Related papers: Combinatorics of canonical bases revisited: String…
For arbitrary reduced words we give formulas for the crystal structures on string and Lusztig data of type A as well as the defining inequalities of the corresponding polytopes revealing certain dualities between them.
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"…
A string polytope is a rational convex polytope whose lattice points parametrize a highest weight crystal basis, which is obtained from a string cone by explicit affine inequalities depending on a highest weight. It also inherits geometric…
The combinatorics of reduced words and commutation classes plays an important role in geometric representation theory. A string polytope is a lattice polytope associated to each reduced word of the longest element $w_0$ in the symmetric…
The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely…
We present n-1 different embeddings of string polytopes of type A. We characterize their compatibility with the crystal structure on the string polytopes, and formulate a conjecture describing how to obtain n-1 different atomic…
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…
Stringy canonical forms are a class of integrals that provide $\alpha'$-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebra, there exist completely rigid stringy integrals, whose…
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…
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 introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but…
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…
Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B into the tensor product of B and k…
We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but…
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…
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…
Let U be the quantum group associated to a Lie algebra of type A_n. The negative part U^- of U has a canonical basis B defined by Lusztig and Kashiwara, with favourable properties. We show how the spanning vectors of the cones defined by…
On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the…
Using the theory of PBW bases, one can realize the crystal $B(\infty)$ for any semisimple Lie algebra over $\mathbf{C}$ using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced…
In the current paper, we give a quiver theoretical interpretation of Mirkovi\'c-Vilonen polytopes in type $A_n$. As a by-product, we give a new proof of the Anderson-Mirkovi\'c conjecture which describes the explicit forms of the actions of…