Related papers: Affine dual equivalence and k-Schur functions

We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of…

Ikeda-Mihalcea-Naruse's double Schubert polynomials represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated…

We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck…

Recently, Blasiak-Morse-Seelinger introduced symmetric functions called Katalan functions, and proved that the $K$-theoretic $k$-Schur functions due to Lam-Schilling-Shimozono form a subfamily of the Katalan functions. They conjectured that…

We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the…

Using the affine web category introduced in a prequel as a building block, we formulate a diagrammatic $\Bbbk$-linear monoidal category, the affine Schur category, for any commutative ring $\Bbbk$. We then formulate diagrammatic categories,…

We introduce two families of symmetric functions with an extra parameter t that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when t = 1. The families are defined by a statistic on…

We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3-point Gromov-Witten…

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified…

We study the torus-equivariant homology $H_*^T(\mathrm{Gr}_G)$ of the affine Grassmannian $\mathrm{Gr}_G$, where $G=\mathrm{Sp}_{2n}(\mathbb{C})$ is the symplectic group. This homology admits a natural ring structure and a Schubert basis,…

We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which…

We generalize the work of Fomin, Greene, Reiner, and Shimozono on balanced labellings in two directions: (1) we define the diagrams of affine permutations and the balanced labellings on them; (2) we define the set-valued version of the…

We introduce the new concept of weighted $K$-$k$-Schur functions -- a novel family within the broader class of Katalan functions -- that unifies and extends both $K$-$k$-Schur functions and closed $k$-Schur Katalan functions. This new…

The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_\lambda^{(k)}$, symmetric functions that form…

Generalizing the polynomial web category, we introduce a diagrammatic $\Bbbk$-linear monoidal category, the affine web category, for any commutative ring $\Bbbk$. Integral bases consisting of elementary diagrams are obtained for the affine…

Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $\lambda$ of $G$, the $T$-fixed point subscheme $(\bar{Gr}_G^\lambda)^T$ of the Schubert variety…

We conjecture affine or Hall-Littlewood analogues of the dual Jacobi-Trudi formulas for orthogonal and symplectic Schur functions indexed by rectangular partitions of maximal height. These conjectures are then used to derive $t$-analogues…

Much of modern Schubert calculus is centered on Schubert varieties in the complete flag variety and on their classes in its integral cohomology ring. Under the Borel isomorphism, these classes are represented by distinguished polynomials…

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and…

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…