Related papers: Crystals for shifted key polynomials
We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial…
In this paper, we give an explicit combinatorial realization of the crystal B(\lambda) for an irreducible highest weight U_q(q(n))-module V(\lambda) in terms of semistandard decomposition tableaux. We present an insertion scheme for…
We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the…
We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic…
We introduce and study a generalization of Schur's $P$-/$Q$-functions associated to a polynomial sequence, which can be viewed as ``Macdonald's ninth variation'' for $P$-/$Q$-functions. This variation includes as special cases Schur's…
Normal crystals (also known as Stembridge crystals) are commonly used to establish the Schur positivity of symmetric functions, as their characters are sums of Schur polynomials. In this paper, we develop a combinatorial framework for a…
Schubert polynomials represent a basis for the cohomology of the complete flag variety and thus play a central role in geometry and combinatorics. In this context, Schubert polynomials are generating functions over various combinatorial…
The aim of this paper is to give a new approach to modified $q$-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling…
In this paper we introduce a $(p, q)$-deformed analogues of the generalized Touchard polynomials and Stirling numbers, the post-quantum analogues of the $q$-deformed generalized Touchard polynomials and Stirling numbers. The connection…
We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity…
The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further…
The classification problem of $P$- and $Q$-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of $P$- and $Q$-polynomial association schemes to multivariate cases, namely…
Explicit examples of {\bf positive} crystalline measures and Fourier quasicrystals are constructed using pairs of stable of polynomials, answering several open questions in the area.
Berenstein and Kazhdan's theory of geometric crystals gives rise to two commuting families of geometric crystal operators acting on the space of complex $m \times n$ matrices. These are birational actions, which we view as a…
In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method…
Results of Morse and Schilling show that the set of increasing factorizations of reduced words for a permutation is naturally a crystal for the general linear Lie algebra. Hiroshima has recently constructed two superalgebra analogues of…
The flagged refined stable Grothendieck polynomials of skew shapes generalize several polynomials like stable Grothendieck polynomials, flagged skew Schur polynomials. In this paper, we provide a combinatorial expansion of the flagged…
The study of $P$-polynomial association schemes (distance-regular graphs) and $Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial association schemes, has been a central theme not only in the theory of association…
The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group $S_n$. Wyser and Yong have described polynomial…
The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for…