Some branching formulas for Kac--Moody Lie algebras

In this paper we give some branching rules for the fundamental representations of Kac--Moody Lie algebras associated to $T$-shaped graphs. These formulas are useful to describe generators of the generic rings for free resolutions of length three described in [JWm16]. We also make some conjectures about the generic rings

Young tableaux, canonical bases and the Gindikin-Karpelevich formula

A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux.Comment: 19 page