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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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Noncritical Resonances

Noncritical Resonances

(p.64) Chapter Ten Noncritical Resonances
Weyl Group Multiple Dirichlet Series

Ben Brubaker

Daniel Bump

Solomon Friedberg

Princeton University Press

This chapter deals with noncritical resonances. A short pattern is resonant at i if lsubscript i plus 1 = bᵢ. This property depends only on the associated prototype, so resonance is actually a property of prototypes. A first (middle) row entry is also called aᵢ critical if it is equal to one of its four neighbors, which are lᵢ, lsubscript i plus 1, bᵢ, and bsubscript i minus 1. We say that the resonance at i is critical if either aᵢ or asubscript i plus 1 is critical. The chapter introduces the relevant theorem, stating that if t is a strict pattern with no critical resonances, then t′ is also strict with no critical resonances. It also chooses a pair of canonical indexings of Γ‎ = Γ‎ₜ and Δ‎′ = Δ‎subscript tprime.

Keywords:   noncritical resonance, prototype, critical resonance, canonical indexings, short pattern

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