Morita equivalences of Ariki–Koike algebras

By R. Dipper, Andrew Mathas. 2000

129 citations recorded by the source corpus.

Abstract. We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a59e511ffa335491d253da97a99847410fff5ba8

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Citations in the corpus

The representation theory of the Ariki-Koike and cyclotomic q-Schur algebras (2002, background): “…are strong ties between the representation theory of cuspidal representations of GLd(q) in characteristic zero, so it is not unreasonable to expect that the modular theory of the cyclotomic Schur algebras and Ariki-Koike algebras also carry information about the modular representations of GLd(q).”
Specialization of linear systems from curves to graphs (2007, methodology): “However, it can also be obtained independently using the Morita equivalences established by [Dipper and Mathas 2002]:”
Weights of multipartitions and representations of Ariki–Koike algebras (2006, background): “This classification has now been proved by Lyle and Mathas [15, Theorem 2.11], but their proof uses the results of the present paper.”
BLOCKS OF CYCLOTOMIC HECKE ALGEBRAS (2006, unknown): “epend very much on whether or not q = 1 and whether or not some of the parameters Q 1,...,Qr are zero. The following result allows us to treat these cases separately. 2.12. Theorem (Dipper and Mathas [13], Theorem 1.5 and Corollary 5.7). Suppose that Q = Q 1 ` Q 2 ` ··· ` Qκ is a partition of Q such”
On the parametrization of the simple modules for Ariki-Koike algebras at roots of unity (2003, background): “Then, Ariki and Mathas (see [2] and [6]) described the non zero Dλ and showed that they are indexed by some “Kleshchev” multipartitions.”
Schur–Weyl duality for higher levels (2006, methodology): “• In section 5, we prove Theorems A and B, modulo one technical fact the proof of which is deferred to section 6. We also formulate a characterization of the Whittaker functor V which is of independent interest, and we explain how to recover the degenerate analogue of a theorem of Dipper and Mathas ”
Categorifications and cyclotomic rational double affine Hecke algebras (2013, unknown): “It is well-known that there are kp-algebra isomorphisms H s kp,d “ À aPC l d Hskp,paq, Hlkp,a “ 1aHskp,d1a and Hskp,paq “ MatSd{Sa ` Hlkp,a ̆ , where Sa “ Sa1 ˆ ̈ ̈ ̈ ˆSal , see [16].”
Cyclotomic Nazarov-Wenzl Algebras (2005, methodology): “modifying the proof of [ DM02 , Theorem 1.1], or [AM00, Theorem 1.3], one can show that under the assumptions of Theorem 8.5 the simple Wr,n(u)–modules are always labelled by the u–Kleshchev multipartitions.”