A quantum Murnaghan–Nakayama rule for the flag manifold

Benedetti Velásquez, Carolina; Bergeron, Nantel; Colmenarejo, Laura; Saliola, Franco; Sottile, Frank

doi:10.5802/alco.427

Benedetti Velásquez, Carolina ¹; Bergeron, Nantel ²; Colmenarejo, Laura ³; Saliola, Franco ⁴; Sottile, Frank ⁵

¹ Carolina Benedetti Universidad de Los Andes Bogotá Colombia
² Nantel Bergeron York University Toronto, Ontario Canada
³ Laura Colmenarejo North Carolina State University Raleigh, North Carolina USA
⁴ Franco Saliola Université du Québec à Montréal Montréal, Québec Canada
⁵ Frank Sottile Texas A&M University College Station, Texas USA

Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 619-653.

Abstract

In this paper, we give a rule for the multiplication of a Schubert class by a tautological class in the (small) quantum cohomology ring of the flag manifold. As an intermediate step, we establish a formula for the multiplication of a Schubert class by a quantum Schur polynomial indexed by a hook partition. This entails a detailed analysis of chains and intervals in the quantum Bruhat order. This analysis allows us to use results of Leung–Li and of Postnikov to reduce quantum products by hook Schur polynomials to the (known) classical product.

Received: 2024-07-15
Revised: 2025-03-05
Accepted: 2025-03-07
Published online: 2025-06-26

DOI: 10.5802/alco.427

Classification: 05E05, 14N15
Keywords: Murnaghan–Nakayama rule, Schubert calculus, Schubert polynomials, quantum cohomology

Author's affiliations:

Benedetti Velásquez, Carolina ¹; Bergeron, Nantel ²; Colmenarejo, Laura ³; Saliola, Franco ⁴; Sottile, Frank ⁵

¹ Carolina Benedetti Universidad de Los Andes Bogotá Colombia
² Nantel Bergeron York University Toronto, Ontario Canada
³ Laura Colmenarejo North Carolina State University Raleigh, North Carolina USA
⁴ Franco Saliola Université du Québec à Montréal Montréal, Québec Canada
⁵ Frank Sottile Texas A&M University College Station, Texas USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Benedetti Vel\'asquez, Carolina and Bergeron, Nantel and Colmenarejo, Laura and Saliola, Franco and Sottile, Frank},
     title = {A quantum {Murnaghan{\textendash}Nakayama} rule for the flag manifold},
     journal = {Algebraic Combinatorics},
     pages = {619--653},
     publisher = {The Combinatorics Consortium},
     volume = {8},
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     year = {2025},
     doi = {10.5802/alco.427},
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}

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Benedetti Velásquez, Carolina; Bergeron, Nantel; Colmenarejo, Laura; Saliola, Franco; Sottile, Frank. A quantum Murnaghan–Nakayama rule for the flag manifold. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 619-653. doi : 10.5802/alco.427. https://alco.centre-mersenne.org/articles/10.5802/alco.427/

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