Related papers: Type A Partially-Symmetric Macdonald Polynomials

We study the structure of tensor products of $\mathfrak{gl}(\infty) = \varinjlim \mathfrak{gl}(n)$-modules $\mathbf L(\mathbf \lambda) \otimes \mathbf F$ where $\mathbf L(\mathbf \lambda)$ is a simple integrable highest weight module and…

In a recent paper with Sahi and Stokman, we introduced quasi-polynomial generalizations of Macdonald polynomials for arbitrary root systems via a new class of representations of the double affine Hecke algebra. These objects depend on a…

A well-known and fundamental property of the Macdonald polynomials $P_\lambda(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an…

We present conjectures giving formulas for the Macdonald polynomials of type B, C, D which are indexed by a multiple of the first fundamental weight. The transition matrices between two different types are explicitly given.

In the theory of symmetric Jack polynomials the coefficients in the expansion of the $p$th elementary symmetric function $e_p(z)$ times a Jack polynomial expressed as a series in Jack polynomials are known explicitly. Here analogues of this…

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…

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former, as well as…

Ballantine--Beck--Feigon--Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals…

We construct special solutions of the quantum Knizhnik-Zamolodchikov equation on the tensor product of the vector representation of the quantum algebra of type A_{N-1}. They are constructed from non-symmetric Macdonald polynomials through…

We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian PDE.…

This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables.…

Let $\Pi$ be the fundamental group of a smooth variety X over $F_p$. Given a non-Archimedean place $\lambda$ of the field of algebraic numbers which is prime to p, consider the $\lambda$-adic pro-semisimple completion of $\Pi$ as an object…

We construct a new family of graded representations $\widetilde{W}_{\lambda}$ for the positive elliptic Hall algebra $\mathcal{E}^{+}$ indexed by Young diagrams $\lambda$ which generalize the standard $\mathcal{E}^{+}$ action on symmetric…

We present explicit formulas for the Macdonald polynomials of types $C_n$ and $D_n$ in the one-row case. In view of the combinatorial structure, we call them "tableau formulas". For the construction of the tableau formulas, we apply some…

It is well known that the $q$-Whittaker polynomials, which are $t=0$ specializations of the Macdonald polynomials $P_\lambda(X;q,t)$, expand positively as the sum of Schur polynomials. Macdonald polynomials have a quasisymmetric refinement:…

Let $V$ be a finite-dimensional vector space over $\mathbb{F}_p$. We say that a multilinear form $\alpha \colon V^k \to \mathbb{F}_p$ in $k$ variables is $d$-approximately symmetric if the partition rank of difference $\alpha(x_1, \dots,…

The lattice cell in the ${i+1}^{st}$ row and ${j+1}^{st}$ column of the positive quadrant of the plane is denoted $(i,j)$. If $\mu$ is a partition of $n+1$, we denote by $\mu/ij$ the diagram obtained by removing the cell $(i,j)$ from the…

We introduce a family of rings of symmetric functions depending on an infinite sequence of parameters. A distinguished basis of such a ring is comprised by analogues of the Schur functions. The corresponding structure coefficients are…

Let $\lambda =\left( \lambda_{1},\lambda_{2},...,\lambda_{r}\right) $ be an integer partition, and $\left[p_{\lambda }\right] $ the $q$-analog of the symmetric power function $%p_{\lambda }$. This $q$-analogue has been defined as a special…

We classify the discriminantly separable polynomials of degree two in each of three variables, defined by a property that all the discriminants as polynomials of two variables are factorized as products of two polynomials of one variable…