Related papers: Crystal Bases and Young Tableaux

Using methods of homological algebra, we obtain an explicit crystal isomorphism between two realizations of crystal bases of the lower part of the quantized enveloping algebra of (almost all) finite dimensional simply-laced Lie algebras.…

For an untwisted affine Lie algebra we prove an embedding of any higher level Demazure module into a tensor product of lower level Demazure modules (e.g. level one in type A) which becomes in the limit (for anti-dominant weights) the…

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…

We review the polyhedral realizations of crystal bases in the former half and in the latter half, we introduce braid-type isomorphisms for some rank 2 finite type crystals. Using this isomorphisms, for semi-simple Lie algebra we can show…

We show that for each reduced expression for the longest word in the Weyl group of type A_n, the corresponding cone arising in Lusztig's description of the canonical basis in terms of tight monomials is simplicial, and construct explicit…

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure…

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…

Fix a simply-laced semisimple Lie algebra. We study the crystal $ B(n\lambda)$, were $\lambda$ is a dominant minuscule weight and $n$ is a natural number. On one hand, $B(n\lambda)$ can be realized combinatorially by height $n$ reverse…

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 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…

Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of linear inequalities. As an application, we…

In earlier work with C.~Monical, we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of…

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commutative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing…

The negative part $U^-$ of a quantised enveloping algebra associated to a simple Lie algebra possesses a canonical basis $\mathcal{B}$ with favourable properties. Lusztig has associated a cone to a reduced expression $\mathbf{i}$ for the…

In this paper, we consider polyhedral realizations for crystal bases $B(\lambda)$ of irreducible integrable highest weight modules of a quantized enveloping algebra $U_q(\mathfrak{g})$, where $\mathfrak{g}$ is a classical affine Lie algebra…

We give an explicit description of the unique crystal isomorphism between two realizations of $B(\infty)$ in type $D$: that using marginally large tableaux and that using PBW monomials with respect to one particularly nice reduced…

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor $N$ in the Iwasawa decomposition of a semi-simple Lie algebra $\mathfrak g$,…

We define new crystal maps on $B(\infty)$ using its polyhedral realization, and show that the crystal $B(\infty)$ equipped with the new crystal maps is isomorphic to Kashiwara's $B(\infty)$ as bicrystals. In addition, we combinatorially…

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0.…

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…