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Cones, crystals, and patterns. (English) Zbl 0908.17010
This work describes an interpretation of the Gelfand-Tsetlin patterns in terms of crystal graphs and generalizes such patterns to arbitrary complex semisimple algebraic groups. For each element of the crystal basis, the author gets a sequence of integers. He then considers the cone \(C\) spanned by all such sequences. In some cases \(C\) has a simple description. He finds a description in terms of the root ordering for all rank two Kac-Moody algebras.

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
11B83 Special sequences and polynomials
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