The Rank One property for free Frobenius extensions

TL;DR

This paper proves that under certain algebraic conditions, the multiplicity matrix for central quotients of prime Frobenius extensions has rank one, offering a new proof for a conjecture related to rational Cherednik algebras.

Contribution

It establishes the rank one property for multiplicity matrices in a broad class of algebras, including Cherednik algebras, using Frobenius extension theory.

Findings

The rank one property holds for central quotients of prime Frobenius extensions.

The result applies to graded Hecke algebras, quantum groups, and non-commutative resolutions.

Provides a new proof of the conjecture for rational Cherednik algebras.

Abstract

A conjecture by the second author, proven by Bonnaf\'e-Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra. In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Skylanin algebras. In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.