Metaplectic Ice

Brubaker, Ben; Bump, Daniel; Chinta, Gautam; Friedberg, Solomon; Gunnells, Paul E.

doi:10.1007/978-0-8176-8334-4_3

Cited by 21 publications

(48 citation statements)

References 24 publications

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“…In our case, we have a meromorphic function, but the polar divisor is a set of hyperplanes, and the theorem is easily extended to this case. Hence once we have meromorphic continuation to (8) we obtain meromorphicity on C 2 . The braid relation σ 1 σ 2 σ 1 (Ψ ) = σ 2 σ 1 σ 2 (Ψ ) is then a consequence.…”

Section: A Methods Of Analytic Continuationmentioning

confidence: 94%

“…Tokuyama's formula may be given another interpretation, as evaluating the partition function of a statistical mechanical system of the free-fermionic six-vertex model. For this, see [15], Chapter 19 of [14], and the paper [8] in this volume. Since more details may be found in these references, we will be brief.…”

Section: Chinta-gunnellsmentioning

confidence: 99%

“…If a 1 a 2 + b 1 b 2 = c 1 c 2 , the site is called free-fermionic. The term comes from physics: in the free-fermionic case, the row transfer matrices (see [8]) are differentiated versions of the Hamiltonians of a quantum mechanical system, the XXZ model, and the quanta for this model are particles of spin 1/2, called fermions.…”

Section: Chinta-gunnellsmentioning

confidence: 99%

“…In [8] in this volume, this is generalized to a system whose partition function is the metaplectic spherical Whittaker function (35). The explanation for this is as follows: there is a map from the set of states of the model to the B λ+ρ crystal.…”

Section: Chinta-gunnellsmentioning

confidence: 99%

“…Let V be a two-dimensional vector space with basis v + and v − . (In the metaplectic case, the scheme proposed in [8] gives its dimension as 2n.) Associate with each site an endomorphism of V ⊗ V as follows.…”

Section: Chinta-gunnellsmentioning

confidence: 99%

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Section: A Methods Of Analytic Continuationmentioning

confidence: 94%

Section: Chinta-gunnellsmentioning

confidence: 99%

Section: Chinta-gunnellsmentioning

confidence: 99%

Section: Chinta-gunnellsmentioning

confidence: 99%

Section: Chinta-gunnellsmentioning

confidence: 99%

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2012

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Crystal Constructions in Number Theory

Puskás

2019

Association for Women in Mathematics Series

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Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics.Date: October 16, 2018.

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Hecke modules from metaplectic ice

Brubaker

Buciumas

Bump

et al. 2017

Sel. Math. New Ser.

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Abstract. We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of padic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.An important tool in studying representations of p-adic groups is the affine Hecke algebra. Ion [28], Brubaker, Bump and Licata [8], Brubaker, Bump and Friedberg [7,9] and other papers considered representations of Hecke algebras related to models of representations of p-adic groups. These works mainly focused on unique functionals such as the Whittaker, spherical and Bessel functionals, but, as we shall show here, the ideas are also applicable to certain non-unique functionals on central extensions of p-adic groups. The goal is to show that these functionals act as Hecke algebra module maps to various target spaces. This results in recursion formulas for Iwahori fixed vectors in the model using Demazure-like operators based on the Hecke action. It allows for effective computation, provides links to the geometry of associated Bott-Samelson varieties, and proves that these functions match important classes of symmetric functions, such as various specializations of non-symmetric Macdonald polynomials.In a separate direction, Brubaker, Bump and Friedberg [6] showed that the values of the spherical Whittaker function for unramified principal series of GL r (F ), where F is a p-adic field, may be realized as partition functions of a solvable lattice model. A similar model for the spherical Whittaker functions for covers of GL r (F ) -a situation in which the Whittaker functional is no longer unique -was given in Brubaker, Bump, Chinta, Friedberg and Gunnells [5]. However, except when the cover is trivial, this model was not solvable in the sense of Baxter [3]. The Boltzmann weights were altered to obtain a solvable model in Brubaker, Buciumas and Bump [4]. This is the "metaplectic ice" of the title. Moreover, [4] provided a new algebraic framework, relating the models to the R-matrices of a Drinfeld twist of the quantum affine Lie superalgebra U q ( gl(n|1)). As a biproduct of this discovery, a relationship between the scattering matrices of the intertwining integrals on Whittaker coinvariants and R-matrices of a Drinfeld twist of U q ( gl(n)) came to light.In the present paper, we will unify these directions by providing simultaneous foundations for both. The search for new foundations is motivated by the following consideration. In [7], we took the point of view of the universal principal series. This means that the induced character becomes C[Λ]...

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Metaplectic Ice

Cited by 21 publications

References 24 publications

Introduction: Multiple Dirichlet Series

Introduction: Multiple Dirichlet Series

Crystal Constructions in Number Theory

Hecke modules from metaplectic ice

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