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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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Outline of the Proof

Outline of the Proof

Chapter:
(p.36) Chapter Six Outline of the Proof
Source:
Weyl Group Multiple Dirichlet Series
Author(s):

Ben Brubaker

Daniel Bump

Solomon Friedberg

Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691150659.003.0006

This chapter presents the proof of the equivalence of the two definitions for the λ‎-parts in terms of Gelfand-Tsetlin patterns. The equivalence of these two descriptions is a deep fact that uses subtle combinatorial manipulations depending in an essential way on the properties of λ‎-th order Gauss sums. This equivalence is the key step in demonstrating functional equations. In outlining the proof, the chapter introduces many concepts and ideas as well as several equivalent forms of the result, called Statements A through G. Each statement is an intrinsically combinatorial identity involving products of Gauss sums, but with each statement the nature of the problem changes. The first reduction, Statement B, changes the focus from Gelfand-Tsetlin patterns to “short” Gelfand-Tsetlin patterns, consisting of just three rows.

Keywords:   functional equation, Gelfand-Tsetlin pattern, Gauss sum, combinatorial identity, Statement A, Statement B

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