689 research outputs found
The 6-vertex model and deformations of the Weyl character formula
We use statistical mechanics -- variants of the six-vertex model in the plane
studied by means of the Yang-Baxter equation -- to give new deformations of
Weyl's character formula for classical groups of Cartan type B, C, and D, and a
character formula of Proctor for type BC. In each case, the corresponding
Boltzmann weights are associated to the free fermion point of the six-vertex
model. These deformations add to the earlier known examples in types A and C by
Tokuyama and Hamel-King, respectively. A special case for classical types
recovers deformations of the Weyl denominator formula due to Okada.Comment: v2: renamed the last family of models and showed their connection to
character formulae for groups of type BC; addressed some issues in the proof
of Lemma 6.2; updated abstrac
A Yang-Baxter equation for metaplectic ice
We will give new applications of quantum groups to the study of spherical
Whittaker functions on the metaplectic -fold cover of , where
is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and
Gunnells had shown that these Whittaker functions can be identified with the
partition functions of statistical mechanical systems. They postulated that a
Yang-Baxter equation underlies the properties of these Whittaker functions. We
confirm this, and identify the corresponding Yang-Baxter equation with that of
the quantum affine Lie superalgebra
, modified by Drinfeld twisting to
introduce Gauss sums. (The deformation parameter is specialized to the
inverse of the residue field cardinality.) For principal series representations
of metaplectic groups, the Whittaker models are not unique. The scattering
matrix for the standard intertwining operators is vector valued. For a simple
reflection, it was computed by Kazhdan and Patterson, who applied it to
generalized theta series. We will show that the scattering matrix on the space
of Whittaker functions for a simple reflection coincides with the twisted
-matrix of the quantum group .
This is a piece of the twisted -matrix for
, mentioned above
Hecke Modules from Metaplectic Ice
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 -adic groups and -matrices for quantum groups.
Instances of such modules arise from (possibly non-unique) functionals on
-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
-matrices of quantum groups depending on the cover degree and associated
root system
Vertex operators, solvable lattice models and metaplectic Whittaker functions
We show that spherical Whittaker functions on an -fold cover of the
general linear group arise naturally from the quantum Fock space representation
of introduced by Kashiwara, Miwa and Stern
(KMS). We arrive at this connection by reconsidering solvable lattice models
known as `metaplectic ice' whose partition functions are metaplectic Whittaker
functions. First, we show that a certain Hecke action on metaplectic Whittaker
coinvariants agrees (up to twisting) with a Hecke action of Ginzburg,
Reshetikhin, and Vasserot. This allows us to expand the framework of KMS by
Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are
necessary for connections to metaplectic forms. Our main theorem interprets the
row transfer matrices of this ice model as `half' vertex operators on quantum
Fock space that intertwine with the action of
.
In the process, we introduce new symmetric functions termed
\textit{metaplectic symmetric functions} and explain how they relate to
Whittaker functions on an -fold metaplectic cover of GL. These resemble
\textit{LLT polynomials} introduced by Lascoux, Leclerc and Thibon; in fact the
metaplectic symmetric functions are (up to twisting) specializations of
\textit{supersymmetric LLT polynomials} defined by Lam. Indeed Lam constructed
families of symmetric functions from Heisenberg algebra actions on the Fock
space commuting with the -action. We explain
that half vertex operators agree with Lam's construction and this
interpretation allows for many new identities for metaplectic symmetric and
Whittaker functions, including Cauchy identities. While both metaplectic
symmetric functions and LLT polynomials can be related to vertex operators on
the -Fock space, only metaplectic symmetric functions are connected to
solvable lattice models.Comment: v3 changes: minor edit
Crystal constructions in Number Theory
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
Colored five-vertex models and Demazure atoms
Type A Demazure atoms are pieces of Schur functions, or sets of tableaux
whose weights sum to such functions. Inspired by colored vertex models of
Borodin and Wheeler, we will construct solvable lattice models whose partition
functions are Demazure atoms; the proof of this makes use of a Yang-Baxter
equation for a colored five-vertex model. As a biproduct, we construct Demazure
atoms on Kashiwara's crystal and give new algorithms for
computing Lascoux-Sch\"utzenberger keys
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