Related papers: Crystal constructions in Number Theory

We study Whittaker functions on nonlinear coverings of simple algebraic groups over a non-archimedean local field. We produce a recipe for expressing such a Whittaker function as a weighted sum over a crystal graph, and show that in type A,…

For a semisimple Lie algebra admitting a good enumeration, we prove a parametrization for the elements in its Weyl group. As an application, we give a coordinate-free comparison between the crystal graph description (when it is known) and…

We present the geometric solutions to some variational problems of statistical mechanics and combinatorics. Together with the Wulff construction, which predicts the shape of the crystals, we discuss the construction which exhibits the shape…

We construct multiple Dirichlet series in several complex variables whose coefficients involve quadratic residue symbols. The series are shown to have an analytic continuation and satisfy a certain group of functional equations. These are…

Generally speaking, this thesis focuses on the interplay between the representations of Lie groups and probability theory. It subdivides into essentially three parts. In a first rather algebraic part, we construct a path model for geometric…

We study the structure of $p$-parts of Weyl group multiple Dirichlet series. In particular, we extend results of Chinta, Friedberg, and Gunnells and show, in the stable case, that the $p$-parts of Chinta and Gunnells agree with those…

We develop the theory of Weyl group multiple Dirichlet series for root systems of type C. For an arbitrary root system of rank r and a positive integer n, these are Dirichlet series in r complex variables with analytic continuation and…

In this paper we continue the study of the higher-rank graphs associated to finite-dimensional complex semisimple Lie algebras, introduced by the author and R. Yuncken, whose construction relies on Kashiwara's theory of crystals. First we…

This paper establishes a combinatorial link between different approaches to constructing Whittaker functions on a metaplectic group over a non-archimedean local field. We prove a metaplectic analogue of Tokuyama's Theorem and give a crystal…

Chinta and Gunnells introduced a rather intricate multi-parameter Weyl group action on rational functions on a torus, which, when the parameters are specialized to certain Gauss sums, describes the functional equations of Weyl group…

Using the theory of PBW bases, one can realize the crystal $B(\infty)$ for any semisimple Lie algebra over $\mathbf{C}$ using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced…

Crystal graphs are powerful combinatorial tools for working with the plactic monoid and symmetric functions. Quasi-crystal graphs are an analogous concept for the hypoplactic monoid and quasi-symmetric functions. This paper makes a…

Several combinatorial problems of (quasi-)crystallography are reviewed with special emphasis on a unified approach, valid for both crystals and quasicrystals. In particular, we consider planar sublattices, similarity sublattices,…

A fundamental problem from invariant theory is to describe the endomorphism algebra of multilinear functions on a representation V invariant under the action of a group G. According to Weyl's classic, a first main (later: fundamental)…

Lusztig's theory of PBW bases gives a way to realize the infinity crystal for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced…

Let k be an algebraically closed field of characteristic p>2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under…

We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple…

Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…

A type of directed multigraph called a W-digraph is introduced to model the structure of certain representations of Hecke algebras, including those constructed by Lusztig and Vogan from involutions in a Weyl group. Building on results of…

As a vital link between group theory and graph theory, Cayley graphs provide a geometric framework for encoding algebraic structures. This study explores the properties of Cayley graphs derived from cyclic groups whose order is the square…