# Reference for class of involutions containing longest element of finite Coxeter group?

Consider a finite (say irreducible) Coxeter group $W$ with a fixed generator set $S$ and rank $n$. This is the same thing as a finite real reflection group, generated by a set of “simple” reflections. Important examples include the Weyl groups of simple Lie algebras, but there are a few other types such as $H_3, H_4$ of ranks 3,4, and most of the dihedral groups $I_2(m)$ of order $2m$. Coxeter showed uniformly the existence of a unique element $w_0 \in W$ of maximum length relative to the generating set $S$. It is always an involution (of order precisely $2$) and its length is the number of “positive roots” (in the setting of Bourbaki or Deodhar).

On the other hand, Richardson and Springer both published in 1982 classifications of conjugacy classes of involutions in a (possibly infinite) Coxeter group $W$: Each involution is conjugate to the element $-1$ in a parabolic subgroup $W_I$ with $I \subset S$ whose longest element in this (Coxeter!) group acts as $-1$ in the standard geometric realization of $W_I$. (Here one knows that the length function of $W$ restricts to the length function of $W_I$ relative to its generating set $I$.) Moreover, $I$ is unique up to $W$-equivalence. In case $-1 \in W$, then $w_0 = -1$.

When $W$ is finite (and irreducible), but $-1 \notin W$, is it written down anywhere for which proper subsets $I \subset S$ the element $w_0$ is conjugate to $-1 \in W_I$?

When I was sorting out some old notes recently I realized that I hadn’t seen this anywhere, even though it would have made a good exercise in my short survey of the results on involutions (in 8.3 of my 1990 book on reflection groups). The answer seems to be easy, given the standard classification of finite Coxeter groups, where it turns out that $-1 \notin W$ just when the type is $A_n$ for $n>1, \; D_n$ for $n \geq 5$ odd, $E_6$, or $I_2(m)$ for $m$ odd (which agrees with $A_2$ when $m=3$). For $A_{2n}$ one gets $W_I$ of type $A_1 \times \dots \times A_1$ ($n$ copies). For $A_{2n+1}$ one has instead $A_1 \times \dots \times A_1$ ($n+1$ copies). For $D_n$ with $n$ odd, $W_I$ has type $D_{n-1}$. For $E_6$, the subgroup $W_I$ has type $D_4$. And for $I_2(m)$ with $m$ odd, either standard generator gives a suitable parabolic subgroup of type $A_1$. But has this been stated anywhere?

Some related remarks:

1. The longest element $w_0$ comes up most often for Weyl groups (finite Coxeter groups with the crystallographic property, types $A - G$ in the classification), often in connection with representations, e.g., the dual $V^*$ of a finite dimensional simple module of highest weight $\lambda$ for a simple Lie algebra having Weyl group $W$ is the simple module of highest weight $-w_0 \lambda$.

2. Without using the classification of connected Coxeter graphs, Coxeter found in his study of a Coxeter element $c$ (product of simple generators in $S$ taken in any order, all such being conjugate) that when the order $h$ of $c$ is even and $c$ is well chosen, $c^{h/2}= w_0$. Moreover, when all degrees of basic polynomial invariants for $W$ are even (including $h$), one has $w_0 = -1$.

3. Using the classification, one finds that $-1 \notin W$ precisely in the irreducible types indicated above. (In these cases, the Coxeter graph has a symmetry $\sigma$ of order 2, with $w_0 = -\sigma$. But $D_n$ for $n$ even also has such a symmetry, even though $w_0 = -1$.)

4. The two papers mentioned above (neither readily accessible online) appeared in Bull. Austral. Math. Soc. 26 (1982), 1-15 (Richardson) and Comm. Algebra 10 (1982), 631-636 (Springer). Both cite a useful paper by Deodhar on “root systems” in Coxeter groups, which appeared in the same issue as Springer’s note, but neither paper cites the other. Richardson’s was submitted in July 1982, but Springer’s in August 1981. So there is a historical mystery, probably impossible to settle now. Though I was acquainted with all three authors, it never occurred to me to raise this question.