Partial normalizations of Coxeter arrangements and discriminants

TL;DR

This paper explores partial normalization spaces of Coxeter arrangements and discriminants, linking their geometry to representation theory and Frobenius structures, and introduces new free divisors through these constructions.

Contribution

It introduces a novel approach to partial normalizations of Coxeter arrangements, connecting them with Frobenius manifolds and Cohen--Macaulay ideals, and identifies new free divisors.

Findings

Derived 3rd order differential relations for Coxeter invariants

Linked normalization spaces to Frobenius manifold structures

Discovered new free divisors from partial normalizations

Abstract

We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin's Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also describe an independent approach to these structures via duality of maximal Cohen--Macaulay fractional ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter group. Finally, we show that our partial normalizations give rise to new free divisors.