The Saito determinant for Coxeter discriminant strata

TL;DR

This paper computes the determinant of the Saito metric on Coxeter discriminant strata, revealing its factorization properties and geometric multiplicities, thus generalizing Coxeter group Jacobian factorizations within Frobenius structures.

Contribution

It provides a formula for the determinant of the Saito metric on Coxeter discriminant strata, extending Coxeter factorization concepts to these geometric subsets.

Findings

Determinant is proportional to a product of linear factors in flat coordinates.

Multiplicities of factors are expressed via Coxeter geometry.

Results generalize Coxeter Jacobian factorization to discriminant strata.

Abstract

Let $W$ be a finite Coxeter group and $V$ its reflection representation. The orbit space $\mathcal{M}_W= V/W$ has the remarkable Saito flat metric defined as a Lie derivative of the $W$-invariant bilinear form $g$. We find determinant of the Saito metric restricted to an arbitrary Coxeter discriminant stratum in $\mathcal{M}_W$. It is shown that this determinant is proportional to a product of linear factors in the flat coordinates of the form $g$ on the stratum. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum. This result may be interpreted as a generalisation to discriminant strata of the Coxeter factorisation formula for the Jacobian of the group $W$. As another interpretation, we find determinant of the operator of multiplication by the Euler vector field in the natural Frobenius structure on the strata.