相关论文: Comparing formulas for type $GL_n$ Macdonald polyn…
The theory of differential equations has an arithmetic analogue in which derivatives are replaced by Fermat quotients. One can then ask what is the arithmetic analogue of a linear differential equation. The study of usual linear…
In this paper we aim to specify some characteristics of the so called family of $q$-Appell Polynomials by using $q$-Umbral calculus. Next in our study, we focus on $q$-Genocchi numbers and polynomials as a famous member of this family. To…
In a previous part of this work, we gave a new tableau formula for the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$, using a weight on tableaux involving the \emph{queue inversion} (quinv) statistic. In this paper we…
In a preceding paper the theory of nonsymmetric Macdonald polynomials taking values in modules of the Hecke algebra of type $A$ (Dunkl and Luque SLC 2012) was applied to such modules consisting of polynomials in anti-commuting variables, to…
The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial $P_\kappa(z)$ are known explicitly. These formulas generalise the known $r=1$ case of the Pieri-type formulas…
We study the enumeration of closed walks of given length and algebraic area on the honeycomb lattice. Using an irreducible operator realization of honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian and show that…
The $m$-symmetric Macdonald polynomials form a basis of the space of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},\dots$ (while having no special symmetry in the variables $x_1,\dots,x_m$).We establish in this article…
The paper is mainly devoted to the irreducibility of the polynomial representation of the double affine Hecke algebra for an arbitrary reduced root systems and generic "central charge" q. The technique of intertwiners in the non-semisimple…
This paper introduces Variable Substitution as a domain-specific graph augmentation technique for graph contrastive learning (GCL) in the context of searching for mathematical formulas. Standard GCL augmentation techniques often distort the…
A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra $\cal G$ is presented. The gravitational model contains n 2-forms and $l \geq n$ scalar fields, wheren is the rank of $\cal G$. The solution is…
In this paper, we study Grothendieck polynomials from a combinatorial viewpoint. We introduce the factorial Grothendieck polynomials, analogues of the factorial Schur functions and present some of their properties, and use them to produce a…
Our main aim with these notes is to introduce the combinatorial and symmetric function tools that relate to the description of the Poincare polynomial of the triply graded Khovanov-Rozansky homology of torus links, a.k.a. the (reduced)…
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of $\gl(n)$ as…
In this elementary paper we prove that the extra vanishing property characterizes the BC interpolation Macdonald polynomials inside a very general class of multivariate interpolation polynomials. It follows that they are the only…
Let $\cp:=(P_1,...,P_s)$ be a given family of $n$-variate polynomials with integer coefficients and suppose that the degrees and logarithmic heights of these polynomials are bounded by $d$ and $h$, respectively. Suppose furthermore that for…
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum…
Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the…
We consider GLq(N)-covariant quantum algebras with generators satisfying quadratic polynomial relations. We show that, up to some inessential arbitrariness, there are only two kinds of such quantum algebras, namely, the algebras with…
Wilson lines, being comparators that render non-local operator products gauge invariant, are extensively used in QCD calculations, especially in small-$x$ calculations, calculations concerning validation of factorisation schemes and in…
We introduce an algebra model to study higher order sum rules for orthogonal polynomials on the unit circle. We build the relation between the algebra model and sum rules, and prove an equivalent expression on the algebra side for the sum…