Related papers: External Littelmann paths for crystals of type A

We construct a subcrystal of the Littelmann's path crystal whose formal character coincides with that of a certain simple integrable module of level zero over the untwisted affine Lie algebra associated to sl_n. We also establish an…

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…

We give a new model for the crystal graphs of an affine Lie algebra g^, combining Littelmann's path model with the Kyoto path model. The vertices of the crystal graph are represented by certain infinitely looping paths which we call skeins.…

A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an…

In this paper we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,\omega)$ of a row-finite vertex weighted graph…

Let $B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(\Lambda_0)$ of $B(\Lambda_0)$ via quiver…

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by…

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…

We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be viewed as a discrete counterpart to the Littelmann path model.…

We explicitly describe the isomorphism between two combinatorial realizations of Kashiwara's infinity crystal in types B and C. The first realization is in terms of marginally large tableaux and the other is in terms of Kostant partitions…

We construct a path model for geometric crystals in the sense of Berenstein and Kazhdan. Our model is in every way similar to Littelmann's and tropicalizes to his path model. This paper lays the foundational material for a subsequent work…

We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C.…

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works…

Let g = n^- + h + n^+ be a symmetrizable Kac-Moody algebra. Let B(\infty) be the Kashiwara crystal of U_q(n^-), let \lambda be a dominant integral weight, let T_\lambda = {t_\lambda} be the crystal with one element of weight \lambda, and…

We study the crystal graphs of irreducible $U_{v}(\hat{sl}}_{e})$-modules of higher level l. Generalizing works of the first author, we obtain a simple description of the bijections between the classes of multipartitions which naturally…

Let $B(\Lambda)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(\Lambda)$ of the crystal $B(\Lambda)$ using quiver…

We prove an inductive formula to construct a path from the highest weight element to any given vertex in the crystal graph of the polytope realization of the Kirillov-Reshetikhin crystal $KR^{i,m}$ of type $A$. For $i \leq 2$ or $i \geq…

Let $\lambda = \sum_{i \in I_{0}} m_{i} \varpi_{i}$, with $m_{i} \in \mathbb{Z}_{\ge 0}$ for $i \in I_{0}$, be a level-zero dominant integral weight for an affine Lie algebra $\mathfrak{g}$ over $\mathbb{Q}$, where the $\varpi_{i}$, $i \in…

We introduce a new generalisation of partitions, multi-grounded partitions, related to ground state paths indexed by dominant weights of Lie algebras. We use these to express characters of irreducible highest weight modules of Kac-Moody…

We show that a multipartition is cylindric if and only if its level rank-dual is a source in the corresponding affine type $A$ crystal. This provides an algebraic interpretation of cylindricity, and completes a similar result for FLOTW…