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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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Statement B and the Yang-Baxter Equation

Statement B and the Yang-Baxter Equation

(p.115) Chapter Nineteen Statement B and the Yang-Baxter Equation
Weyl Group Multiple Dirichlet Series

Ben Brubaker

Daniel Bump

Solomon Friedberg

Princeton University Press

This chapter reinterprets Statements A and B in a different context, and yet again directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 19.10. The p-parts of Weyl group multiple Dirichlet series, with their deformed Weyl denominators, may be expressed as partition functions of exactly solved models in statistical mechanics. The transition to ice-type models represents a subtle shift in emphasis from the crystal basis representation, and suggests the introduction of a new tool, the Yang-Baxter equation. This tool was developed to prove the commutativity of the row transfer matrix for the six-vertex and similar models. This is significant because Statement B can be formulated in terms of the commutativity of two row transfer matrices. This chapter presents an alternate proof of Statement B using the Yang-Baxter equation.

Keywords:   statistical mechanics, Statement A, Statement B, Weyl group multiple Dirichlet series, Weyl denominator, partition function, ice-type model, Yang-Baxter equation, row transfer matrix, six-vertex model

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