On $A_1^2$ restrictions of Weyl arrangements

TL;DR

This paper investigates the $A_1^2$ restrictions of Weyl arrangements, providing a combinatorial description of their exponents and bases for derivation modules, advancing understanding of restriction exponents in algebraic combinatorics.

Contribution

It offers a new combinatorial characterization of the exponents and derivation bases for $A_1^2$ restrictions of Weyl arrangements, extending prior work on $A_1$ restrictions.

Findings

Provides a combinatorial description of $A_1^2$ restriction exponents.

Describes bases for derivation modules using related roots.

Advances understanding of restriction exponents in Weyl arrangements.

Abstract

Let $\mathcal{A}$ be a Weyl arrangement in an $\ell$-dimensional Euclidean space. The freeness of restrictions of $\mathcal{A}$ was first settled by a case-by-case method by Orlik and the second author (1993), and later by a uniform argument by Douglass (1999). Prior to this, Orlik and Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik, Solomon and the second author (1986), asserts that the exponents of any $A_1$ restriction, i.e., the restriction of $\mathcal{A}$ to a hyperplane, are given by $\{m_1,\ldots, m_{\ell-1}\}$, where $\exp(\mathcal{A})=\{m_1,\ldots, m_{\ell}\}$ with $m_1 \le \cdots\le m_{\ell}$. As a next step towards conceptual understanding of the restriction exponents we will investigate the $A_1^2$ restrictions, i.e., the restrictions of $\mathcal{A}$ to the subspaces of type $A_1^2$. In this paper, we give a combinatorial description of the exponents and describe bases for the modules of derivations of the $A_1^2$ restrictions in terms of the classical notion of related roots by Kostant (1955).