2026/07/27 by David J. Benson, Kay Jin Lim
Mathematics · #math.RT
We study the representation theory of the affine nilCoxeter algebra A of type An-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 \mathfrakSn. The algebra R=C^\mathfrakSn is equal to the centre of A, and isomorphic to C/\mathfrakp for each of the minimal primes \mathfrakp. We prove that the ring R is isomorphic to k[X1,…,Xn-1]μn, where μn is the finite group scheme of nth roots of unity, acting so that Xi 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 Cl(R)≅ℤ/n, and every projective R-module is free.