Root systems, affine subspaces, and projections

TL;DR

This paper investigates the combinatorial and algebraic structures of root systems, focusing on affine subspace intersections and projections, providing new proofs, partial resolutions, and enumerative insights into root system properties.

Contribution

It introduces novel combinatorial and algebraic methods to analyze root systems, including new proofs, partial solutions to open problems, and enumeration results.

Findings

Refined combinatorial proofs of Oshima's Lemma and Kostant's result

Partial progress on a problem by Hopkins and Postnikov

New enumerative formulas for root systems

Abstract

We tackle several problems related to a finite irreducible crystallographic root system $\Phi$ in the real vector space $\mathbb E$. In particular, we study the combinatorial structure of the subsets of $\Phi$ cut by affine subspaces of $\mathbb E$ and their projections. As byproducts, we obtain easy algebraic combinatorial proofs of refinements of Oshima's Lemma and of a result by Kostant, a partial result towards the resolution of a problem by Hopkins and Postnikov, and new enumerative results on root systems.