The variety of nilpotent matrices is $F$-regular

Abstract

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.

Disclosure

“raised by Craig Huneke at RayFest, University of Nebraska-Lincoln, April 2025; we are grateful to Craig and to Linquan Ma for useful discussions. Some of the results were suggested by examples computed with Macaulay2 [GS] and Magma [BCP]; ChatGPT-5.5 Pro was used towards finding suitable references. References [Al] A. A. Albert, Symmetric and alternate matrices in an arbitrary field, I, Trans. Amer. Math. Soc. 43 (1938), 386–436. 10”

PDF page 11

Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T18:54:01.204655Z

Structural counts