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 dime...
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 maxi...
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 prin...
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...
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....
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 class...
Liam Buttitta, Ada Stelzer· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
