Positivity and the canonical basis of tensor products of finite-dimensional irreducible representations of quantum sl(k)

TL;DR

This paper explores the positivity properties of the canonical basis in tensor products of quantum sl(k) representations through categorification, revealing that coefficients are positive due to tilting modules and projective functors.

Contribution

It establishes a categorification framework linking tilting modules to the canonical basis, demonstrating positivity of coefficients in tensor product representations of quantum sl(k).

Findings

Coefficients of the canonical basis are positive under Chevalley generator action.

Tilting modules correspond to the canonical basis in categorification.

Projective functors preserve tilting modules, ensuring positivity.

Abstract

In a categorification of tensor products of fundamental representations of quantum sl(k) via highest weight categories, the indecomposable tilting modules descend to the canonical basis. Since projective functors map tilting modules to tilting modules, the coefficients of the canonical basis of tensor products of finite dimensional, irreducible representations under the action of the Chevalley generators are positive.