 1 for all s, t ∈ S〉, where mss = 1 (i.e. all the generators have order two) and mst = mts ∈ {2, 3, . . . ,∞} if s 6= t. One reads (st) = 1 as no relation between s and t imposed. The matrix m is called the Coxeter matrix of W. We usually ignore S and call W a Coxeter group. The subgroup of W generated by T ⊂ S is denoted WT . Such a subgroup is called a standard parabolic or parabolic for short. Traditionally a Coxeter matrix is depicted in a following decorated graph called Dynkin diagram. Its vertex set is S and any two distinct vertices are joined by an edge with label mst. By convention • we omit an edge if mst = 2, 2000 Mathematics Subject Classification: 20F55 Partially supported by a KBN grant 2 P03A 017 25. 2 S. R. Gal • we omit a label if mst = 3, • we draw double edge instead that labeled by 4 and • we draw dashed edge instead that labeled by ∞. We will use the following convention for the Cayley graph: the group acts on the left, therefore {g1, g2} is an edge provided g −1 1 g2 ∈ S. We say that an edge {g1, g2} is marked by the generator g 1 g2. Theorem 1.1 [D2]. Let W be a group generated by a set of involutions S. Let CW be the Cayley graph of (W,S). Then (W,S) is a Coxeter system if and only if for any element s ∈ S the fixpoint set (CW) separates CW (i.e. CW − (CW) has two components being interchanged by s). Proof: It is a part of Theorem 2.3.3 in [D2] (cf. [D2, Definitions 2.2.1 and 2.2.10]). Definition. A conjugate of a generator from S is called a reflection. The set of all reflections is denoted by R(W). Corollary 1.2. Reflections are intrinsically defined by a Coxeter system. Precisely, if T ⊂ S then R(WS) ∩WT = R(WT ). Proof: Assume that r ∈ R(WS)∩WT . Since the action of r is conjugate to that of some generator, the fixpoint set of r also separatesCW . Then r and the neutral element e are in different components of CWS − (CWS) , and thus in different components of CWT − (CWT ) . Take any path joining e and r in CWT . Such a path has to intersect the separating fixpoint set (CWT ) r in a midpoint of some edge {w,wt}. Thus rw = wt and, what follows, r = wtw ∈ R(WT ). Definition. The length of an element w of any group with respect to a given generating set S is denoted by l(w) and is equal to the minimal length of a word in S representing w. The word is called reduced if its length is equal to the length of the element it represents. Lemma 1.3 [T]. If (W,S) is a Coxeter system than any word can be reduced (into any reduced word representing the same element of W) in a sequence of moves of the following form: (a) removing a subword of a form ss, or (b) replacing an alternating subword st . . . of length mst by an alternating subword ts . . . of the same length. Note: The move (a) is not symmetric, i.e. we do not need creation of a pair ss. In particular any two reduced words representing the same element may be joined by moves of the form (b) only. 2. Normal closure of a standard parabolic If mst = 2n+ 1 is odd then s and t are conjugates since the relation reads then (st)s(st) = t. If one wants to find a normal closure of a subgroup WT it is convenient to incorporate in T all the generators conjugate to those already in T . This clarifies the assumption of the following On Normal Subgroups of Coxeter Groups 3 Proposition 2.1. Assume that T ⊂ S is such that for any t ∈ T and s ∈ S − T the exponent mst is even (or ∞). Then (1) there exists a well defined homomorphism φT :W → WS−T that is an identity on WS−T and sends WT to the identity element, (2) the normal closure WT of WT is the kernel of φT , (3) WT is generated by ST , a set of all the conjugates of T with respect to WS−T , (4) the Cayley graph C WT of (WT , ST ) is made from the Cayley graph CW of W by collapsing all the edges marked by the generators not in T , (5) (WT , ST ) is a Coxeter system. Proof : Let t ∈ T and s 6∈ T , then the relation st . . . = ts . . ., after substituting the identity for s, becomes trivial exactly when mst is even. Thus (1) follows. This also proves that the relation “s and t are conjugate” is the equivalence closure of the relation “mst is odd”. Any element w of W may be written as follows w0t1w1 . . . tnwn = ( w0t1w −1 0 ) ( w0w1t2(w0w1) −1 )