Finite reflection groups and linear preserver problems
Let \(G\) be a Coxeter group and \(V\) a Euclidean space such that \(G\subset \text{End}\,V\). Let \({\mathcal L}(G)\) be the set of linear transformations \(\Phi\) in \(\text{ End}\,V\) such that \(\Phi(G)=G\). Let \(P,Q\) be in the normalizer \(N(G)\) of \(G\) in the orthogonal group \(O(V)\) and assume that \(PQ\in G\). Let \({\mathcal RE}(G)\) be the set of transformations \(X\to PXQ\) and \(X\to PX^*Q\), where \(X^*\) is the adjoint of \(X\). Then \({\mathcal RE}(G)\subset{\mathcal L}(G)\). The authors give a complete description of the set \({\mathcal L}(G)\) for each of the four infinite families of finite irreducible Coxeter groups, i.e., of \(A_n\), \(B_n\), \(D_n\), and \(I_2(n)\). They show that \({\mathcal RE}(G)={\mathcal L}(G)\) except for \(G=B_n\).
scientific article; zbMATH DE number 2103626