# Standard involutions conjugate to the negative of a standard involution in a Coxeter group

Consider a finite irreducible Coxeter group $$W$$ with a fixed generator set $$S$$. Every involution in $$W$$ is conjugate to a standard involution $$c_I$$, for some subset $$I\subset S$$. For example, this standard involution can be defined as the element $$−1$$ in a parabolic subgroup $$W_I$$; see [Ka01, §27], [Hu90, §8.3] or this question. Moreover, there are explicit algorithms for determining conjugacy among these standard involutions.

Supposing that $$-1 \in W$$ (which is true for most types), we also have an involution $$-c_I$$ for any subset $$I\subset S$$. My question is: given $$I$$, how can we determine the subset $$J$$ for which $$c_J$$ is conjugate to $$-c_I$$?

It's not possible to determine this based on dimensions of eigenspaces alone. For example, for $$W$$ of type $$B_5$$ or $$C_5$$, it's unclear to me which of the (non-conjugate) involutions $$-c_{\{1\}}$$ and $$-c_{\{5\}}$$ is conjugate to $$c_{\{1,3,4,5\}}$$ and which is conjugate to $$c_{\{2,3,4,5\}}$$.

[Ka01] Richard Kane, Reflection Groups and Invariant Theory
[Hu90] James E. Humphreys, Reflection Groups and Coxeter groups

Concerning the particular example in the question, $$-c_{\{1\}}$$ is conjugate to $$c_{\{1,3,4,5\}}$$, and $$-c_{\{5\}}$$ is conjugate to $$-c_{\{1,2,3,4\}}$$. This can almost be deduced from the dimension of the -1-eigenspaces, but with the following twist. The Dynkin diagram of type $$A_{10}$$ “folds” onto the Dynkin diagram of type $$C_5$$. This “folding” induces a surjection of corresponding sets of simple roots $$\pi\colon \Sigma_{A_{10}} \twoheadrightarrow \Sigma_{B_5}$$ and an associated embedding $$\iota\colon W(C_{5}) \hookrightarrow W(A_{10})$$ of the Weyl group of type $$C_5$$ into the Weyl group of type $$A_{10}$$. (See this question.) The embedding sends $$-1\in W(C_{5})$$ to the longest element $$w_0\in W(A_{10})$$, and its image is the subgroup $$W_0 \subset W(A_{10})$$ of elements that commute with $$w_0$$. More generally:

Claim 1: The embedding $$\iota$$ sends the longest word $$w_I = c_I$$ of the parabolic subgroup associated with a subset of simple roots $$I\subset \Sigma_{B_5}$$ to the longest word $$w_{\pi^{-1}I}$$ of the parabolic subgroup corresponding to the "unfolded" subset $$\pi^{-1}I\subset \Sigma_{A_{10}}$$.

We thus obtain a very explicit bijection between $$\mathcal C :=$$ {conjugacy classes of involutions in $$W(C_{5})$$} and $$\mathcal A :=$$ {$$W_0$$-conjugacy classes of involutions in $$W_0$$}:

Under this bijection, multiplication with $$-1$$ on $$\mathcal C$$ corresponds to multiplication with $$w_0$$ on $$\mathcal A$$. Now let $$l^-(w)$$ denote the dimension of the -1-eigenspace of $$w$$. On the representatives of involutions $$w$$ listed above, the tuple $$((l^-(w),l^-(\iota w))$$ takes the following values: $$(0,0)$$, $$(1,2)$$, $$(1,1)$$, $$(2,4)$$, $$(2,3)$$, $$(2,2)$$, $$(3,5)$$, $$(3,4)$$, $$(3,3)$$, $$(4,4)$$, $$(4,3)$$, $$(5,5)$$. These are all distinct! Therefore, $$-w$$ (for $$w\in \mathcal C$$) and $$w_0w$$ (for $$w\in\mathcal A$$) are uniquely determined by the tuple $$(l^-(-w),l^-(w_0w))$$.

Claim 2: By computing the tuples $$(l^-(-w),l^-(w_0w))$$, we find that:

The second and fourth line answer the question.

## Generalizations

It is straightforward to extend this analysis to $$W(B_n) = W(C_n)$$ for arbitrary $$n$$, and to $$W(F_4)$$ (using the folding of $$E_6$$ onto $$F_4$$). With some more care, the conjugacy class of $$-c_I$$ can also be computed for almost all $$I$$ for $$W(D_{2n})$$, but with a remaining ambiguity concerning the involutions corresponding to:

I do not know how to treat $$W(E_7)$$ and $$W(E_8)$$.

## Details

Claim 1 follows from the explicit description of foldings in the first section of Steinberg, Endomorphisms of Linear Algebraic Groups (1968); see in particular 1.12(c), 1.32(a) and Corollary 1.33. I would be interested whether there is a more direct, group theoretic argument for this claim.

For the calculation of $$l^-(w_0w_I)$$ that is implicit in claim 2, decompose the ambient vector space into the span of $$I$$ and its orthogonal complement, and observe that the restriction of $$w_I$$ to the complement is trivial.

Here is another approach to identify the conjugay class of the specific involution $$-c_{\{1\}}$$ in the question. The approach is complementary to my other answer in that it also allows the determination of the conjugacy classes of $$-c$$ for all involutions in $$W(E_7)$$ and in $$W(E_8)$$.

There are precisely two conjugacy classes of involutions in $$W(C_5)$$ whose $$-1$$-eigenspace is one-dimensional. These are represented by:

Likewise, there are precisely two conjugacy classe of involutions whose $$-1$$-eigenspace is four-dimenensional, and these are represented by:

The question is whether $$-c_{\{1\}} \sim c_{\{1,3,4,5\}}$$ and $$-c_{\{5\}} \sim c_{\{2,3,4,5\}}$$, or whether conversely $$-c_{\{1\}} \sim c_{\{2,3,4,5\}}$$ and $$-c_{\{5\}} \sim c_{\{1,3,4,5\}}$$.

As it turns out, for the approach I'm about to outline, it's best to start with $$c_{\{5\}}$$. The fixed-point space of $$c_{\{5\}}$$ contains the sub-root system generated by the simple roots $$\alpha_1,\alpha_2,\alpha_3$$, since all of these are perpendicular to $$\alpha_5$$. So the fixed-point space of $$c_{\{5\}}$$ contains a closed sub-root system of type $$A_3$$. Equivalently, the $$-1$$-eigenspace of $$-c_{\{5\}}$$ contains a sub-root system of type $$A_3$$, and hence so does the $$-1$$-eigenspace of any conjugate of $$-c_{\{5\}}$$.

Now consider the two candidate standard representatives of the conjugacy class of $$-c_{\{5\}}$$ from $$(*)$$ above. The $$-1$$-eigenspace of the first candidate is spanned by a closed sub-root system of type $$A_1\times C_3$$. The classification of closed sub-root system of Borel and de Siebenthal implies that this sub-root system does not contain any sub-root system of type $$A_3$$. (The maximal closed sub-root systems of a root system of type $$C_3$$ are of types $$A_1\times C_2$$ and $$A_2$$.) So the possiblity $$-c_{\{5\}} \sim c_{\{1,3,4,5\}}$$ can be excluded, and the correct answer must be: $$- c_{\{5\}} \sim c_{\{2,3,4,5\}}$$ $$- c_{\{1\}} \sim c_{\{1,3,4,5\}}$$

Similar arguments resolve the only ambiguities in $$E_7$$ and $$E_8$$. In $$E_7$$, we find that

as the $$-1$$-eigenspaces of the two involutions in the top line contain a sub-root system of type $$A_2$$, while the $$-1$$-eigenspace of the involution on the lower right cannot. In $$E_8$$, we find that

as the $$-1$$-eigenspaces of $$-c$$ and $$c$$ contain a sub-root system of type $$A_2$$, while the $$-1$$-eigenspace of $$c'$$ does not.