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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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(p.22) Chapter Three Duality
Weyl Group Multiple Dirichlet Series

Ben Brubaker

Daniel Bump

Solomon Friedberg

Princeton University Press

This chapter shows that the boxing and circling decorations of the BZL patterns are in a sense dual to each other. The circling and boxing rules seem quite different from each other, but they are actually closely related, and the involution also sheds light on this fact. The chapter describes a natural box-circle duality: a bijection between the bᵢ and the lᵢ in which bᵢ is circled if and only if the corresponding lᵢ is boxed. It also considers a striking property of the crystal graph and proceeds by obtaining two BZL patterns in which circled entries in one correspond to boxed entries in the other, and illustrates this with an example.

Keywords:   boxing, circling, BZL pattern, involution, box-circle duality, bijection, crystal graph

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