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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 Implies Statement A

Statement B Implies Statement A

(p.51) Chapter Seven Statement B Implies Statement A
Weyl Group Multiple Dirichlet Series

Ben Brubaker

Daniel Bump

Solomon Friedberg

Princeton University Press

This chapter recalls the use of the Schützenberger involution on Gelfand-Tsetlin patterns to prove that Statement B implies Statement A. These statements will be discussed two more times in the later chapters of the book. Chapter 18 reinterprets both Statements A and B in terms of crystals, and directly proves that the reinterpreted Statement B implies the reinterpreted Statement A in Theorem 18.2. Then Chapter 19 again 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 current chapter shows that the Schützenberger involution qᵣ can be formulated in terms of operations on short Gelfand-Tsetlin patterns. To facilitate the inductive proof, relevant equations are used.

Keywords:   crystal, Schützenberger involution, Gelfand-Tsetlin pattern, Statement B, Statement A

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