\(\Theta\)-semisimple classes of type D in \(\mathrm{PSL}_{n}(q)\) (Q2810927)

A rack \((X, \triangle)\) is a non-empty finite set \(X\) together with a function \(\triangle: X \times X \rightarrow X\) such that \(i \triangle (\cdot): X \rightarrow X\) is a bijection for all \(i \in X\) and \(i \triangle(j \triangle k) = (i \triangle j) \triangle(i \triangle k)\), for all \(i, j, k \in X\). A subrack \(Y\) of \(X\) is a subset of \(X\) such that \(Y \triangle Y \subseteq Y\). A rack is said to be indecomposable if it cannot be decomposed as the disjoint union of two subracks. A rack \(X\) is of type \(D\) when there exists a decomposable subrack \(Y = R \cup S\) of \(X\) and elements \(r \in R\), \(s \in S\) such that \(r \triangle (s \triangle (r \triangle s)) \neq s\). Let \(H\) be a group and \(\psi \in \mathrm{Aut}(H)\). A \(\psi\)-twisted conjugate class of \(h \in H\) is an orbit of \(H\) for the action of \(H\) on itself by \(h \bullet_{\psi}x = hx \psi (h)^{-1}\) denoted by \({O}_h^{\psi}\). Then, \({O}_h^{\psi}\) is a rack with respect to \(\psi\) with \(y \triangle z = y \psi (zy^{-1})\).NEWLINENEWLINE Let \(K\) be a stable subgroup of \(H\) under \(\psi\), \({O}_h^{\psi, K}\) the orbit of \(h\) under the restriction of the \(\bullet_{\psi}\)-action to \(K\). For \(n \in \mathbb{N}\), \(p \in \mathbb{N}\) a prime, \(\mathbb{K} = \overline{\mathbb{F}_p}\), \(q = p^m\) for some \(m \in \mathbb{N}\), \(G = \mathrm{PSL}_n(q)\), \(n \gg 2\), or \(q \neq 2,3\), the authors present some general techniques to deal with twisted conjugacy classes in finite groups and establish a systematic approach to twisted classes in \(G\). Conditions are given under which \({O}_h^{\psi, G}\) is of type \(D\) for different \(\psi\) and \(h\). It is also shown that \(O_1^{\theta, \mathrm{PSL}_4(3)}\) is not type \(D\) where \(\theta\) is a standard graph automorphism of the Dynkin diagram induced by an algebraic group automorphism of \(\mathrm{SL}_n(\mathbb{K})\).NEWLINENEWLINE Theorem. Let \(p\) be an odd prime and \(\psi = d \circ \theta \in \mathrm{Aut}(G)\) for a diagonal automorphism \(d\) induced by an algebraic group automorphism of \(\mathrm{SL}_n(\mathbb{K})\). If \(n \geq 5, \;q \geq 7\), then any \((\psi, p)\)-regular \(((|h \psi|, p) = 1)\) class \({O}_h^{\psi, G}\) is of type \(D\) with some exceptions. The study is also connected with the classification problem of finite-dimensional pointed Hopf algebras.