Related papers: Cauchy identities for staircase matrices

The $t=0$ specialization of the Mimachi-Noumi Cauchy-type identity rewrites certain infinite product in terms of specialized nonsymmetric Macdonald polynomials of type $GL_n$. We interpret the infinite product as a character of the space of…

Lascoux has given a triangular version of the Cauchy identity where Schur polynomials are replaced by Demazure characters and Demazure atoms. He has then used the staircase expansion to recover expansions for all Ferrers shapes, where the…

In this paper, we present an independent proof of the Cauchy identities for staircase matrices, originally discovered in arXiv:2411.03117, using the combinatorics of the Bruhat poset and the bubble-sort procedure. Additionally, we derive…

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.

We prove two identities of Hall-Littlewood polynomials, which appeared recently in a paper by two of the authors. We also conjecture, and in some cases prove, new identities which relate infinite sums of symmetric polynomials and partition…

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $\mathsf{F}_\lambda$ arise in the context of $\mathfrak{sl}(2)$ higher spin six vertex models, and are multiparameter deformations of the classical Hall-Littlewood…

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…

Let $\mathbb{K}$ denote a field and let $\mathfrak{X}$ denote a finite non-empty set. Let $\text{Mat}_\mathfrak{X}(\mathbb{K})$ denote the $\mathbb{K}$-algebra consisting of the matrices with entries in $\mathbb{K}$ and rows and columns…

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 give a simple crystal theoretic interpretation of the Lascoux's expansion of a non-symmetric Cauchy kernel $\prod_{i+ j\leq n+1}(1-x_iy_j)^{-1}$, which is given in terms of Demazure characters and atoms. We give a bijective proof of the…

We give another proof for the (-1)-enumeration of self-complementary plane partitions with at least one odd side-length by specializing a certain Schur function identity. The proof is analogous to Stanley's proof for the ordinary…

We present a broader framework for the Cauchy identity derived from the determinant expansion of collocation matrices. This approach yields an infinite family of identities, where the original Cauchy identity stands as a particular case. To…

Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2$, $2n$, $2n+2$, $2n$, $2n+2$, $2n$ and a central triangular hole of size $2$ that…

This paper introduces noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand,…

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…

Let Sym denote the algebra of symmetric functions and $P_\mu(\,\cdot\,;q,t)$ and $Q_\mu(\,\cdot\,;q,t)$ be the Macdonald symmetric functions (recall that they differ by scalar factors only). The $(q,t)$-Cauchy identity $$ \sum_\mu…

We show that, for a certain class of partitions and an even number of variables of which half are reciprocals of the other half, Schur polynomials can be factorized into products of odd and even orthogonal characters. We also obtain related…

A fundamental result by L. Solomon in algebraic combinatorics and representation theory states that Mackey formulas for products of characters of a symmetric group, or equivalently the computation of tensor products of representations…

We introduce the edge Schur functions $E^{\lambda}$ that are defined as a generating series over edge labeled tableaux. We formulate $E^{\lambda}$ as the partition function for a solvable lattice model, which we use to show they are…

We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the…