On the Poincaré series and cardinalities of finite reflection groups

Let W be a crystallographic reflection group with length function t(-). We give a short and elementary derivation of the identity ZweW q(W) = l(l qht(@)+l)/(1 qht(al)), where the product ranges over positive roots a, and ht(a) denotes the sum of the coordinates of a with respect to the simple roots. We also prove that in the noncrystallographic case, this identity is valid in the limit q -* 1; i.e., WI W (ht(a) + 1)/ht(a). 0. INTRODUCTION Let W be a finite real reflection group of rank n with the usual associated data: root system b C Rn, positive roots D+, simple roots a1,... an, corresponding simple reflections s1,... , s-Sn E W, and inner product ( , ). For w E W, we let ?(w) denote the minimum length among all expressions w = si1... sil, sgn(w) = (-1)'(w) = det(w), and 4X(w) = {ae E 4+: wa e -4+}. For roots ae E @, we let aV denote the co-root 2a/(a, a), and use the notation a > 0 to indicate ae E 4+. Recall that b is said to be crystallographic if (a, ,3V) E Z for all a,, ,B E b. Standard references are [B] and [H]. The purpose of this paper is two-fold. First, we give a very short and elementary proof of a product formula for the Poincare series EwEW qe(W) in the crystallographic case. Our proof is based on Macdonald's approach [M], in which the formula is deduced by specializing a more general identity in the group ring of the root lattice (Theorem 1). However the proof we give is, in our opinion, significantly simpler in the details. Indeed, it involves nothing more sophisticated than elementary manipulations and the fact that for all w E W, ?(w) = 14(w)l. Eric Opdam [private communication] has pointed out that a similar proof appears in the unpublished Ph. D. thesis of Rene Beerends [Be]. Our proof is arguably more elementary than Beerends' in that the issue of divisibility of expressions by the Weyl denominator does not arise. Our second objective is to show that, although this particular product formula for the Poincare series is not valid in the non-crystallographic case, it is valid in the limit q -+ 1, and this leads to a new formula for the cardinality of W (Theorem 3). Of course, there is an explicit product formula for the Poincare series of any real reflection group due to Solomon (e.g., [H, ?3.15]), and the cardinality of any such Received by the editors October 9, 1996 and, in revised form, March 29, 1997. 1991 Mathematics Subject Classification. Primary 20H15, 20F55. The author was partially supported by a grant from the NSF. @)1998 American Mathematical Society 3177 This content downloaded from 207.46.13.52 on Sat, 22 Oct 2016 05:31:11 UTC All use subject to http://about.jstor.org/terms 3178 JOHN R. STEMBRIDGE group is the product of the degrees of the fundamental invariants, but these facts require lengthier arguments. 1. THE POINCARE SERIES In the following, it will be convenient to work in the group ring of R', with the basis elements written as formal exponentials eA for A E R . 1.1. Define p = Z,>0 a/2. Since si permutes D+ {ai}, it follows that sjp = p aj. On the other hand, the reflection sj acts via sjp = p (p, a)')aj, so we have (1) (p,ao')=1 (1 o i=l Rearranging the term indexed by w, we obtain eWP JJ (1 _ uve-wa) = JJ (ewa/2 _ uae-wa/2) ca>O cx>O = sgn(w) JJ (uae-wa/2 ewa/2) J (ew/2 _ u,e-wa/2 wa O =sgn(w) eP JJ (ua -ewa) JJ (1uae-wa), wa O where each product is implicitly restricted to a > 0. Here we have used only the fact that sgn(w) = (-1)1'(W)1, a corollary of the fact that ?(w) = 14(w)l (e.g., [H, ?1.7]). Thus (2) can be rewritten in the form (3) S JJ (un, ewa) JJ (1-uae-w') = e-P E Fi(u)/A(,ui). wEW wa O i= 1.3. Define 6 E RT so that (a?i,) = 1 for 1 (sj,ui, 6) = (,ui, 6) (,ui, ay'), or equivalently, (,ui, ay') > 0 for 1 (l j , where A is a sum of positive roots. It follows that for each i we must have ,ui = p A for some A of this form. Combining (1) and (4), we obtain that for such A, we must have (A, ay) O for some polynomial F(u). 1.5. One may determine F(u) by extracting the coefficient of eP in (2), or equivalently the constant term in (3). However, (3) is a polynomial in the variables e-t,.. ., e.. , so the constant term can be obtained via the specialization e-ci 0, yielding (6) F(u)= E 1 ua. wCW acx (w) 1.6. Since the expression eP Hoa>0o(1_e-a) = Ha>O (ea/2 _e-a/2) is antisymmetric with respect to the action of W, by setting ua = 1 in (5) and (6) and cancelling a factor of IWI, we obtain the Weyl denominator formula (7) A(p) = eP (1 e-a) = I| (ea/2 -e-a/2) >O >o Combining (5)-(7) yields Theorem 1 (Macdonald). If 1 is crystallographic, then (8) E sgn(w) eWP (1 u e-wa) = ep [1 (1 e-e) E I 1ua. wcW a>O a>cwO wCW aC (w) 1.7. Following [M], we now specialize Theorem 1 by setting ua = q and el' = q-(A16). For each w : 1, there must exist a simple root ai such that w-1ai O ca>O wcW Since ?(w) = lb(w)j, this proves Corollary 2 (Macdonald). If b is crystallographic, then E q(W) 1 qht(a)l wEW > qht(a) W a>o 2. THE ORDERS OF REAL REFLECTION GROUPS Let us now discard the hypothesis that 41b is crystallographic. Although the formula for the Poincare series given in Corollary 2 cannot be valid at this level of generality (it is not even a rational function of q), we claim that it is valid in the limit q --1; i.e., Theorem 3. If W is a real reflection group, then (9) l WI rI ht(ca) + 1 iwi ~~ht(av) This content downloaded from 207.46.13.52 on Sat, 22 Oct 2016 05:31:11 UTC All use subject to http://about.jstor.org/terms 3180 JOHN R. STEMBRIDGE For example, if W is the dihedral group of order 10, then the coordinates of the positive roots with respect to the simple roots are (1, 0), (0, 1), (-y, 1), (1, -y), and (-y, y), where -y = (1 + V5-)/2 denotes the golden ratio. Thus (9) takes the form iWI= 2 .2 -(2 + -y) -(2 + -Y) -(1 + 2-y) 1 1 (1 +ty (1 + -y) -2-y In the following, it will be convenient to drop the pretense that et' denotes a "formal" exponential, and instead identify e/1 with the function v l-4 e(II,v) on Rn. The group ring of R' is now a function space, so let A(,) (v) denote the evaluation of A (IL) at v. Lemma 4. If m = J1l, then for all IL,i-v E Rn, lim A(1)( isfinite. Proof. Since G(q) =A /\(pu)(vlog q) is analytic at q = 11 it suffices to show that the Taylor series of G vanishes through terms of order less than m. For this, consider that -qiG(q) = i sgn(w)q(w J wE sgn(w) in the limit q -1. However, as a function of v, the latter expression is a polynomial function of degree 0(a, v) (e.g., [H, 3.13]), and hence have degree at least m. O Proof of Theorem 3. Setting u, = 1 in (2), we obtain |WI * eP JI ( -e-ae) = ZFi(1)A(wj), a>O i= by reasoning similar to 1.6. Evaluating this identity at -6 log q yields IWI = q(P"6) E F(l) I( lq) i=11(q where 11(q) = 0 (1 qht(a)). In the limit q -1, we therefore have WI =EciFi(l) i=1 where ci limq-1 A(1aj)(-6logq)/ll(q), a limit whose existence follows from Lemma 4. On the other hand, if we apply the reasoning of 1.7 to (2), we obtain q-(P,6) f (1 qht(a)+l) = EFi(q)A(pa)(-6logq), a>O i= or equivalently, I 1 qho,l AP6 q (ti ) (6 log q) 1l qht(a)+ q (P,b) Fi(q a>0 1 qhta i 1(q) This content downloaded from 207.46.13.52 on Sat, 22 Oct 2016 05:31:11 UTC All use subject to http://about.jstor.org/terms POINCARE SERIES AND CARDINALITIES OF REFLECTION GROUPS 3181 Thus in the limit q -1, we obtain 1 ht (a) = E ciFi(1) = IWI . a>O ~~~~i==1