Affine Insertion and Pieri Rules for the Affine Grassmannian

By Thomas Lam, Luc Lapointe, Jennifer Morse, Mark Shimozono. 2006

90 citations recorded by the source corpus.

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. 2) A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H_*(Gr). 3) A combinatorial interpretation of the pairing between homology and cohomology of the affine Grassmannian. These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously. Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/313e786904ae8b845018b12b2fd9d4e6e0467718

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Schubert polynomials for the affine Grassmannian (2006, methodology): “There is a bijection w <- X(w) from affine Grassmannian permutations {w G W0} to partitions {A I Ai < n} obtained by taking the code of the permutation; see [17,19].”
The q, t-Catalan numbers and the space of diagonal harmonics : with an appendix on the combinatorics of Macdonald polynomials (2007, methodology): “Building on conjectures of Shimozono and joint work of Lam, Lapointe, Morse and Shimozono, [LLMS06], Lam [Lam] has shown that the the k-Schur and variants of the k-Schur, known as dual k-Schur, are polynomial representatives for the Schubert classes in homology and cohomology, respectively, of the a”
k-Schur Functions and Affine Schubert Calculus (2013, methodology): “In addition, some conjectures of [81] were proven in [13], in particular that “skew shaped” strong Schur functions are symmetric.”