A k-tableau characterization of k-Schur functions

By Luc Lapointe, Jennifer Morse. 2005

78 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/41458460632aee5378e07f76d19150959eafc6fd

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Citations in the corpus

K-theory Schubert calculus of the affine Grassmannian (2009, background): “iar with aﬃne Stanley symmetric functions may take this as the deﬁnition of Fv(x).) The set {Fv(x) |v∈WI af}were called aﬃne Schur functions in [Lam06], and are equal to the dual k-Schur functions of [LM07]. Proposition 7.13. The set {Gv(x) |v∈WI af}is a “basis” of Λˆ(n). In other words, Λˆ(n) = Q v”
Skew quasisymmetric Schur functions and noncommutative Schur functions (2010, background): “the symmetric group [41, 61], k-Schur functions connected to the enumeration of Gromov-Witten invariants [37], cylindric Schur functions [44], shifted Schur functions related to the representation theory of GL(n) [48], and factorial Schur functions that are special cases of double Schubert polynomia”
Affine Insertion and Pieri Rules for the Affine Grassmannian (2006, methodology): “The k-Schur functions s (k) u (x) are defined in [21] as the symmetric functions satisfying a certain Pieri rule.”
Modules of the 0-Hecke algebra and quasisymmetric Schur functions (2014, background): “Their pervasive nature has also led to their generalization in a variety of ways including Schur P functions arising in the representation theory of the double cover of the symmetric group [41, 54], k-Schur functions connected to the enumeration of Gromov-Witten invariants [35], shifted Schur functi”
k-Schur Functions and Affine Schubert Calculus (2013, methodology): “Lapointe, Lascoux, and Morse [92] conjectured that the Macdonald symmetric functions expand positively in terms of k-Schur functions.”