Tableaux on k+1-cores, reduced words for affine permutations, and k-Schur expansions

By Luc Lapointea, Jennifer Morseb. 2004

98 citations recorded by the source corpus.

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 a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young lattice with certain tableaux on $k+1$ cores. This algorithm reveals that the $k$-Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group $\tilde S_{k+1}$ by a maximal parabolic subgroup. From this, the conjectured $k$-Pieri rule implies that the $k$-Kostka matrix connecting the homogeneous basis $\{h_\la\}_{\la\in\CY^k}$ to $\{s_\la^{(k)}\}_{\la\in\CY^k}$ may now be obtained by counting appropriate classes of tableaux on $k+1$-cores. This suggests that the conjecturally positive $k$-Schur expansion coefficients for Macdonald polynomials (reducing to $q,t$-Kostka polynomials for large $k$) could be described by a $q,t$-statistic on these tableaux, or equivalently on reduced words for affine permutations.

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

View Paper: https://www.semanticscholar.org/paper/5e1d97ff57bd0b6cc1b83a88da608357d251dcd3

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Citations in the corpus

Quantum cohomology of G/P and homology of affine Grassmannian (2007, methodology): “[19] L. Lapointe and J. Morse: Quantum cohomology and the k-Schur basis, Trans.”
Affine Insertion and Pieri Rules for the Affine Grassmannian (2006, methodology): “If u ∈ S̃n is 0-Grassmannian then weak tableaux of shape u are the k-tableaux (with k = n− 1) first defined in [20].”
QUANTUM COHOMOLOGY AND THE k-SCHUR BASIS (2005, methodology): “The skew k-tableaux are well-defined since µ ⊆ ν implies that c(µ) ⊆ c(ν )( e.g. [ 22 ], Prop. 14).”
K-theory Schubert calculus of the affine Grassmannian (2009, background): “See [LLMS, LM05 ] for the homological version.”