Related papers: A Generalization of Schur's $P$- and $Q$-Functions

We study Schur Q-polynomials evaluated on a geometric progression, or equivalently q-enumeration of marked shifted tableaux, seeking explicit formulas that remain regular at q=1. We obtain several such expressions as multiple basic…

We produce skew Pieri Rules for Hall--Littlewood functions in the spirit of Assaf and McNamara. The first two were conjectured by the first author. The key ingredients in the proofs are a q-binomial identity for skew partitions and a Hopf…

We study a variant of the majorization relation. In particular we consider inequalities involving some Schur-concave symmetric polynomials related to the multinomial expansion. We also discuss how these topics were motivated by conjectures…

We revisit a double-scaled limit of the superconformal index of ${\cal N}=2$ superconformal field theories (SCFTs) which generalizes the Schur index. The resulting partition function, $\hat {\cal Z}(q,\alpha)$, has a standard $q$-expansion…

Introduced by Okounkov and Reshetikhin, the Schur process is known to be a determinantal point process, meaning that its correlation functions are minors of a single correlation kernel matrix. Previously, this was derived using…

We study a multi-symmetric generalization of the classical Schur functions called the multi-symmetric Schur functions. These functions form an integral basis for the ring of multi-symmetric functions indexed by tuples of partitions and are…

For a simple Lie algebra $\mathfrak{g}$ and an irreducible faithful representation $\pi$ of $\mathfrak{g}$, we introduce the Schur polynomials of $(\mathfrak{g},\pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the…

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…

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…

Let $k$ and $m$ be positive integers and $\lambda/\mu$ a skew partition. We compute the principal specialization of the skew Schur polynomials $s_{\lambda /\mu}(x_1, \ldots, x_{k})$ modulo $q^m-1$ under suitable conditions. We interpret the…

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of…

The poly-Bernoulli numbers and its relative are defined by the generating series using the polylogarithm series, and we call them type $B$ and $C$, respectively. As a generalization of these poly-Bernoulli numbers, we introduce Schur type…

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…

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the…

We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. We then discuss a family of polynomials…

The reduced Schur functions are studied. Their relations to the basic representation of $A^(1)_{r-1}$ and modular representations of the symmetric groups are clarified. Littlewood-Richardson coefficients appear in the linear relations among…

Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$,…

We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $\omega$ on the ring of…

Let $k$ be a commutative $\mathbb{Q}$-algebra. We study families of functors between categories of finitely generated $R$-modules which are defined for all commutative $k$-algebras $R$ simultaneously and are compatible with base changes.…

In this paper, we investigate a series of W-type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W-constraints for tau functions of higher…