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Weyl Group Multiple Dirichlet SeriesType A Combinatorial Theory (AM-175)$
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Ben Brubaker, Daniel Bump, and Solomon Friedberg

Print publication date: 2011

Print ISBN-13: 9780691150659

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691150659.001.0001

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Crystals and Gelfand-Tsetlin Patterns

Crystals and Gelfand-Tsetlin Patterns

Chapter:
(p.10) Chapter Two Crystals and Gelfand-Tsetlin Patterns
Source:
Weyl Group Multiple Dirichlet Series
Author(s):

Ben Brubaker

Daniel Bump

Solomon Friedberg

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691150659.003.0002

This chapter translates the definitions of the Weyl group multiple Dirichlet series into the language of crystal bases. It reinterprets the entries in these arrays and the accompanying boxing and circling rules in terms of the Kashiwara operators. Thus, what appeared as a pair of unmotivated functions on Gelfand-Tsetlin patterns in the previous chapter now takes on intrinsic representation theoretic meaning. The discussion is restricted to crystals of Cartan type Aᵣ. The Weyl vector, denoted by ρ‎, is considered as an element of the weight lattice, and the bijection between Gelfand-Tsetlin patterns and tableaux is described. The chapter also examines the λ‎-part of the multiple Dirichlet series in terms of crystal graphs.

Keywords:   crystal base, Weyl group multiple Dirichlet series, Kashiwara operator, boxing, circling, Gelfand-Tsetlin pattern, crystal, Weyl vector, tableaux, crystal graph

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