A k-tableau characterization of k-Schur functions

Abstract

We study k-Schur functions characterized by k-tableaux, proving combinatorial properties such as a k-Pieri rule and a k-conjugation. This new approach relies on developing the theory of k-tableaux, and includes the introduction of a weight-permuting involution on these tableaux that generalizes the Bender-Knuth involution. This work lays the groundwork needed to prove that the set of k-Schur Littlewood-Richardson coefficients contains the 3-point Gromov-Witten invariants; structure constants for the quantum cohomology ring.

Schubert polynomials for the affine Grassmannian of the symplectic group

“…For G = SL n (C), Lam [15] identified the Schubert basis of H * (Gr SLn(C) ) with symmetric functions, called k-Schur functions, of Lapointe, Lascoux and Morse [18]; these arose in the study of Macdonald polynomials. The Schubert basis of H * (Gr SLn(C) ) are the dual k-Schur functions [19] which are generalized by the affine Stanley symmetric functions [14]. In [16] Pieri rules were given for the multiplication of Bott's generators on the Schubert bases of Bott's realization of H * (Gr SLn(C) ) and H * (Gr SLn(C) ).…”

Affine dual equivalence and $k$-Schur functions

“…The k-Schur function s (k) λ (X; t) is the weighted generating function of starred strong tableaux of shape ρ(λ), where ρ is the bijection between k-bounded partitions and k + 1-cores introduced in [22]. In was also shown that the rank of ρ(λ) in the n-core poset equals |λ| and it is conjectured that the leading term of s (k) λ (X; t) in the Schur function expansion is s λ (X).…”

“…Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono [17] as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A [15] and at t = 1 it is equivalent to the k-tableaux characterization of Lapointe and Morse [22]. In this paper, we extend Haiman's dual equivalence relation on standard Young tableaux [12] to all starred strong tableaux.…”

“…In this paper, we advocate for the geometrically inspired definition as the weighted generating function of starred strong tableaux presented by Lam, Lapointe, Morse and Shimozono [17]. This definition at t = 1 is equivalent to the k-tableaux characterization in [22] which has been shown to represent the Schubert basis in the homology of the affine Grassmannian of type A [15]. Furthermore, the starred strong tableaux are a natural generalization of standard tableaux which appear throughout combinatorics.…”

“…At this time, a number of conjecturally equivalent definitions for k-Schur functions exist [17,18,19,20,21,22], making the term "k-Schur function" rather ambiguous. In this paper, we advocate for the geometrically inspired definition as the weighted generating function of starred strong tableaux presented by Lam, Lapointe, Morse and Shimozono [17].…”