On the Combinatorics of Crystal Graphs, II. The Crystal Commutator

Combinatorics of canonical bases revisited: Type A

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remov…

Crystal bases and combinatorics of infinite rank quantum groups

The tensor powers of the vector representation associated to an infinite rank quantum group decompose into irreducible components with multiplicities independant of the infinite root system considered. Although the irreducible modules obtained in this way are not of highest weight, they admit a crystal basis and a canonical basis. This permits in particular to obtain for each familly of classical Lie algebras a Robinson-Schensted correspondence on biwords defined on infinite alphabets. We then depict a structure of bi-crystal on these bi-words. This RS-correspondence yields also a plactic alge…

On Mirković-Vilonen cycles and crystals combinatorics

Let $G$ be a complex reductive group and let $G^\vee$ be its Langlands dual. Let us choose a triangular decomposition $\mathfrak g^\vee=\mathfrak n^\vee_-\oplus\mathfrak h^\vee\oplus\mathfrak n^\vee_+$ of the Lie algebra $G^\vee$. Braverman, Finkelberg and Gaitsgory show that the set of all Mirković-Vilonen cycles in the affine grassmannian $\mathscr G=G\bigl(\mathbb C((t))\bigr)/G\bigl(\mathbb C[[t]]\bigr)$ is a crystal isomorphic to the crystal of the canonical basis of $U(\mathfrak n^\vee_+)$. Starting from the string parameter of an element of the canonical basis, we give an explicit descr…

The $\mathfrak{sl}_\infty$-crystal combinatorics of higher level Fock spaces

For integers $e,\ell\geq 2$, the level $\ell$ Fock space has an $\mathfrak{sl}_\infty$-crystal structure arising from the action of a Heisenberg algebra, intertwining the $\widehat{\mathfrak{sl}_e}$-crystal. The vertices of these crystals are charged $\ell$-partitions. We give the combinatorial rule for computing the arrows anywhere in the $\mathfrak{sl}_\infty$-crystal. This allows us to pinpoint the location of any charged $\ell$-partition. As an application, we compute the support of the spherical representation of a cyclotomic rational Cherednik algebra, and in particular, the set of param…

On the Combinatorics of Crystal Graphs, I. Lusztig's Involution

In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations …

Combinatorics of canonical bases revisited: String data in type $A$

We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.

Combinatorics of crystal graphs and Kostka-Foulkes polynomials for the root systems $B_{n},C_{n}$ and $D_{n}.$

We use Kashiwara-Nakashima's combinatorics of crystal graphs associated to the roots sytems $B_{n}$ and $D_{n}$ to extend the results of \QCITE{cite}{}{lec3} and \QCITE{cite}{}{Mor} by showing that Morris type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima's tableaux of types $B_{n},C_{n}$ and $D_{n}$ generalizing Lascoux-Sch\UNICODE{0xfc}tzenberger's charge and from which it is possible to compute the Kostka-Foulkes polynomials $K_{λ,μ}(q)$ with restrictive conditions on $(λ,μ)$ . This statistic is different fro…

Combinatorics of the Casselman-Shalika formula in type A

In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableaux model for the crystal and obtain the same coefficients using data from each individual tableaux; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a q-po…

Combinatorics of $(\ell,0)$-JM partitions, $\ell$-cores, the ladder crystal and the finite Hecke algebra

The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $λ$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irr…

Combinatorics of generalized exponents

We give a purely combinatorial proof of the positivity of the stabilized forms of the generalized exponents associated to each classical root system. In finite type A_{n-1}, we rederive the description of the generalized exponents in terms of crystal graphs without using the combinatorics of semistandard tableaux or the charge statistic. In finite type C_n, we obtain a combinatorial description of the generalized exponents based on the so-called distinguished vertices in crystals of type A_{2n-1}, which we also connect to symplectic King tableaux. This gives a combinatorial proof of the positi…

Combinatorics in affine flag varieties

The Littelmann path moel gives a realization of the crystals of integrable representations of symmetrizable Kac-Moody Lie algebras. Recent work of Gaussent-Littelmann and others has demonstrated a connection between this model and the geometry of the loop Grassmannian. The alcove walk model is a version of the path model which is intimately connected to the combinatorics of the affine Hecke algebra. In this paper we define a refined alcove walk model which encodes the points of the affine flag variety. We show that this combinatorial indexing naturally indexes the "cells" in generalized Mirkov…

寒武纪格是通过2-簇组合学实现的分数Calabi-Yau

Cambrian lattices are fractionally Calabi-Yau via 2-cluster combinatorics

本文通过2-簇组合学证明了寒武纪格的导出范畴为分数Calabi-Yau，扩展了Rognerud关于A型寒武纪格的结果，并研究了晶格类型下的 torsion 类与2-簇tilting对象的组合学。

Geometric and Combinatorial Realizations of Crystals of Enveloping Algebras

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a simply-laced Kac-Moody algebra. Using an enumeration of the irreducible components of Lusztig's quiver varieties in finite and affine type A by combinatorial data, we compute the geometrically defined crystal structure in terms of this combinatorics. We conclude by comparing the combinatorial realization of the crystal graph thus obtained with other combinatorial mo…

某些 PBW 基上晶体结构的组合描述（扩展摘要）

Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)

本文研究单复李代数 PBW 基上的无穷晶体，通过限定 Weyl 群最长元素约化表达式的条件，使 ADE 型晶体算子可用简单括号规则描述，并建立其与 A、D 型表格组合学的联系。

($\ell,0)$-Carter partitions, a generating function, and their crystal theoretic interpretation

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-JM partitions" (see \cite{F}) that are also $\ell$-regular. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas (\cite{JM}). The condition of being an $(\ell,0)$-JM partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-regular partition is that it indicates the irreducibility of the corresponding specialized Specht module over the finite Hecke algebra (see \cite{JM}). We use…

关于$B$型与$C$型弦多面体的组合学研究

On combinatorics of string polytopes in types $B$ and $C$

本文通过给出B型与C型弦锥的显式描述（类比A型的Gleizer-Postnikov描述），研究了B、C型弦多面体的组合性质，并刻画了特定最高权下与C型Gelfand-Tsetlin多面体幺模等价的C型弦多面体。

$B(λ)$ 与 $B(\infty)$ 的一个新的 Young 墙实现

A new Young wall realization of $B(λ)$ and $B(\infty)$

本文利用 Young 墙的新组合学，为量子仿射代数构造了任意水平最高权晶体 $B(λ)$ 和 $B(\infty)$ 的新实现，并证明了约化 Young 墙晶体与 $B(λ)$ 同构。

Classification of the finite-dimensional unitary representations of type B rational Cherednik algebras

We compare crystal combinatorics of the level $2$ Fock space with the classification of unitary representations of type $B$ rational Cherednik algebras to show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica.

管道梦上的 Crystal Chute Moves

Crystal Chute Moves on Pipe Dreams

本文通过对 rc-图上的 chute moves 施加限制，在 Schubert 多项式的单项式上定义了 Demazure crystal 结构，并提供了将其分解为 key 多项式之和的方法。