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
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
induces a bijection
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.