Crystal Bases and Young Tableaux

Crystal Bases and Young Tableaux

Correspondence between Young Walls and Young Tableaux Realizations of Crystal Bases for the Classical Lie Algebras

We give a 1-1 correspondence with the Young wall realization and the Young tableau realization of the crystal bases for the classical Lie algebras. Introduction Young tableaux and Young walls play important roles in the interplay, which can be explained in a beautiful manner using the crystal base theory for quantum groups, between the fields of representation theory and combinatorics. Indeed, ...

Young Tableaux, Canonical Bases, and the Gindikin-karpelevich Formula

A combinatorial description of the crystal B(∞) for finitedimensional 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 ta...

Young Tableaux and Crystal B(∞) for Finite Simple Lie Algebras

We study the crystal base of the negative part of a quantum group. An explicit realization of the crystal is given in terms of Young tableaux for types An, Bn, Cn, Dn, and G2. Connection between our realization and a previous realization of Cliff is also given.

Young tableaux

This paper gives a qualitative description how Young tableaux can be used to perform a Clebsch-Gordan decomposition of tensor products in SU(3) and how this can be generalized to SU(N).