For a finite Coxeter group W and w an element of W the excess of w is defined to be e(w) = min{ℓ(x)+ ℓ(y)− ℓ(w) | w = xy, x 2 = y 2 = 1} where ℓ is the length function on W . Here we investigate the behaviour of e(w), and a related concept reflection excess, when restricted to standard parabolic subgroups of W . Also the set of involutions inverting w is studied. (MSC2000: 20F55)
IntroductionThis paper, continuing the investigations begun in [5] and [6], studies further properties of excess in Coxeter groups. First we recall the definition of excess. Suppose W is a Coxeter group with length function ℓ, and setThen for w ∈ W, the excess of w isThe main result in [5] asserts that every element in W is W -conjugate to an element whose excess is zero. In a similar vein, [6] shows that if W is a finite Coxeter group, then every Wconjugacy class possesses at least one element which simultaneously has minimal length in the conjugacy class and excess equal to zero. The present paper explores other properties of excess in finite Coxeter groups. So from now on we assume W is finite. Since every element in a finite Coxeter group may be written (in possibly many ways) as xy where x 2 = y 2 = 1, e(w) is defined for all w ∈ W . Moreover, every element w ∈ W may be written as xy, where x 2 = y 2 = 1 and L(w) = L(x) + L(y), where L is the reflection length function on W . This latter fact is (essentially) established in Carter [3] (see also Lemma 2.4 of [6]). This leads to the related notion of reflection excess. For w ∈ W its reflection excess E(w) is defined byClearly E(w) ≥ e(w). However E(w) and e(w) can be markedly different -see for example Proposition 3.3 of [6].The first issue we address here is how excess and reflection excess behave on restriction to standard parabolic subgroups of W -as is well-known, such subgroups are Coxeter groups in their own right. If W J is a standard parabolic subgroup of W and w ∈ W J , we let e J (w) (respectively E J (w)) be the excess of w (respectively reflection excess of w) considered as an element of W J . * The authors wish to acknowledge partial support for this work from the Department of Economics, Mathematics and Statistics at Birkbeck and the Manchester Institute for Mathematical Sciences (MIMS). * a * b) of spartan pairs. With this information we are then able to complete, in Theorems 3.6 and 3.7, the proof of Theorem 1.2. All is not lost for type D, as Theorem 3.8 demonstrates, with various conditions which guarantee that e J (w) = e(w).Our final section investigates N(I w ) for w ∈ W . Proposition 4.2 and Lemma 4.3 reveal that, under certain circumstances, N(I w ) = Φ + (though this is not always the case) and the balance of this section presents a proof of Theorem 1.3.2
