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 when restricted to each block of the algebra.
In this paper, we show that if is a prime algebra that is a free Frobenius extension over a regular central subalgebra , and the centre of is normal Gorenstein, then each central quotient of by a maximal ideal of satisfies the rank-one property with respect to the Cartan matrix of . 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.
Une conjecture du deuxième auteur, qui a été prouvée par Bonnafé–Rouquier, dit que la matrice de multiplicité des bébé modules de Verma de l’algèbre rationnelle restreinte de Cherednik est de rang un sur lorsqu’elle est restreinte à chaque bloc de l’algèbre.
Dans cet article nous montrons que si est une algèbre première qui est une extension libre de Frobenius sur une sous-algèbre centrale régulière , et si le centre de est Gorenstein normal, alors chaquequotient central de par un idéal maximal de satisfait la propriété de rang un par rapport à la matrice de Cartan de . Les exemples où le résultat est applicable incluent les algèbres de Hecke graduées, les algèbres de Hecke affines étendues, les algèbres enveloppantes quantifiées aux racines de l’unité, les résolutionscrépantes non commutatives des domaines de Gorenstein et les algèbres PI Sklyanin à dimension et .
En particulier, puisque la matrice de multiplicité pour les algèbres de Cherednik rationnelles restreintes a la propriété de rang un si et seulement si sa matrice de Cartan l’a aussi, notre résultat fournit une preuve différente de la conjecture originale.
Revised:
Accepted:
Published online:
Gwyn Bellamy 1; Ulrich Thiel 2
@article{CRMATH_2023__361_G8_1341_0, author = {Gwyn Bellamy and Ulrich Thiel}, title = {The {Rank-One} property for free {Frobenius} extensions}, journal = {Comptes Rendus. Math\'ematique}, pages = {1341--1348}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.502}, language = {en}, }
Gwyn Bellamy; Ulrich Thiel. The Rank-One property for free Frobenius extensions. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1341-1348. doi : 10.5802/crmath.502. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.502/
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