Related papers: A Littlewood-type identity for Robbins polynomials
One interesting combinatorial feature of classical determinantal varieties is that the character of their coordinate rings give a natural truncation of the Cauchy identity in the theory of symmetric functions. Natural generalizations of…
A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.…
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…
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson…
We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over saturated chains in the Bruhat order. We derive several…
We study Hall-Littlewood polynomials using an integrable lattice model of $t$-deformed bosons. Working with row-to-row transfer matrices, we review the construction of Hall-Littlewood polynomials (of the $A_n$ root system) within the…
We introduce certain generalisations of the characters of the classical Lie groups, extending the recently defined factorial characters of Foley and King. In this extension, the factorial powers are replaced with an arbitrary sequence of…
Our results revolve around a new operation on partitions, which we call overlap. We prove two overlap identities for so-called Littlewood-Schur functions. Littlewood-Schur functions are a generalization of Schur functions, whose study was…
Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate $q$, is…
We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2. Alternating sign matrices, subject to certain symmetry…
We introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce…
A theorem due to Tokuyama expresses Schur polynomials in terms of Gelfand-Tsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley's formula for the Schur $q$-polynomials and Gelfand's…
Littlewood polynomials are polynomials with each of their coefficients in $\{-1,1\}$. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro…
Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's…
We generalize a theorem of Littlewood concerning the factorization of Schur polynomials when their variables are twisted by roots of unity. We show that a certain family of flagged skew Schur polynomials admit a similar factorization. These…
Schubert polynomials were discovered by A. Lascoux and M. Sch\"utzenberger in the study of cohomology rings of flag manifolds in 1980's. These polynomials generalize Schur polynomials, and form a linear basis of multivariate polynomials. In…
The (dual) Cauchy identity has an easy algebraic proof utilising a commutation relation between the up and (dual) down operators. By using Fomin's growth diagrams, a bijective proof of the commutation relation can be "bijectivised" to…
The Cauchy identities play an important role in the theory of symmetric functions. It is known that Cauchy sums for the $q$-Whittaker and the skew Schur polynomials produce the same factorized expressions modulo a $q$-Pochhammer symbol. We…
We obtain new combinatorial formulae for modified Hall--Littlewood polynomials, for matrix elements of the transition matrix between the elementary symmetric functions and Hall-Littlewood's ones, and for the number of rational points over…
In 2007, the first author gave an alternative proof of the refined alternating sign matrix theorem by introducing a linear equation system that determines the refined ASM numbers uniquely. Computer experiments suggest that the numbers…