Related papers: Equivariant Kazhdan-Lusztig polynomials of $q$-nif…
We decompose the indicator function of each $(a, b)$-Catalan matroid polytope as a weighted sum of indicator function of matroid polytopes that correspond to direct sums of uniform matroids. Catalan matroids lie in the interior of the…
We propose a combinatorial interpretation of the coefficient of $q$ in Kazhdan- Lusztig polynomials and we prove it for finite simply-laced Weyl groups.
In this paper, we investigate the relation between a $q$-matroid and its associated matroid called the projectivization matroid. The latter arises by projectivizing the groundspace of the $q$-matroid and considering the projective space as…
Kazhdan-Lusztig-Stanley polynomials are a combinatorial generalization of Kazhdan-Lusztig polynomials of for Coxeter groups that include g-polynomials of polytopes and Kazhdan-Lusztig polynomials of matroids. In the cases of Weyl groups,…
A $q$-matroid is the analogue of a matroid which arises by replacing the finite ground set of a matroid with a finite-dimensional vector space over a finite field. These $q$-matroids are motivated by coding theory as the representable…
We derive a formula for the entries of the (unitriangular) transition matrices between the standard monomial and dual canonical bases of the irreducible polynomial representations of U_q(gl_n) in terms of Kazhdan-Lusztig polynomials.
For classical matroids, the direct sum is one of the most straightforward methods to make a new matroid out of existing ones. This paper defines a direct sum for $q$-matroids, the $q$-analogue of matroids. This is a lot less straightforward…
We explain a strategy for a proof of the positivity of all coefficients of Kazhdan-Lusztig-polynomials for arbitrary Coxeter groups by constructing spaces whose dimensions we conjecture to be these coefficients.
Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the…
In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan--Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate…
A matroid is uniform if and only if it has no minor isomorphic to $U_{1,1}\oplus U_{0,1}$ and is paving if and only if it has no minor isomorphic to $U_{2,2}\oplus U_{0,1}$. This paper considers, more generally, when a matroid $M$ has no…
Motivated by the $Z$-polynomials of matroids, Ferroni, Matherne, Stevens, and Vecchi introduced the inverse $Z$-polynomial of a matroid. In this paper, we prove several fundamental properties of the inverse $Z$-polynomial, including…
In this paper we show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of…
The paper is devoted to the generalization of Lusztig's q-analog of weight multiplicities to the Lie superalgebras gl(n,m) and spo(2n,M). We define such q-analogs K_{lambda,mu}(q) for the typical modules and for the irreducible covariant…
In this paper, we give a formula for the number of lattice points in the dilations of Schubert matroid polytopes. As applications, we obtain the Ehrhart polynomials of uniform and minimal matroids as special cases, and give a recursive…
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…
We consider the Ehrhart polynomial of hypersimplices. It is proved that these polynomials have positive coefficients and we give a combinatorial formula for each of them. This settles a problem posed by Stanley and also proves that uniform…
The present paper continues the work of [10] and [6]. For any symmetrizable generalized Cartan Matrix $C$ and the corresponding quantum group $\mathbf{U}$, we consider the associated quiver $Q$ with an admissible automorphism $a$. We…
In this paper we complete the proof of the X=K conjecture, that for every family of nonexceptional affine algebras, the graded multiplicities of tensor products of symmetric power Kirillov-Reshetikhin modules known as one-dimensional sums,…
We present a comprehensive generalization of Lusztig's braid group symmetries for quasi-split iquantum groups. Specifically, we give 3 explicit rank one formulas for symmetries acting on integrable modules over a quasi-split iquantum group…