Simple modules for affine nilCoxeter algebras

Abstract

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.