Burge对应与晶体图

Crystal bases of modified quantized enveloping algebras and a double RSK correspondence

The crystal base of the modified quantized enveloping algebras of type $A_{+\infty}$ or $A_\infty$ is realized as a set of integral bimatrices. It is obtained by describing the decomposition of the tensor product of a highest weight crystal and a lowest weight crystal into extremal weight crystals, and taking its limit using a tableaux model of extremal weight crystals. This realization induces in a purely combinatorial way a bicrystal structure of the crystal base of the modified quantized enveloping algebras and hence its Peter-Weyl type decomposition generalizing the classical RSK correspon…

RSK type correspondence of Pictures and Littlewood-Richardson Crystals

We present a Robinson-Schensted-Knuth type one-to-one correspondence between the set of pictures and the set of pairs of Littlewood-Richardson crystals.

RSK correspondence and classically irreducible Kirillov-Reshetikhin crystals

We give a new combinatorial model of the Kirillov-Reshetikhin crystals of type $A_n^{(1)}$ in terms of non-negative integral matrices based on the classical RSK algorithm, which has a simple description of the affine crystal structure without using a promotion operator. We also have a similar description of the Kirillov-Reshetikhin crystals associated to exceptional nodes in the Dynkin diagrams of classical affine or non-exceptional type.

Crystals and trees: quasi-Kashiwara operators, monoids of binary trees, and Robinson--Schensted-type correspondences

Kashiwara's crystal graphs have a natural monoid structure that arises by identifying words labelling vertices that appear in the same position of isomorphic components. The celebrated plactic monoid (the monoid of Young tableaux), arises in this way from the crystal graph for the $q$-analogue of the general linear Lie algebra $\mathfrak{gl}_{n}$, and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence. The authors previously constructed an analogous `quasi-crystal' structure for the related hypo…

Crystallizing the hypoplactic monoid: from quasi-Kashiwara operators to the Robinson--Schensted--Knuth-type correspondence for quasi-ribbon tableaux

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so prov…

Schensted-type correspondences and plactic monoids for types $B_{n}$ and $D_{n}$

We use Kashiwara's theory of crystal bases to study plactic monoids for $U_{q}(so_{2n+1})$ and $U_{q}(so_{2n})$. Simultaneously we describe a Schensted type correspondence in the crystal graphs of tensor powers of vector and spin representations and we derive a Jeu de Taquin for type $B$ from the Sheats sliding algorithm.

A Crystal Analysis of $P$-Arrays

Gasharov introduced the combinatorial objects known as $P$-arrays to prove $s$-positivity for the chromatic symmetric functions of incomparability graphs of (3+1)-free posets. We define a crystal, a directed colored graph with some additional axioms, on the set of $P$-arrays. The components of the crystal have $s$-positive characters, thereby refining the $s$-positivity theorems of Gasharov, as well as Shareshian and Wachs. The crystal hints at a possible generalization of the Robinson-Schensted correspondence applied to $P$-arrays.

Schensted-type correspondence, plactic monoid and jeu de taquin for type C

We use Kashiwara's theory of crystal bases to study the plactic monoid for type C. Then we describe the correponding sliding and bumping algorithms.

q-Whittaker函数、有限域与若尔当标准形

q-Whittaker functions, finite fields, and Jordan forms

该研究通过组合方法结合有限域上的部分旗与若尔当标准形，建立q-Burge对应证明q-Whittaker函数的柯西恒等式，还将其应用于预投射代数模的计数，建立了对称函数组合与表示论的新联系。

Admissible Pictures and Littlewood-Richardson Crystals

We present a one-to-one correspondence between the set of admissible pictures and the Littlewood-Richardson crystals. As a simple consequence, we shall show that the set of pictures does not depend on the choice of admissible orders.

RSK对应的一个经由$A$型箭图表示的扩展推广

An extended generalization of RSK correspondence via $A$ type quiver representations

本文通过$A$型箭图表示，对任意Coxeter元素构造了从填充到反向平面划分的双射，统一并推广了RSK对应的两种先前推广，发展了箭图表示的组合推论。

类型A_n的晶基多面体实现的年轻表格描述

Young tableau descriptions for the polyhedral realizations of crystal bases in type $A_n$

本文通过表格模型的组合性质，建立了类型A_n晶基B(λ)和B(∞)的多面体实现与逆半标准年轻表格的对应关系，并给出了Gelfand-Tsetlin模式上的晶结构。

Schensted type correspondence for type $G_{2}$ and computation of the canonical basis of a finite dimensional $U_{q}(G_{2})$-module

We use Kang-Misra's combinatorial description of the crystal graphs for $U_{q}(G_{2})$ to introduce the plactic monoid for type $G_{2}$. Then we describe the corresponding insertion algorithm which yields a Schensted type correspondence. Next we give a simple algorithm for computing the canonical basis of any finite dimensional $U_{q}(G_{2})$-module.

Crystal bases and three-dimensional $ \mathcal{N}=4 $ Coulomb branches

We establish and develop a correspondence between certain crystal bases (Kashiwara crystals) and the Coulomb branch of three-dimensional $ \mathcal{N} =4 $ gauge theories. The result holds for simply-laced, non-simply laced and affine quivers. Two equivalent derivations are given in the non-simply laced case, either by application of the axiomatic rules or by folding a simply-laced quiver. We also study the effect of turning on real masses and the ensuing simplification of the crystal. We present a multitude of explicit examples of the equivalence. Finally, we put forward a correspondence betw…

通过 $A$ 型箭图表示的组合学对 RSK 的扩展推广

An extended generalization of RSK via the combinatorics of type $A$ quiver representations

本文通过揭示 $A$ 型箭图表示的组合学性质，构造了一族由对称群中 Coxeter 元参数化的双射，将整数划分 $λ$ 的任意填充映射到反向平面划分，并在特定参数选择下恢复了经典的 Robinson--Schensted--Knuth 对应。