2. Conjugacy ClassesΒΆ

Let us recall that the automorphism \(\phi\) induces a natural equivalence relation on the group \(W\), which we call \(\phi\)-conjugacy. In particular, we say two elements \(x,y \in W\) are \(\phi\)-conjugate if there exists an element \(z \in W\) such that

\[x = z^{-1}y\phi(z).\]

The equivalence classes under this relation are known as the \(\phi\)-conjugacy classes of \(W\), which we denote by \(H^1(\phi,W)\).

Definition

We say \(C \subset W\phi\) is a \(\phi\)-conjugacy class if any one of the following equivalent conditions is satisfied:

  • \(C\) is a \(W\)-orbit under the natural conjugation action of \(W\) on \(W\phi\).
  • \(C\) is a conjugacy class of \(\widetilde{W}\) contained in the coset \(W\phi\).
  • the set \(\{w \in W \mid (w, \phi) \in C\}\) is a \(\phi\)-conjugacy class of \(W\).

We now consider how we may reduce the problem of determining \(\phi\)-conjugacy classes to the case where \(W\) is irreducible. First let us write \(W\) as a direct product \(W^{(1)} \times \cdots \times W^{(r)}\) such that each \(W^{(i)}\) is an orbit of \(\phi\) acting on the irreducible factors of \(W\). We will denote by \(\phi^{(i)}\) the restriction of \(\phi\) to the orbit \(W^{(i)}\) then the following is obvious

Lemma

The natural product map

\begin{align*} W^{(1)} \times \cdots \times W^{(r)} &\to W\\ (w_1,\dots,w_r) &\mapsto w_1\cdots w_r \end{align*}

induces a bijection

\[H^1(\phi^{(1)},W^{(1)}) \times \cdots H^1(\phi^{(r)},W^{(r)}) \to H^1(\phi, W)\]

In other words \(W^{(i)} = W^{(i)}_1 \times \cdots \times W^{(i)}_{m_i}\) and \(\phi\) cyclically permutes the factors \(W^{(i)}_j\).

The following function will produce the \(\phi\)-conjugacy classes of any Coxeter coset.