The Rank-One property for free Frobenius extensions
A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra.In this paper, we show that if $H$ is a prime algebra tha...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-10-01
|
| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.502/ |
| _version_ | 1797638562928132096 |
|---|---|
| author | Bellamy, Gwyn Thiel, Ulrich |
| author_facet | Bellamy, Gwyn Thiel, Ulrich |
| author_sort | Bellamy, Gwyn |
| collection | DOAJ |
| description | A conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra.In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank-one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras.In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture. |
| first_indexed | 2024-03-11T13:05:22Z |
| format | Article |
| id | doaj.art-3aa8f0e079c245438613033758138458 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T13:05:22Z |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-3aa8f0e079c2454386130337581384582023-11-03T14:36:25ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-10-01361G81341134810.5802/crmath.50210.5802/crmath.502The Rank-One property for free Frobenius extensionsBellamy, Gwyn0Thiel, Ulrich1School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow, G12 8QQ.Department of Mathematics, University of Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, GermanyA conjecture by the second author, proven by Bonnafé–Rouquier, says that the multiplicity matrix for baby Verma modules over the restricted rational Cherednik algebra has rank one over $\mathbb{Q}$ when restricted to each block of the algebra.In this paper, we show that if $H$ is a prime algebra that is a free Frobenius extension over a regular central subalgebra $R$, and the centre of $H$ is normal Gorenstein, then each central quotient $A$ of $H$ by a maximal ideal $\mathfrak{m}$ of $R$ satisfies the rank-one property with respect to the Cartan matrix of $A$. Examples where the result is applicable include graded Hecke algebras, extended affine Hecke algebras, quantized enveloping algebras at roots of unity, non-commutative crepant resolutions of Gorenstein domains and 3 and 4 dimensional PI Sklyanin algebras.In particular, since the multiplicity matrix for restricted rational Cherednik algebras has the rank-one property if and only if its Cartan matrix does, our result provides a different proof of the original conjecture.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.502/ |
| spellingShingle | Bellamy, Gwyn Thiel, Ulrich The Rank-One property for free Frobenius extensions Comptes Rendus. Mathématique |
| title | The Rank-One property for free Frobenius extensions |
| title_full | The Rank-One property for free Frobenius extensions |
| title_fullStr | The Rank-One property for free Frobenius extensions |
| title_full_unstemmed | The Rank-One property for free Frobenius extensions |
| title_short | The Rank-One property for free Frobenius extensions |
| title_sort | rank one property for free frobenius extensions |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.502/ |
| work_keys_str_mv | AT bellamygwyn therankonepropertyforfreefrobeniusextensions AT thielulrich therankonepropertyforfreefrobeniusextensions AT bellamygwyn rankonepropertyforfreefrobeniusextensions AT thielulrich rankonepropertyforfreefrobeniusextensions |