Related papers: From Macdonald Polynomials to a Charge Statistic b…
The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is…
Kostka-Foulkes polynomials are Lusztig's $q$-analogues of weight multiplicities for irreducible representations of semisimple Lie algebras. It has long been known that these polynomials have non-negative coefficients. A statistic on…
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. Ram and…
A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for…
We give a new construction of Lascoux-Sch\"uetzenberger's charge statistic in type A which is motivated by the geometric Satake equivalence. We obtain a new formula for the charge statistic in terms of modified crystal operators and an…
Multiline queues are versatile combinatorial objects that play a key role in understanding the remarkable connection between the asymmetric simple exclusion process (ASEP) on a circle and Macdonald polynomials. Specializing the results of…
We construct atomic decompositions for crystals of type $C_{2}$ and define a charge statistic on them, thus providing positive combinatorial formulas for Kostka-Foulkes polynomials associated to them together with a natural geometric…
We give a new construction of Lascoux-Sch\"utzenberger's charge statistic in type A which is motivated by the geometric Satake equivalence We obtain a new formula for the charge statistic in terms of modified crystal operators and an…
Lascoux stated that the type A Kostka-Foulkes polynomials K_{lambda,mu}(t) expand positively in terms of so-called atomic polynomials. For any semisimple Lie algebra, the former polynomial is a t-analogue of the multiplicity of the dominant…
The non-negative integer cocharge statistic on words was introduced in the 1970's by Lascoux and Sch\"utzenberger to combinatorially characterize the Hall-Littlewood polynomials. Cocharge has since been used to explain phenomena ranging…
In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a…
A fully relativistically covariant formulation of the classical Maxwell electrodynamics of an arbitrarily-moving point charge is presented, purely in terms of gauge invariant potentials without entailing any gauge fixing. A new,…
The Macdonald polynomials expanded in terms of a modified Schur function basis have coefficients called the $q,t$-Kostka polynomials. We define operators to build standard tableaux and show that they are equivalent to creation operators…
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 initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula"…
Demazure characters of type A, which are equivalent to key polynomials, have been decomposed by Lascoux and Sch\"{u}tzenberger into standard bases. We prove that the resulting polynomials, which we call Demazure atoms, can be obtained from…
Enriched versions of type A Schubert polynomials are constructed with coefficients in a polynomial ring in variables $c_1, c_2, \ldots$. Specializing these variables to $0$ recovers the double Schubert polynomials of Lascoux and…
We consider products of two Macdonald polynomials of type A, indexed by dominant weights which are respectively a multiple of the first fundamental weight and a weight having zero component on the k-th fundamental weight. We give the…
A recent breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of a pair of statistics on fillings of Young diagrams. The…
Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials,…