# Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of $\mathbb{Z}^N$. Denote by $A_0$ the 0-weight sub-algebra of $A$. Let $d_j$ be the dimension of the vector space of polynomials of degree $j$ in $A_0$. Is the series $\sum_j d_jt^j$ known for $N>3$?

• This series is a coefficient of $1=t_1^0\dots t_N^0$ in a double product $$\prod_{i,j=1}^N \frac1{1-\frac{t_i}{t_j}x}.$$ It is nothing but reformulation which may be helpful for somebody to make a more explicit answer. – Fedor Petrov Feb 20 '16 at 18:19