Related papers: Canonical Basis and Macdonald Polynomials

As shown in our paper [JCTA 177 (2021), Paper No. 105305], the chromatic quasi-symmetric function of Shareshian-Wachs can be lifted to ${\bf WQSym}$, the algebra of quasi-symmetric functions in noncommuting variables. We investigate here…

A decomposition of a higher order linear differential operator with polynomial coefficients into a direct sum of two factor operators is obtained. This leads to a lower echelon matrix representation for operators of the above mentioned type…

The method to solve inhomogeneous linear differential equations that is usually taught at school relies on the fact that the right hand side function is the product of a polynomial and an exponential and that the linear spaces of those…

A new type of sl_3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl_3 basic hypergeometric series is a…

We give an elementary proof of the development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions.

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.

A problem in representation theory of $p$-adic groups is the computation of the \textit{Casselman basis} of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall…

We present natural families of coordinate algebras of noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the Yang-Baxter equations. As a…

We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an…

A weight function which $q$-generalizes the ground state wave function of the multi-component Calogero-Sutherland quantum many body system is introduced. Conjectures, and some proofs in special cases, are given for a constant term identity…

In this contribution we consider sequences of monic polynomials orthogonal with respect to the standard Freud-like inner product involving a quartic potential $\left\langle…

We introduce an analogue $K_n(x,z;q,t)$ of the Cauchy-type kernel function for the Macdonald polynomials, being constructed in the tensor product of the ring of symmetric functions and the commutative algebra $\mathcal{A}$ over the…

It has been suggested that the chiral symmetry can be implemented only in classical Lagrangians containing higher covariant derivatives of odd order. Contrary to this belief, it is shown that one can construct an exactly soluble…

We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of…

We introduce and study a family of $(q,t)$-deformed discrete $N$-particle beta ensembles, where $q$ and $t$ are the parameters of Macdonald polynomials. The main result is the existence of a large-$N$ limit transition leading to random…

We establish new properties of inhomogeneous spin $q$-Whittaker polynomials, which are symmetric polynomials generalizing $t=0$ Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come…

We show that the transition matrices between the standard and the canonical bases of infinitely many weight subspaces of the higher-level q-deformed Fock spaces are equal.

We give an explicit combinatorial formula for the Schur expansion of Macdonald polynomials indexed by partitions with second part at most two. This gives a uniform formula for both hook and two column partitions. The proof comes as a…

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make…

An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…