$q$-Rationals and Finite Schubert Varieties

Nicholas Ovenhouse

doi:10.5802/crmath.446

Algebra, Combinatorics

q

-Rationals and Finite Schubert Varieties

Nicholas Ovenhouse ¹

¹ Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven CT, USA

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 807-818

Abstract

The classical $q$ -analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give $q$ -analogues of rational numbers. Some combinatorial interpretations are already known, namely as the rank generating functions for certain partially ordered sets. We give a new interpretation, showing that the numerators of $q$ -rationals count the sizes of certain varieties over finite fields, which are unions of open Schubert cells in some Grassmannian.

Received: 2022-09-28
Revised: 2022-11-21
Accepted: 2022-11-21
Published online: 2023-05-11

DOI: 10.5802/crmath.446

Classification: 05A15, 05A17

Author's affiliations:

Nicholas Ovenhouse ¹

¹ Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven CT, USA

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Nicholas Ovenhouse},
     title = {$q${-Rationals} and {Finite} {Schubert} {Varieties}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {807--818},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     doi = {10.5802/crmath.446},
     language = {en},
}

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Nicholas Ovenhouse. $q$-Rationals and Finite Schubert Varieties. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 807-818. doi: 10.5802/crmath.446

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