On the presentation of Hecke-Hopf algebras for non-simply-laced type

TL;DR

This paper provides explicit presentations of Hecke-Hopf algebras for non-simply-laced Coxeter groups with specific parameters, proving some conjectures and providing counterexamples for others.

Contribution

It offers explicit presentations of Hecke-Hopf algebras for certain parameters and proves or disproves related conjectures in the non-simply-laced case.

Findings

Confirmed conjecture for crystallographic non-simply-laced Coxeter groups when m_{ij} is 4 or 5.

Proved conjecture for m_{ij} equals 4.

Counterexample showing conjecture does not hold when m_{ij} equals 6.

Abstract

Hecke-Hopf algebras were defined by A. Berenstein and D. Kazhdan. We give an explicit presentation of an Hecke-Hopf algebra when the parameter $m_{ij},$ associated to any two distinct vertices $i$ and $j$ in the presentation of a Coxeter group, equals $4,$ $5$ or $6$. As an application, we give a proof of a conjecture of Berenstein and Kazhdan when the Coxeter group is crystallographic and non-simply-laced. As another application, we show that another conjecture of Berenstein and Kazhdan holds when $m_{ij},$ associated to any two distinct vertices $i$ and $j,$ equals $4$ and that the conjecture does not hold when some $m_{ij}$ equals $6$ by giving a counterexample to it.