Equivariant quantum cohomology of the Grassmannian via the rim hook rule

Bertiger, Anna; Milićević, Elizabeth; Taipale, Kaisa

doi:10.5802/alco.14

Bertiger, Anna ¹; Milićević, Elizabeth ²; Taipale, Kaisa ³

¹ Microsoft, Seattle, WA, USA
² Department of Mathematics & Statistics, Haverford College, Haverford, PA, USA
³ Department of Mathematics, University of Minnesota, Minneapolis, MN, USA

Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 327-352.

Abstract

A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the product of two classes in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provided a way to compute quantum products of Schubert classes in the Grassmannian of $k$ -planes in complex $n$ -space by doing classical multiplication and then applying a combinatorial rim hook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao’s puzzle rule then gives an effective algorithm for computing all equivariant quantum Littlewood–Richardson coefficients. Interestingly, this rule requires a specialization of torus weights modulo $n$ , suggesting a direct connection to the Peterson isomorphism relating quantum and affine Schubert calculus.

Received: 2017-08-03
Revised: 2018-01-21
Accepted: 2018-02-08
Published online: 2018-06-28

MR Zbl

DOI: 10.5802/alco.14

Classification: 14N35, 14N15, 14M15, 55N91, 05E05
Mots-clés : Schubert calculus, equivariant quantum cohomology, core partition, abacus diagram, factorial Schur polynomial

Author's affiliations:

Bertiger, Anna ¹; Milićević, Elizabeth ²; Taipale, Kaisa ³

¹ Microsoft, Seattle, WA, USA
² Department of Mathematics & Statistics, Haverford College, Haverford, PA, USA
³ Department of Mathematics, University of Minnesota, Minneapolis, MN, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Bertiger, Anna; Milićević, Elizabeth; Taipale, Kaisa. Equivariant quantum cohomology of the Grassmannian via the rim hook rule. Algebraic Combinatorics, Volume 1 (2018) no. 3, pp. 327-352. doi : 10.5802/alco.14. https://alco.centre-mersenne.org/articles/10.5802/alco.14/

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