Non-gatherable triples for classical affine root systems

TL;DR

This paper provides a comprehensive classification of minimal non-gatherable triangle triples in lambda-sequences for affine classical root systems, enhancing understanding of their combinatorial structure and implications for affine Hecke algebra representations.

Contribution

It offers a complete description of non-gatherable triples in affine classical root systems and relates these findings to the non-affine case, clarifying their combinatorial and algebraic significance.

Findings

Complete classification of non-gatherable triples in affine classical root systems

Identification of combinatorial obstacles in affine Hecke algebra representations

Connections established between affine and non-affine root system classifications

Abstract

This paper contains a complete description of minimal non-gatherable triangle triples in the lambda-sequences for the affine classical root systems and some claims for arbitrary (reduced) affine root systems. It continues our previous paper devoted to the non-affine case; interestingly, the affine theory clarifies the classification in the non-affine case. The lambda-sequences are associated with reduced decompositions (words) in affine Weyl groups. The existence of the non-gatherable triples is a combinatorial obstacle for using the technique of intertwiners in the theory of irreducible representations of the (double) affine Hecke algebras, complementary to their algebraic-geometric theory.