A Stable Range for Dimensions of Homogeneous O(n)-Invariant Polynomials on the n × n Matrices

Abstract

The complex orthogonal group O(n) acts on the n×n matrices, Mn, by restricting the adjoint action of GL(n,C). This action provides us with an action on the ring of complex valued polynomial functions on the n×n matrices, P(Mn). The polynomials of degree d, denoted Pd(Mn), form a finite dimensional representation of O(n) and provide a graded module structure on P(Mn) as well as the subring of invariant polynomials, P(Mn)O(n). For 0≤d≤n, it is shown that dim Pd(Mn)O(n) is equal to the coefficient of qd in Π∞k=1(1/(1−qk))ck, where ck is the number of k vertex cyclic graphs with directed edges counted up to dihedral symmetry. The above formula provides a combinatorial interpretation of an initial segment of the Hilbert series for this ring.