On Dipper–Mathas’s Morita equivalences

Jun Hu¹ ; Kai Zhou²

¹ Department of Mathematics Beijing Institute of Technology 100081, Beijing, P.R. China
² Department of Mathematics Zhejiang University 310027, Hangzhou, P.R. China

Colloquium Mathematicum, Tome 149 (2017) no. 1, pp. 103-123

Résumé

Dipper and Mathas have proved that every Ariki–Koike algebra (i.e., nondegenerate cyclotomic Hecke algebra of type $G(\ell ,1,n)$) is Morita equivalent to a direct sum of tensor products of some smaller Ariki–Koike algebras which have $q$-connected parameter sets. They proved this result by explicitly constructing a progenerator which induces this equivalence. In this paper we use the nondegenerate affine Hecke algebra $\mathcal {H}^{\rm aff}_n$ to derive Dipper–Mathas’s Morita equivalence as a consequence of an equivalence between the block $\mathcal {H}^{\rm aff}_n\mbox {-mod}[{\boldsymbol \gamma }]$ of the category of finite-dimensional modules over $\mathcal {H}^{\rm aff}_n$ and the block $\mathcal {H}^{\rm aff}_{n_1}\otimes \dots \otimes \mathcal {H}^{\rm aff}_{n_r}\mbox {-mod}[({\boldsymbol \gamma }^{(1)},\dots ,{\boldsymbol \gamma }^{(r)})]$ of the category of finite-dimensional modules over the parabolic subalgebra $\mathcal {H}^{\rm aff}_{n_1}\otimes \dots \otimes \mathcal {H}^{\rm aff}_{n_r}$ under certain conditions on ${\boldsymbol \gamma },{\boldsymbol \gamma }^{(1)},\ldots ,{\boldsymbol \gamma }^{(r)}$. Similar results for the degenerate versions of these algebras are also obtained.

Détail

DOI : 10.4064/cm6711-7-2016

Keywords: dipper mathas have proved every ariki koike algebra nondegenerate cyclotomic hecke algebra type ell morita equivalent direct sum tensor products smaller ariki koike algebras which have q connected parameter sets proved result explicitly constructing progenerator which induces equivalence paper nondegenerate affine hecke algebra mathcal aff derive dipper mathas morita equivalence consequence equivalence between block mathcal aff mbox mod boldsymbol gamma category finite dimensional modules mathcal aff block mathcal aff otimes dots otimes mathcal aff mbox mod boldsymbol gamma dots boldsymbol gamma category finite dimensional modules parabolic subalgebra mathcal aff otimes dots otimes mathcal aff under certain conditions boldsymbol gamma boldsymbol gamma ldots boldsymbol gamma similar results degenerate versions these algebras obtained