Non-Gatherable Triples for Non-Affine Root Systems

TL;DR

This paper characterizes minimal non-gatherable triangle triples in lambda-sequences for classical root systems, highlighting their role as combinatorial obstacles in the representation theory of affine Hecke algebras.

Contribution

It provides a complete description of these triples for $F_4$ and $E_6$, advancing understanding of their combinatorial structure.

Findings

Identifies minimal non-gatherable triples in classical root systems.

Shows these triples are obstacles in using intertwiners for representation descriptions.

Completes the classification for $F_4$ and $E_6$.

Abstract

This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the classical root systems, $F_4$ and $E_6$. Such sequences are associated with reduced decompositions (words) in affine and non-affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners for an explicit description of the irreducible representations of the (double) affine Hecke algebras, complementary to their algebraic-geometric theory.