相关论文: Perfect Crystals for U_q(D_4^{(3)})

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the…

We give a realization of crystal graphs for basic representations of the quantum affine algebra U_q(C_n^{(1)}) using combinatorics of Young walls. The notion of splitting blocks plays a crucial role in the construction of crystal graphs.

Hatayama et al. conjectured fermionic formulas associated with tensor products of U'_q(g)-crystals B^{r,s}. The crystals B^{r,s} correspond to the Kirillov--Reshetikhin modules which are certain finite dimensional U'_q(g)-modules. In this…

We shall realize certain affine geometric crystal of type $D_4^{(3)}$ associated with the fundamental representation $W(\pi_1)$ explicitly . By its explicit form, we see that it has a positive structure.

We study products of the affine geometric crystal of type A corresponding to symmetric powers of the standard representation. The quotient of this product by the R-matrix action is constructed inside the unipotent loop group. This quotient…

We consider reduced imaginary Verma modules for the untwisted quantum affine algebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginary crystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlier work…

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its…

The main purpose of this paper is to give a combinatorial realization of Kirillov-Reshetikhin (KR simply) crystals $B^{r, s}$ for type $\text{E}_n^{(1)}$ with a minuscule node $r$ and $s \ge 1$. To do this, we describe explicitly the…

We study perfect crystals for the standard modules of the affine Lie algebra $A_1^{(1)}$ at all levels using the theory of multi-grounded partitions. We prove a family of partition identities which are reminiscent of the Andrews-Gordon…

A discrete set in the Euclidian space is almost periodic, if the measure with the unite masses at points of the set is almost periodic in the weak sense. We prove the following result: if A is a discrete almost periodic set and the set A-A…

Let $U_q(\hat{\cal G})$ be an infinite-dimensional quantum affine Lie algebra. A family of central elements or Casimir invariants are constructed and their eigenvalues computed in any integrable irreducible highest weight representation.…

We study cyclically presented groups of type $\mathfrak{F}$ to determine when they are perfect. It turns out that to do so, it is enough to consider the Prishchepov groups, so modulo a certain conjecture, we classify the perfect Prishchepov…

Let $\mathcal{O}^{int}_q(m|n)$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type $B, C, D$ introduced in the author's previous work. It is a natural counterpart of the category of finitely…

We describe a cluster algebra algorithm for calculating q-characters of Kirillov-Reshetikhin modules for any untwisted quantum affine algebra. This yields a geometric q-character formula for tensor products of Kirillov-Reshetikhin modules.…

Colored planar rook algebra is a semigroup algebra in which the basis element has a diagrammatic description. The category of finite dimensional modules over this algebra is completely reducible and suitable functors are defined on this…

The alcove model of the first author and A. Postnikov uniformly describes highest weight crystals of semisimple Lie algebras. We construct a generalization, called the quantum alcove model. In joint work of the first author with S. Naito,…

Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of…

We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in…

We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated…

Using the methods of Kang et al. and recent results on the characters of Kirillov-Reshetikhin modules by Nakajima and Hernandez, the existence of Kirillov-Reshetikhin crystals B^{r,s} is established for all nonexceptional affine types. We…