Related papers: Skew divided difference operators and Schubert pol…

Enriched versions of type A Schubert polynomials are constructed with coefficients in a polynomial ring in variables $c_1, c_2, \ldots$. Specializing these variables to $0$ recovers the double Schubert polynomials of Lascoux and…

We prove a common generalization of the fact that the weighted number of maximal chains in the strong Bruhat order on the symmetric group is ${n \choose 2}!$ for both the code weights and the Chevalley weights. We also define weights which…

Gelfand-Tsetlin polytopes are classical objects in algebraic combinatorics arising in the representation theory of $\mathfrak{gl}_n(\mathbb{C})$. The integer point transform of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$ projects to…

Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for…

We give the formula for multiplying a Schubert class on an odd orthogonal or symplectic flag manifold by a special Schubert class pulled back from a Grassmannian of maximal isotropic subspaces. This is also the formula for multiplying a…

We give a combinatorial proof that Postnikov and Stanley's formula for dual Schubert polynomials in terms of weighted chains in Bruhat order is equivalent to a classical Cauchy identity for polynomials. This gives a natural interpretation…

We prove Jacobi-Trudi-type determinantal formulas for skew dual Grothendieck polynomials which are $K$-theoretic deformations of Schur polynomials. We also prove a bialternant-type formula analogous to the classical definition of Schur…

We consider one-point commuting difference operators of rank one. The coefficients of these operators depend on a functional parameter, shift operators being included only with positive degrees. We study these operators in the case of…

First, we study the subskewfield of rational pseudodifferential operators over a differential field K generated in the skewfield of pseudodifferential operators over K by the subalgebra of all differential operators. Second, we show that…

In this paper we introduce a notion of duality for matrix valued orthogonal polynomials with respect to a measure supported on the nonnegative integers. We show that the dual families are closely related to certain difference operators…

In this paper, we investigate the differential smoothness of skew PBW extensions over commutative polynomial rings on one and two indeterminates.

We present an iterative technique to obtain skew-orthogonal polynomials with quartic weight, arising in the study of symplectic ensembles of random matrices.

This paper constitutes a first attempt to do analysis with skew polynomials. Precisely, our main objective is to develop a theory of residues for skew rational functions (which are, by definition, the quotients of two skew polynomials). We…

We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded…

We give an elementary approach utilizing only the divided difference formalism for obtaining expansions of Schubert polynomials that are manifestly nonnegative, by studying solutions to the equation $\sum Y_i\partial_i=\mathrm{id}$ on…

Using the vertex operator representations for symplectic and orthogonal Schur functions, we define two families of symmetric functions and show thatthey are the skew symplectic and skew orthogonal Schur polynomials defined implicitly by…

Pseudodifferential parabolic equations with an operator square root arise in wave propagation problems as a one-way counterpart of the Helmholtz equation. The expression under the square root usually involves a differential operator and a…

We give an explicit natural identification between the quiver coefficients of Buch and Fulton, decomposition coefficients for Schubert polynomials, and the Schubert structure constants for flag manifolds. This is also achieved in K-theory…

It is shown that, under some natural assumptions, the tensor product of differentially smooth algebras and the skew-polynomial rings over differentially smooth algebras are differentially smooth.

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…