Consider for example commutative polynomial algebra $C[x,y]: xy=yx$, look on $F_p$-matrices satisfying that relation - the number over $F_p$ will be given by polynomial in $p$ (classical result due to Feit and Fine and that is quite rare fact for algebraic varieties).
Question 1: What are the some conditions/examples of algebras/groups which will ensure that number of $F_p$ $n\times n$ matrices representing the algebra is given by polynomial for all $n$ and generic $p$ ?
Some examples (not all proved in full generality, but no big doubts ) : commutative polynomial algebra (e.g. MO) , its Koszul dual - Grassman alegbra, more generally take q-commuting (for example q=-1 MO, MO ) polynomials, most probably Manin's relations (which are "Manin's non-commutative endomorphisms") of (q)-polynomial algebras: MO. Fundamental group of genus g closed Riemann surface Σ : Hausel, Rodriguez-Villegas.
From the examples above it is natural to hope for:
Question 2: If quadratic algebra $A$ is "polynomial count" in the sense above , would be it true for its Koszul dual ? For "Manin's non-commutative endomorphisms" (MO) $End(A)$ ? More generally for algebras $A,B:$ will it be true for $End(A,B)$ (same as Manin's black product of $A$ and $B$, i.e. Koszul dual to its Segre product) ?
Typically the generating function over sizes of matrices for such counting polynomials is given by some beautiful formulas in spirit of q-exponential formula (which are related to things like Donaldson-Thomas theory , e.g. O.Schiffmann). See also: Yifeng Huang "Generating function from counting mutually annihilating matrices and alike" https://yifeng-huang-math.github.io/files/talk_gocc.pdf
Question 3: How to compute such generating function of count polynomials over $n$ in terms of initial algebra $A$ ?
PS
The number of examples might be probably extended by producing new algebras from the old ones by various nilpotency conditions: e.g. start with algebra $A$, consider its Manin's $End(A)$ and impose additional relation $M^l = 0$ (see MO ), one may probably extend it to polynomial nilpotent conditions: MO .
