$q$-Rationals and Finite Schubert Varieties
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,...
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Académie des sciences
2023-05-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.446/ |
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| author | Ovenhouse, Nicholas |
| author_facet | Ovenhouse, Nicholas |
| author_sort | Ovenhouse, Nicholas |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-fd33967822f942e18464f0bc5196438b |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-05-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-fd33967822f942e18464f0bc5196438b2025-02-07T11:07:37ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-05-01361G480781810.5802/crmath.44610.5802/crmath.446$q$-Rationals and Finite Schubert VarietiesOvenhouse, Nicholas0Department of Mathematics, Yale University, 10 Hillhouse Ave, New Haven CT, USAThe 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.446/ |
| spellingShingle | Ovenhouse, Nicholas $q$-Rationals and Finite Schubert Varieties Comptes Rendus. Mathématique |
| title | $q$-Rationals and Finite Schubert Varieties |
| title_full | $q$-Rationals and Finite Schubert Varieties |
| title_fullStr | $q$-Rationals and Finite Schubert Varieties |
| title_full_unstemmed | $q$-Rationals and Finite Schubert Varieties |
| title_short | $q$-Rationals and Finite Schubert Varieties |
| title_sort | q rationals and finite schubert varieties |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.446/ |
| work_keys_str_mv | AT ovenhousenicholas qrationalsandfiniteschubertvarieties |