Related papers: Demazure crystals for specialized nonsymmetric Mac…
We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and…
We show how combinatorial star products can be used to obtain strict deformation quantizations of polynomial Poisson structures on $\mathbb R^d$, generalizing known results for constant and linear Poisson structures to polynomial Poisson…
The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of…
We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_{\lambda}(X;q,t)=\sum_{\sigma\in…
Inhomogeneous analogues of symmetric and nonsymmetric Macdonald polynomials were introduced by F. Knop and the author. In the symmetric case A. Okounkov has recently proved a beautiful expansion formula which can be viewed as a…
One approach to Schubert calculus is to realize Schubert classes as concrete combinatorial objects such as Schubert polynomials. Through an identification of the cohomology ring of the type A full flag variety with the polytope ring of the…
In this paper we derive a counterpart of the well-known Ram-Yip formula for symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new formula is in terms of a generalization of the Lakshmibai-Seshadri paths (originating in…
We show that several families of polynomials defined via fillings of diagrams satisfy linear recurrences under a natural operation on the shape of the diagram. We focus on key polynomials, (also known as Demazure characters), and Demazure…
For any admissible pair of irreducible reduced crystallographic root systems, we present discrete orthogonality relations for a finite-dimensional system of Macdonald polynomials with parameters on the unit circle subject to a truncation…
In 1996, Knop and Sahi introduced a remarkable family of inhomogeneous symmetric polynomials, defined via vanishing conditions, whose top homogeneous parts are exactly the Macdonald polynomials. Like the Macdonald polynomials, these…
We prove an explicit formula for the Poincar\'e polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
The fact that Schubert polynomials are the weighted counting functions for reduced RC-graphs, also known as reduced pipe dreams, was established using their generating functions inside an appropriate Demazure algebra. Here we investigate…
We establish new explicit connections between classical (scalar) and matrix Gegenbauer polynomials, which result in new symmetries of the latter and further give access to several properties that have been out of reach before: generating…
This paper studies the properties of Demazure atoms and characters using linear operators and also tableaux-combinatorics. It proves the atom-positivity property of the product of a dominating monomial and an atom, which was an open…
We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of…
Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We…
We establish the equality of the specialization $P_\lambda(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_\lambda(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine…
We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We…
New sequences of discrete orthogonal polynomials associated with the modified Bessel function $K_\mu(z)$ or Macdonald function are considered. The corresponding weight function is $\lambda^k \rho_{k+\nu+1}(t)/ k!$, where $\ k \in…