Journal of Lie Theory Volume 14 (2004) 69–71 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

Lengths of Involutions in Coxeter Groups

Sarah B. Perkins and Peter J. Rowley

Communicated by K.-H. Neeb

Abstract. Let t be an involution in a Coxeter group W. We determine the minimal and maximal (in the case of finite W) length of an involution in the conjugacy class of t. Mathematics Subject Classification 20F55.

Let W be a finitely generated Coxeter group whose distinguished set – the set of fundamental reflections – is R. The length l(w) of a non-trivial element w in W is defined to be

l(w) = min{l ∈ ℕ : w = r_{1}r_{2} ⋯ r_{l} some r_{i} ∈ R}

and l(1) = 0. Suppose t is an involution in W, and let C = t^{W} be the conjugacy class of t in W. The aim of this short paper is to determine the minimal and maximal (in which case W is assumed finite) length of an involution in C.

Associated to any Coxeter group W is the root system Φ, which is the disjoint union of its positive and negative roots (denoted Φ^{+} and Φ^{−} respectively). The fundamental reflections r ∈ R are in one-to-one correspondence with the fundamental roots α_{r}, r ∈ R and W acts faithfully on Φ (see [1]). For w ∈ W, define N(w) := {α ∈ Φ^{+} : w · α ∈ Φ^{−}}, I(w) := {α ∈ Φ^{+} : w · α = −α} and Fix(w) := {α ∈ Φ^{+} : w · α = α}. It is well known that for each w ∈ W, l(w) = |N(w)|. For J ⊆ R, write W_{J} for the (Coxeter) group generated by J, Φ_{J} for its root system and, when it is finite, w_{J} for the unique longest element of W_{J}. Our main result is given in

Theorem 1.1. Suppose t is an involution in W, and put C = t^{W}. We have

(i) min_{s∈C}{l(s)} = |I(t)| and if x is of minimal length in C, then x = w_{J} for some J ⊆ R.

(ii) If W is finite, then max_{s∈C}{l(s)} = |Φ^{+}| − |Fix(t)| and for y of maximal length in C, y = w_{K}w_{R} for some K ⊆ R.

Put another way, Theorem 1.1 is saying that the maximum and minimum length in a conjugacy class of involutions may be obtained by examining the action on Φ

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag of any one involution in that class. We remark that part (i) appears as Theorem A (a) in [3]. We include a (shorter, and different) proof here to emphasise the similarity between parts (i) and (ii).

Proof. Let t be an involution and C = t^{W}. Note that for any t′ ∈ C, |I(t′)| = |I(t)| and |Fix(t′)| = |Fix(t)|, because t · α = ±α if and only if t^{g} · (g · α) = ±(g · α), for each g ∈ W. It is clear from this that the length of any involution in C is at least |I(t)| and at most |Φ^{+}| − |Fix(t)|. Let r ∈ R with α_{r} ∉ N(t), and suppose α_{r} ∉ Fix(t). It is well known that for any w ∈ W, r ∈ R, l(wr) > l(w) if and only if w · α_{r} ∈ Φ^{+}. We have t · α_{r} ∈ Φ^{+} \ {α_{r}}, so rt·α_{r} ∈ Φ^{+}. Therefore l(rtr) > l(rt). Now rt = (tr)^{−1}, hence l(rt) = l(tr) > l(t), since α_{r} ∉ N(t). Thus l(rtr) > l(t). Suppose now that α_{r} ∈ N(t) with α_{r} ∉ I(t). We have l(rtr) < l(rt) because rt · α_{r} ∈ Φ^{−}, and l(rt) = l(tr) < l(r) because α_{r} ∈ N(t). Thus l(rtr) < l(t).

We have shown that if α_{r} ∉ N(t), then either l(rtr) > l(t) or α_{r} ∈ Fix(t), and that if α_{r} ∈ N(t), then either l(rtr) < l(t) or α_{r} ∈ I(t). Thus for each x of minimal length in C, there exists J ⊆ R with α_{r} ∈ I(x) for each r ∈ J and α_{r} ∉ N(x) when r ∉ J. Let r ∈ J. Then w_{J}x · α_{r} = −w_{J} · α_{r} ∈ Φ^{+}. If r ∉ J then w_{J}x · α_{r} ∈ Φ^{+} unless x · α_{r} ∈ Φ_{J}^{+}. But this would imply that x^{2} ·α_{r} = −x·α_{r} ≠ α_{r}, which is impossible. Thus N(w_{J}x) = Ø and hence x = w_{J}. Now N(x) = Φ_{J}^{+} = I(x) and so x has length |I(t)| in C, which is minimal.

Similarly, when W is finite, for y of maximal length in C there exists K ⊆ R with α_{r} ∈ Fix(y) whenever r ∈ K, and α_{r} ∈ N(y) for r ∉ K. We claim that Fix(y) = Φ_{K}^{+} . Certainly Φ_{K}^{+} ⊆ Fix(y). For the reverse inclusion, let α = ∑_{r∈R} λ_{r}α_{r} ∈ Fix(y) (where each λ_{r} ≥ 0). Now y · α_{r} ∈ Φ^{−} for all r ∈ R \ K, so ∑_{r∈R\K} λ_{r}y · α_{r} is a negative linear combination of roots, say − ∑_{r∈R} μ_{r}α_{r} for some μ_{r} ≥ 0. We have ∑_{r∈R} λ_{r}α_{r} = α = y ·α = ∑_{r∈K}(λ_{r} −μ_{r})α_{r} −∑_{r∈R\K} μ_{r}α_{r}. For r ∈ R\K then, we see that λ_{r} = −μ_{r}. Hence λ_{r} = μ_{r} = 0. Therefore α ∈ Φ_{K}^{+} and so Fix(y) ⊆ Φ_{K}^{+} .

Now for r ∈ K, w_{K}y · α_{r} = w_{K} · α_{r} ∈ Φ^{−}. If r ∉ K, w_{K}y · α_{r} ∈ Φ^{+} only when y · α_{r} ∈ Φ_{K}^{−}, which is impossible. Consequently N(w_{K}y) = Φ^{+}, that is y = w_{K}w_{R} and l(y) = |N(y)| = |Φ^{+}| − |Φ_{K}^{+}| = |Φ^{+}| − |Fix(y)| and this is the maximum possible length of an involution in C.

We remark that it is necessary, as Proposition 1.3 shows, to assume, when W is irreducible, that W is finite in order for max_{s∈C}{l(s)} to be defined. We require the following lemma, which follows from the fact that the geometric representation of W is irreducible and faithful (see [1]).

Lemma 1.2. ([2], Lemma 2.3) Let W be an irreducible Coxeter group and let α ∈ Φ. Then W acts faithfully on the orbit W · α.

Proposition 1.3. Suppose W is an infinite irreducible Coxeter group. Then each conjugacy class of involutions in W contains elements of arbitrarily large length.

Proof. Let t be an involution of W. Then, by Theorem 1.1, I(t) is non-empty, so there exists α ∈ Φ^{+} with t · α = −α. Let β ∈ W · α. Then β = w · α for some w ∈ W. Now t^{w} · β = wtw^{−1} · (w · α) = −β, whence β ∈ N(t^{w}). Thus W · α ⊆ ∪_{w∈W} N(t^{w}). Each element t^{w} has finite length, but W · α is infinite, by Lemma 1.2, hence the conjugacy class of t must be infinite. Consequently, since there can only be finitely many elements of a given length in W, the conjugacy class of t must contain elements of arbitrarily large length.

References

[1] J. E. Humphreys, “Reflection Groups and Coxeter Groups, ” Cambridge Studies in Advanced Mathematics 29, 1990.

[2] Perkins, S. B., and P. J. Rowley, Bad Upward Elements in Infinite Coxeter Groups, Manchester Centre for Pure Mathematics Preprint, 2000/6.

[3] Richardson, R. W., Conjugacy classes of involutions in Coxeter groups, Bull. Austral. Math. Soc. 30 (1982), 1–15.

S. B. Perkins and P. J. Rowley Department of Mathematics, UMIST, PO Box 88, Manchester M60 1QD United Kingdom Sarah.Perkins@umist.ac.uk

Received June 30, 2002 and in final form October 13, 2003