Vladimir Sergeichuk
Institute of Mathematics, Kiev, Ukraine
Operators on unitary spaces.
Abstract: A unitary representation of a quiver is given by assigning to each vertex a unitary space and to each arrow a linear operator. I recall Littlewood's algorithm for reducing a square complex matrix to a canonical form with respect to unitary similarity and extend it to matrices of unitary representations of quivers. Describe the set of dimensions of indecomposable unitary representations of a quiver. Give the number of parameters in an indecomposable unitary representation of a given dimension.
Each square complex matrix is unitarily similar to an upper triangular matrix with lexicographically ordered eigenvalues. Let A and B be two upper triangular n-by-n matrices in general position with lexicographically ordered eigenvalues, Then A and B are unitarily similar if and only if ||h(Ak)||=|| h(Bk)|| for all complex polynomials h(x) and k=1,…,n, where Ak and Bk are the principal k-by-k submatrices of A and B and || || is the Frobenius norm.