Related papers: Combinatorics of canonical bases revisited: Type A

We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length…

We construct a type $A_{n-1}^{(1)}$ geometric crystal on the variety ${\rm Gr}(k,n) \times \mathbb{C}^\times$, and show that it tropicalizes to the disjoint union of the Kirillov-Reshetikhin crystals corresponding to rectangular tableaux…

We study the description of the crystal structure on the set of Mirkovi\'c-Vilonen polytopes. Anderson and Mirkovi\'c defined an operator and conjectured that it coincides with the Kashiwara operator. Kamnitzer proved the conjecture for…

We give a combinatorial description of the Springer correspondence for classical Lie algebras $\mathfrak{g}$ of type $B,C$ or $D$ and their duals $\mathfrak{g}^*$ in characteristic 2. The combinatorics used here is of the same kind as those…

A combinatorial description of the crystal $\mathcal{B}(\infty)$ for finite-dimensional simple Lie algebras in terms of Young tableaux was developed by J. Hong and H. Lee. Using this description, we obtain a combinatorial rule for…

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal…

We give (conjectural) analogs of Berenstein-Zelevinsky data for affine type $A$. Moreover, by using these affine analogs of Berenstein-Zelevinsky data, we realize the crystal basis of the negative part of the quantized universal enveloping…

We construct a crystal basis for the negative half of the quantum group U associated to the standard super Cartan datum of gl(m|1), which is compatible with known crystals on Kac modules and simple modules. We show that these crystals admit…

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…

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure…

Let G be either an orthogonal, a symplectic or a unitary group over a local field F and let P = MN be a maximal parabolic subgroup. Then the Levi subgroup M is the product of a group of the same type as G and a general linear group, acting…

We construct common triangular bases for almost all the known (quantum) cluster algebras from Lie theory. These bases provide analogs of the dual canonical bases, long anticipated in cluster theory. In cases where the generalized Cartan…

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…

We introduce a new basis of the Temperley-Lieb algebra. It is defined using a bijection between noncrossing partitions and fully commutative elements together with a basis introduced by Zinno, which is obtained by mapping the simple…

Lusztig $q$-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type $C$…

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given…

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie…

Crystal basis theory for the queer Lie superalgebra was developed by Grantcharov et al. and it was shown that semistandard decomposition tableaux admit the structure of crystals for the queer Lie superalgebra or simply…

For symmetric Kashiwara crystals of type $A$ and rank $e=2$, and for the canonical basis elements that we call external, corresponding to weights on the outer skin of the Kashiwara crystal, we construct the canonical basis elements in a…

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…