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 $n$-fold cover of $GL(r,F)$, where $F$ 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 $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ 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 $R$-matrix of the quantum group $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{\sqrt{v}}(\widehat{\mathfrak{gl}}(1|n))$, mentioned above

Monitoring occupancy and abundance of New England cottontails using non-invasive genetic tools

The New England cottontail (Sylvilagus transitionalis ) is a species of conservation concern. Efficient monitoring methods are needed to guide and assess conservation decisions in an adaptive management framework. I used genetic tools and non-invasively collected fecal DNA to determine New England cottontail detection rates during presence/absence surveys and to identify the environmental and behavioral factors that influence detection. I found New England cottontail detection rates to be high (\u3e90%) when surveys were conducted under ideal conditions. Prior knowledge of cottontail activity, low snow depth, and allowing 2-4 days without high winds following a snowfall are the most important factors positively associated with cottontail detection. I also found that increased patch size reduces detection when search efforts are limited to 20 minutes. I used genetic mark-recapture methods to produce baseline abundance estimates for New England cottontail populations across their range. I used microsatellite genotyping in conjunction with single session mark-recapture models in the program CAPWIRE to estimate New England cottontail abundance on 17 occupied patches in Maine, New Hampshire, and New York. Precision of estimates was reasonable for most small sites and several large sites, but decreased with increasing subsampling distance. I also evaluated the methodology used and recommended changes to future survey efforts to improve efficiency and precision. These recommendations include allowing at least three days to pass following a snow fall before conducting a population survey, and sampling pellets intensively on sites to provide a better chance of obtaining an adequate number of recaptures. The tools developed herein will be useful in future occupancy monitoring and abundance estimation needed for the adaptive management of New England cottontail populations

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 $p$-adic 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

Vertex operators, solvable lattice models and metaplectic Whittaker functions

We show that spherical Whittaker functions on an $n$-fold cover of the general linear group arise naturally from the quantum Fock space representation of $U_q(\widehat{\mathfrak{sl}}(n))$ 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 $U_q(\widehat{\mathfrak{sl}}(n))$. In the process, we introduce new symmetric functions termed \textit{metaplectic symmetric functions} and explain how they relate to Whittaker functions on an $n$-fold metaplectic cover of GL$_r$. 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 $U_q(\widehat{\mathfrak{sl}}(n))$-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 $q$-Fock space, only metaplectic symmetric functions are connected to solvable lattice models.Comment: v3 changes: minor edit

Eisenstein Series, Crystals, and Ice

Automorphic forms are generalizations of periodic functions; they are functions on a group that are invariant under a discrete subgroup. A natural way to arrange this invariance is by averaging. Eisenstein series are an important class of functions obtained in this way. It is possible to give explicit formulas for their Fourier coe cients. Such formulas can provide clues to deep connections with other elds. As an example, Langlands' study of Eisenstein series inspired his far-reaching conjectures that dictate the role of automorphic forms in modern number theory. In this article, we present two new explicit formulas for the Fourier coe cients of (certain) Eisenstein series, each given in terms of a combinatorial model: crystal graphs and square ice. Crystal graphs encode important data associated to Lie group representations while ice models arise in the study of statistical mechanics. Both will be described from scratch in subsequent sections. We were led to these surprising combinatorial connections by studying Eisenstein series not just on a group, but more generally on a family of covers of the group. We will present formulas for their Fourier coe cients which hold even in this generality. In the simplest case, the Fourier coe cients of Eisenstein series are described in terms of symmetric functions known as Schur polynomials, so that is where our story begins.National Science Foundation (U.S.) (DMS-0844185)National Science Foundation (U.S.) (DMS-1001079)National Science Foundation (U.S.) (DMS-0844185)United States. National Security Agency (NSA grant H98230-10-1-0183

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