Explicit rational forms for the Poincaré series of the trace rings of generic matrices (Q1178338)

Let \(V\) be an \(n\)-dimensional vector space over \(C\). Let \(G=SL(V)\), \(W=(V\otimes V^*)^ m\), \(R=SW\), \(\bar R=\text{End}(V)\otimes R\). Let \(Z_{m,n}=R^ G\) and \(T_{m,n}=\bar R^ G\) for the obvious \(G\)- actions. \(Z_{m,n}\) and \(T_{m,n}\) are called the commutative and non- commutative trace ring of m generic \(n\times n\) matrices. There is a natural \(N^ m\)-grading on \(R\) and \(\bar R\), and \(Z_{m,n}\) and \(T_{m,n}\) are graded subrings so their Hilbert series can be defined. In this paper formulas for these Hilbert series are given. The author uses the Molien-Weyl formula which leads to an expression in terms of generating functions for flows in certain graphs. These generating functions are computed with graph theory.