We give an elementary proof that the coordinate ring of the variety of nilpotent matrices is $F$-regular; over an infinite field $K$, this ring also arises as the nullcone for the conjugation action of the general linear group $\textrm{GL}_n(K)$ on the polynomial ring $K[X]$, where $X$ is an $n\times n$ matrix of indeterminates. We prove that the divisor class group of the coordinate ring is the cyclic group $\mathbb{Z}/n\mathbb{Z}$. We then study the case of symmetric nilpotent matrices, where the picture is completely different: the coordinate ring is not normal for $n\geqslant 2$; for $K$ algebraically closed of characteristic other than two, we prove that the coordinate ring is an integral domain precisely when $n$ is odd.
We study the representation theory of the affine nilCoxeter algebra $A$ of type $\tilde A_{n-1}$, over a field $k$ of any characteristic. Our main theorem states that this is a Noetherian prime affine PI algebra of PI degree $n!$. As a consequence, the simple $A$-modules are all finite dimensional, and the maximum dimension of a simple module is $n!$ over a suitable finite extension of $k$. To achieve this, we investigate a large commutative subalgebra $C$ which is finitely generated as an algebra and over which $A$ is finitely generated as a module. We show that the associated primes of $C$ are minimal primes, and there are $n!$ of them, regularly permuted by $\mathfrak{S}_n$. The algebra $R=C^{\mathfrak{S}_n}$ is equal to the centre of $A$, and isomorphic to $C/\mathfrak{p}$ for each of the minimal primes $\mathfrak{p}$. We prove that the ring $R$ is isomorphic to $k[X_1,\dots,X_{n-1}]^{\mu_n}$, where $\mu_n$ is the finite group scheme of $n$th roots of unity, acting so that $X_i$ has degree $i$ modulo $n$. The ring $R$ is Cohen--Macaulay, and is Gorenstein if and only if $n$ is odd or $n=2$. It is a toric ring, with divisor class group $\mathsf{Cl}(R)\cong\mathbb{Z}/n$, and every projective $R$-module is free.
We investigate the properties of multivariate polynomials evaluated at points with commuting coordinates over division rings and octonion algebras. Given a division ring $D$, this set of points is denoted by $D_c^n$, and in the special case of $D=\mathbb{H}$, Alon and Paran showed that its points correspond to the maximal left ideals of $\mathbb{H}[x_1,\dots,x_n]$. Here we show that if a polynomial $f$ vanishes at $\vec{a} \in D_c^n$, then any left multiple $gf$ also vanishes at $\vec{a}$. Consequently, over a central division algebra or an octonion algebra, any root of $f$ in $D_c^n$ is also a root of its (reduced) norm. We apply these evaluation properties to discrete algebraic dynamics, proving that if a point in $D_c^n$ is a fixed point of an $n$-tuple $T=(f_1,\dots,f_n)$ of polynomials in $n$ variables, then it is a fixed point of $T^{\circ m}$ for any positive integer $m$.
For $\rho \in (0,1]$, let $R(\rho) = \mathbb{Z}[[z]] \cap O(B(0,\rho))$ denote the ring of power series with integer Taylor coefficients converging on the open disk $B(0,\rho)$. We prove that these rings are pairwise non-isomorphic as abstract rings. Three ingredients drive the proof: the ideal $(z)$ is the unique principal ideal with quotient $\mathbb{Z}$, so any isomorphism sends $z$ to a generator $g$ of $(z)$; every isomorphism is substitution by $g$, because it respects the $(z)$-adic filtration; and a Hadamard gap series with a natural boundary at $|w| = \rho_1$ forces the image $g(B(0,\rho_2))$ into $B(0,\rho_1)$, after which the Schwarz lemma and integrality of coefficients force $g = \pm z$ and $\rho_1 = \rho_2$.
In this work, we show that if $h(x)$ is an irreducible monic integer polynomial of degree $2q$ (with $q$ prime), whose defining field extension of $\mathbb{Q}$ contains a Galois extension of degree $q$, then there is a positive density of integers $n$ such that $h(n)$ is squarefree; in particular, $h(n)$ is squarefree for infinitely many integers $n$. As an application, we prove that the family of exceptional cubic fields contains an infinite subfamily whose unit shapes converge to the hexagonal lattice. To the best of our knowledge, this is the first example of a family of non-Galois totally real cubic fields whose unit shapes converge to the hexagonal lattice.
Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0<k<\min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood--Richardson tableaux.
Liam Buttitta, Ada Stelzer· 0 citations
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