Tensor product multiplicities, canonical bases and totally positive varieties

TL;DR

This paper introduces polyhedral combinatorial formulas for tensor product multiplicities in semisimple Lie algebras, utilizing new $ ext{ii}$-trail concepts and exploring the link between canonical bases and totally positive varieties.

Contribution

It provides explicit polyhedral expressions for tensor multiplicities and clarifies the relationship between different canonical basis parametrizations using geometric and tropical methods.

Findings

Polyhedral formulas express multiplicities as lattice points in convex polytopes.

New $ ext{ii}$-trail combinatorics simplifies the computation of tensor multiplicities.

Explicit relationship between Lusztig's and Kashiwara's canonical basis parametrizations.

Abstract

We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approaches were shown by Lusztig to be equivalent to each other, they lead to different combinatorial parametrizations of the canonical bases. One of our main results is an explicit description of the relationship between these parametrizations. Our approach to the above problems is based on a remarkable observation by G. Lusztig that combinatorics of the canonical basis is closely related to geometry of the totally positive varieties. We formulate this relationship in terms of two mutually inverse transformations: "tropicalization" and "geometric lifting."