Generic movement of eigenvalues for equivariant self-adjoint matrices
In the numerical treatment of bifurcation problems one of the main tasks is to monitor the spectrum of matrices in a parametrized family. If the original problem possesses a geometrical symmetry with respect to a group of transformations, and if the discretization of the problem also respects this symmetry, then the Jacobian matrices arising in the linearization of the problem are also equivariant, i.e., \(\gamma A_\lambda = A_{\lambda \gamma}\) for all \(\gamma \in \Gamma\) where \(\Gamma \subset \text{GL} (n)\) is a compact Lie group. Let \(M^n_\Gamma\) denote the subset of self-adjoint \(n \times n\) real matrices which are equivariant with respect to \(\Gamma\). A subspace \(V\) of \(\mathbb{R}^n\) is \(\Gamma\) invariant if \(\gamma V \subset V\) holds for all \(\gamma \in \Gamma\). For a \(\Gamma\) invariant subspace, there is a unique decomposition into isotypic components \(V = W_1 \otimes \cdots \otimes W_k\) such that the subspaces \(W_j\) are invariant for each linear map \(A : V \to V\) which is equivariant. If the isotypic decomposition of an eigenspace of a matrix \(B \in M^n_\Gamma\) contains more than one irreducible subspace, this is called a \(\Gamma\) resonance for \(B\). A sample result of the paper is: Suppose the family \(A_\lambda \subset M^n_\Gamma\) undergoes a \(\Gamma\) resonance for \(\lambda = 0\) and \(E\) is the corresponding eigenspace. Then generically, \(E = U_1 \otimes U_2\), where \(U_1\) and \(U_2\) are \(\Gamma\) irreducible. The authors then discuss the cases when \(U_1\) and \(U_2\) are and are not isomorphic.
scientific article; zbMATH DE number 758547