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$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(t)$ (with say, integer coefficients) such that $r_n(q) = p_n(q)$ for all prime powers $q$?

Note: for the free algebra $\GFq\langle x, y\rangle$ this is well known, since it is the number of $n$-dimensional representations of the quiver with one vertex and two loops. In fact, it is also known now that this polynomial has non-negative integer coefficients (this follows from the work of Mozgovoy or Hausel, Letellier and Rodriguez-Villegas).

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n(t)$ (with say, integer coefficients) such that $r_n(q) = p_n(q)$ for all prime powers $q$?

Here are the values that I know (see arXiv:1212.6157v2):

• $r_1(q) = q^2$
• $r_2(q) = q^4 + q^3 + q^2$
• $r_3(q) = q^6 + q^5 + 2q^4 + q^3 + 2q^2$
• $r_4(q) = q^8 + q^7 + 3q^6 + 3q^5 + 5q^4 + 3q^3 + 3q^2$

Note: for the free algebra $\GFq\langle x, y\rangle$ this is well known, since it is the number of $n$-dimensional representations of the quiver with one vertex and two loops. In fact, it is also known now that this polynomial has non-negative integer coefficients (this follows from the work of Mozgovoy or Hausel, Letellier and Rodriguez-Villegas).