Eigenvalues connected with Brauer's centralizer algebras (Q909796)

The authors continue to study the situation described in the preceding paper [see the preceding review Zbl 0695.20026]. Use the notation of the previous review. The simple components of the algebras \({\mathfrak A}_ f^{(n)}\) and \({\mathfrak B}_ f^{(n)}\) modulo their radicals are indexed by pairs (\(\lambda\),\(\mu)\) with \(\mu\) a partition of f and \(\lambda\) a partition of some number \(m\leq f\) such that f-m is even. The dimension of the matrix ring corresponding to the pair (\(\lambda\),\(\mu)\) is equal to the rank of a certain matrix \(Z_ n(\lambda,\mu)\). For the case where \(\mu\) has exactly \(| \lambda |\) odd parts which all are distinct the authors show that \(Z_ n(\lambda,\mu)\) is a scalar \(h_ n(\lambda,\mu)E\) where \(h_ n(\lambda,\mu)\) is a polynomial with integer roots.