Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?

Question

Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating function over $k$. That goes back to Feit, Fine 1960, and variations of the problem attracted certain interest recent years (some refs here).

The question about $n$-tuples of commuting (and variations) seems to be less studied. Computational experiments suggest the following pattern. Number of $n$-tuples of commuting matrices over $F_p$ is given:

• polynomials in $p$

• dependence on $n$ of these polynomials is quite simple: for example for 2x2 matrices: $p^{2n}(p^2+p+1) - p^{n+1}(p+1)$, i.e. there are some "basic" polynoms $P_i(p)$ which does NOT depend on $n$ and dependence on $n$ only appears by multiplying these basic polynoms by some power of $p$ - so the general pattern is: $\sum_i p^{a_i(n)} P_i(p)$.

Question 1: Is that observation true ? Is it true for variations - we can consider matrices to be nilpotent, unitriangular, (anti)-symmetric, etc. ? (computations suggest it is true). More generally one can consider "Manin matrix variety" which is generalization of the "commuting variety" and computations suggest the same pattern, is it true ?

Question 2: If it is true what is the meaning of these polynoms $P_i(p)$ ? What can be their number and general form for $k \times k$ matrices ? Is there any procedure which can convert generating function for pairs of commuting matrices to generating function for $n$-tuples of commuting matrices ?

Some more examples (and see more below):

• $p^{2n}(p+1) -p^{n+1}$ Uppertriangular 2x2
• $p^{3n}(p^3+3p^{2}+p^{1}+1) -3p^{2n}(p^3+3p^{2}+p^{1}) + 2p^{n+3}$ Uppertriangular 3x3
• $p^{n}(p+1)+1 -p$ Nilpotent 2x2
• $p^{2n}(p^4+p^{3}+2p^{2}+p^{1}+1) -p^{n}(p^4+2p^{3}+2p^{2}+p^{1}) + p^3$ Nilpotent 3x3

The csv-file with results of computations can be found here or google sheet or see screenshot below. The main computations done in the notebook, for Manin matrices here.

Note 1: for anticommuting matrices the pattern seems to be more complicated.

Note 2: For standard commuting matrices (i.e. without constraints of nilpotency/unitriangular/etc) the generating function over matrix sizes $k$ is probably given by the q-analogue of the exponential formula (Yoshida 1992) (see MOenter link description here ). But formally speaking Yoshida results are proven for finite groups, and not $Z^n$ which is necessary for that application. Most probably that is just technical issue, but would be better to be sure. Results for variations - nilpotency/unitriangular/etc or Manin matrices seems not covered by that q-exponential formula.

Note 3: Typically value at p=1 for polynoms is $1$ that can be explained - except all zeros tuples we have free actions of $F_p^{*}$ - so polynom minus 1 should be divided by $(p-1)$. But also values at $p=-1$ mostly 1 - explanation is not clear.

Note 4: Here is related question , but it remains unanswered in n-tuple part for years.

Thanks: The polynomials above were computed with the help of Peter Taylor code appeared in the mathoverflow answer on related question.

Answer 1

Let $p,k$ be fixed and fix a linear subspace $M$ of $M_k(\mathbb F_p)$. Let $\Lambda$ be the set of linear subspaces $V$ of $M$ which satisfy the condition that every two elements of $V$ commutes with each other. Then $\Lambda$ is a lattice (ordered by inclusion).

For each $V\in\Lambda$ let $g_V(n)$ be the number of $n$-tuples of elements in $V$ which span $V$. Let $f(n)$ be the number of commuting $n$-tuples in $M$. Then we have: $$ f(n) = \sum_{V\in\Lambda} g_V(n) $$ (because every commuting $n$-tuple spans exactly one subspace from $\Lambda$) and for every $V\in\Lambda$, $$ p^{n\dim V} = \sum_{\substack{W\in\Lambda\\W\subset V}} g_W(n) $$ using Mobius inversion we can write $$ g_V(n) = \sum_{\substack{W\in\Lambda\\W\subset V}}\mu(V,W)p^{n\dim W} $$ therefore $$ f(n) = \sum_{W\in\Lambda}p^{n\dim W}\sum_{\substack{V\in\Lambda\\V\supset W}}\mu(V,W) $$ and your second observation follows.