A stable range for dimensions of homogeneous \(O(n)\)-invariant polynomials on the \(n\times n\) matrices (Q5945131)

The complex orthogonal group \(O(n)\) acts on the space \(M_n\) of \(n\times n\) matrices by conjugation. Denote by \(R_n\) the ring of \(O(n)\)-invariant polynomial functions on \(M_n\). The main object of study here is the Hilbert series \(H_n(q)=\sum_{m=0}^\infty h_{n,m}q^m\) of the graded algebra \(R_n\). A generating system of \(R_n\) was given by \textit{K. S. Sibirkij} [Sib. Mat. Zh. 9, 152-164 (1968; Zbl 0273.15024)] and by \textit{C. Procesi} [Adv. Math. 19, 306-381 (1976; Zbl 0331.15021)]. In the present paper a combinatorial consequence of this is formulated as follows. Let \(c_k\) denote the number of \(k\) vertex cyclic graphs with directed edges, counted up to dihedral symmetry. Then for all \(m\) we have \(h_{n,m}\leq d_m\), where \(d_m\) is the coefficient of \(q^m\) in \(\prod_{k=1}^\infty(1-q^k)^{-c_k}\). The author then expresses both \(h_{n,m}\) and \(d_m\) as sums of Littlewood-Richardson coefficients, using the Cauchy-Littlewood identity and the Cartan-Helgason theorem in the case of \(h_{n,m}\), and the representation theory of the symmetric group and some elementary combinatorial arguments in the case of \(d_m\). Comparing these expressions the equality \(h_{n,m}=d_m\) for \(m\leq n\) is obtained. The paper contains some numerical data, for example, for \(n=1,2,3\), the Hilbert series \(H_n(q)\) is expressed as a rational function.NEWLINENEWLINENEWLINEReviewer's remark: The fact that \(h_{n,m}=d_m\) for \(m\leq n\) follows also from Proposition 8.3 (c) in the above mentioned paper of Procesi, stating that (after multilinearization) there are no non-trivial relations among the given generators of \(R_n\).