The freeness of ideal subarrangements of Weyl arrangements

TL;DR

This paper proves that all ideal subarrangements of Weyl arrangements are free and provides explicit exponents, confirming a conjecture and extending classical formulas in Lie theory.

Contribution

It establishes the freeness of ideal subarrangements of Weyl arrangements and determines their exponents via the dual partition of the height distribution, confirming a conjecture by Sommers-Tymoczko.

Findings

All ideal subarrangements are free arrangements.

Exponents are given by the dual partition of the height distribution.

Recovers the classical formula for the entire Weyl arrangement.

Abstract

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by Sommers-Tymoczko. In particular, when an ideal subarrangement is equal to the entire Weyl arrangement, our main theorem yields the celebrated formula by Shapiro, Steinberg, Kostant, and Macdonald. Our proof of the main theorem heavily depends on the theory of free arrangements and thus greatly differs from the earlier proofs of the formula.