On sets of irreducible polynomials closed by composition

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions

Irreducible compositions of degree two polynomials over finite fields have regular structure

Let $q$ be an odd prime power and $D$ be the set of monic irreducible polynomials in $\mathbb F_q[x]$ which can be written as a composition of monic degree two polynomials. In this paper we prove that $D$ has a natural regular structure by showing that there exists a finite automaton having $D$ as accepted language. Our method is constructive.Comment: To appear in The Quarterly Journal of Mathematic

Components of Gr\"obner strata in the Hilbert scheme of points

We fix the lexicographic order $\prec$ on the polynomial ring $S=k[x_{1},...,x_{n}]$ over a ring $k$. We define \Hi^{\prec\Delta}_{S/k}, the moduli space of reduced Gr\"obner bases with a given finite standard set $\Delta$, and its open subscheme \Hi^{\prec\Delta,\et}_{S/k}, the moduli space of families of #\Delta points whose attached ideal has the standard set $\Delta$. We determine the number of irreducible and connected components of the latter scheme; we show that it is equidimensional over ${\rm Spec}\,k$; and we determine its relative dimension over ${\rm Spec} k$. We show that analogous statements do not hold for the scheme \Hi^{\prec\Delta}_{S/k}. Our results prove a version of a conjecture by Bernd Sturmfels.Comment: 49 page

Combinatorial methods of character enumeration for the unitriangular group

Let \UT_n(q) denote the group of unipotent $n\times n$ upper triangular matrices over a field with $q$ elements. The degrees of the complex irreducible characters of \UT_n(q) are precisely the integers $q^e$ with $0\leq e\leq \lfloor \frac{n}{2} \rfloor \lfloor \frac{n-1}{2} \rfloor$, and it has been conjectured that the number of irreducible characters of \UT_n(q) with degree $q^e$ is a polynomial in $q-1$ with nonnegative integer coefficients (depending on $n$ and $e$). We confirm this conjecture when $e\leq 8$ and $n$ is arbitrary by a computer calculation. In particular, we describe an algorithm which allows us to derive explicit bivariate polynomials in $n$ and $q$ giving the number of irreducible characters of \UT_n(q) with degree $q^e$ when $n>2e$ and $e\leq 8$. When divided by $q^{n-e-2}$ and written in terms of the variables $n-2e-1$ and $q-1$, these functions are actually bivariate polynomials with nonnegative integer coefficients, suggesting an even stronger conjecture concerning such character counts. As an application of these calculations, we are able to show that all irreducible characters of \UT_n(q) with degree $\leq q^8$ are Kirillov functions. We also discuss some related results concerning the problem of counting the irreducible constituents of individual supercharacters of \UT_n(q).Comment: 34 pages, 5 table