相关论文: Lectures on geometric realizations of crystals

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…

We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore…

Let $i$ be a reduced expression of the longest element in the Weyl group of type $A$, which is adapted to a Dynkin quiver with a single sink. We present a simple description of the crystal embedding of Young tableaux of arbitrary shape into…

The use of machine learning methods for accelerating the design of crystalline materials usually requires manually constructed feature vectors or complex transformation of atom coordinates to input the crystal structure, which either…

Students in introductory physics courses struggle to understand virtual image formation by a plane mirror and the proper construction of ray diagrams. This difficulty, if not sufficiently addressed, results in further problems throughout…

The polyhedral realization of crystal base has been introduced by A.Zelevinsky and the second author([T.Nakashima, A.Zelevinsky, Adv. Math. 131, no. 1 (1997)]), which describe the crystal base $B(\infty)$ as a polyhedral convex cone in the…

These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected…

This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent…

Crystal structure design is important for the discovery of new highly functional materials because crystal structure strongly influences material properties. Crystal structures are composed of space-filling polyhedra, which affect material…

In this review, we summarize the recent progress on the crystal melting models and the quiver algebras regarding the BPS counting. We shall consider the constructions of crystals for generic quivers and discuss the so-called double quiver…

We introduce a fermionic formula associated with any quantum affine algebra U_q(X^{(r)}_N). Guided by the interplay between corner transfer matrix and Bethe ansatz in solvable lattice models, we study several aspects related to…

An explicit description of the virtualization map for the (modified) Nakajima monomial model for crystals is given. We give an explicit description of the Lusztig data for modified Nakajima monomials in type $A_n$.

Realizations of algebras in terms of canonical or bosonic variables can often be used to simplify calculations and to exhibit underlying properties. There is a long history of using such methods in order to study symmetry groups related to…

We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are…

We continue work begun in \cite{ptab} which introduced \emph{perforated tableaux} as a combinatorial model for crystals of type $A_{n-1}$, emphasizing connections to the classical Robinson-Schensted-Knuth (RSK) correspondence and Lusztig…

Let $\mathfrak{g}$ be a Lie algebra all of whose regular subalgebras of rank 2 are type $A_{1}\times A_{1}$, $A_{2}$, or $C_{2}$, and let $B$ be a crystal graph corresponding to a representation of $\mathfrak{g}$. We explicitly describe the…

Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…

Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(\lambda)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$,…

Based on lectures delivered at (1) the AMS meeting at USC, Nov. 1992 (2) Conference on Quantum Aspects of Black Holes, ITP, UC- Santa %Barbara, %%June 1993. (3) 25th Summer Institute, Ecole Normale Superieure, % Paris, Aug. 1992. To appear…

This set of lecture notes is an expanded version of a mini-course the author gave in March of 2025 for the program ``Representation Theory \& Noncommutative Geometry" at the Institut Henri Poincar\'e, Paris. The goal is to provide a survey…