Related papers: Cyclic sieving, skew Macdonald polynomials and Sch…

In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial…

We give a combinatorial formula for the non-symmetric Macdonald polynomials E_{\mu}(x;q,t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials J_{\mu}(x;q,t). We prove the…

If $\mathcal{P}$ is a lattice polytope (i.e., $\mathcal{P}$ is the convex hull of finitely many integer points in $\mathbb{R}^d$), Ehrhart's famous theorem (1962) asserts that the integer-point counting function $|t \mathcal{P} \cap…

In this paper we prove a new combinatorial formula for the modified Macdonald polynomials $\widetilde{H}_\lambda(X;q,t)$, motivated by connections to the theory of interacting particle systems from statistical mechanics. The formula…

In this contribution we consider the sequence $\{Q_{n}^{\lambda}\}_{n\geq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences \begin{equation*} \langle p,q\rangle…

We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur…

Every classical orthogonal polynomial system $p_n(x)$ satisfies a three-term recurrence relation of the type \[ p_{n+1}(x)=(A_nx+B_n)p_n(x)-C_np_{n-1}(x)~ (n=0,1,2,\ldots, p_{-1}\equiv 0), \] with $C_nA_nA_{n-1}>0$. Moreover, Favard's…

The skew Schubert polynomials are those which are indexed by skew elements of the Weyl group, in the sense of arXiv:0812.0639. We obtain tableau formulas for the double versions of these polynomials in all four classical Lie types, where…

Macdonald polynomials are orthogonal polynomials associated to root systems, and in the type A case, the symmetric kind is a common generalization of Schur functions, Macdonald spherical functions, and Jack polynomials. We use the…

We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables:…

We derive an explicit sum formula for symmetric Macdonald polynomials. Our expression contains multiple sums over the symmetric group and uses the action of Hecke generators on the ring of polynomials. In the special cases $t=1$ and $q=0$,…

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t) =…

We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the…

We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and polynomials. The first states that the cycle type Eulerian quasisymmetric function $Q_{\lambda,j}$ is Schur-positive, and moreover that the…

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1,…

The symmetric Macdonald polynomials are able to be constructed out of the non-symmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their non-symmetric…

A well-known result of Stanley's shows that given a graph $G$ with chromatic symmetric function expanded into the basis of elementary symmetric functions as $X_G = \sum c_{\lambda}e_{\lambda}$, the sum of the coefficients $c_{\lambda}$ for…

The theory of quantum symmetric pairs is applied to $q$-special functions. Previous work shows the existence of a family $\chi$-spherical functions indexed by the integers for each Hermitian quantum symmetric pair. A distinguished family of…

Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that…

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified…