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

The complex orthogonal group O(n) acts on the n × n matrices, Mn, by restricting the adjoint action of GL(n, ). This action provides us with an action on the ring of complex valued polynomial functions on the n × n matrices, (Mn). The polynomials of degree d, denoted d(Mn), form a finite dimensional representation of O(n) and provide a graded module structure on (Mn) as well as the subring of invariant polynomials, (Mn)O(n). For 0 ≤ d ≤ n, it is shown that dim d(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.