A homomorphism between Bott-Samelson bimodules

TL;DR

This paper proves the existence of specific homomorphisms between Bott-Samelson bimodules, which was previously assumed, by establishing conditions based on vanishing two-colored quantum binomial coefficients.

Contribution

It provides a rigorous proof for the existence of homomorphisms between Bott-Samelson bimodules under certain conditions, advancing the categorification of the Hecke algebra.

Findings

Homomorphisms between Bott-Samelson bimodules are proven to exist.

The proof relies on the vanishing of specific two-colored quantum binomial coefficients.

This result solidifies foundational assumptions in the categorification framework.

Abstract

In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott-Samelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.