Quantum matrix algebras of the $GL(m|n)$ type: The structure and spectral parameterization of the characteristic subalgebra

We continue the study of quantum matrix algebras of the $GL(m|n)$ type. We find three alternative forms of the Cayley–Hamilton identity; most importantly, this identity can be represented in a factored form. The factorization allows naturally dividing the spectrum of a quantum supermatrix into subsets of “even” and “odd” eigenvalues. This division leads to a parameterization of the characteristic subalgebra (the subalgebra of spectral invariants) in terms of supersymmetric polynomials in the eigenvalues of the quantum matrix. Our construction is based on two auxiliary results, which are independently interesting. First, we derive the multiplication rule for Schur functions $s_\lambda(M)$, that form a linear basis of the characteristic subalgebra of a Hecke-type quantum matrix algebra; the structure constants in this basis coincide with the Littlewood–Richardson coefficients. Second, we prove a number of bilinear relations in the graded ring $\Lambda$ of symmetric functions of countably many variables.