Given a quiver automorphism with nice properties, we give a presentation of the fixed point subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and ...Kleshchev between the cyclotomic Hecke algebra of type G(r, 1, n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In addition, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras.
This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew ...cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras.
As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell ,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell ,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting.
As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell ,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.
To any Young diagram we can associate the multiset of residues of all its nodes. This paper is concerned with the inverse problem: given a multiset of elements of
Z
/
e
Z
, does it comes from a Young ...diagram? We give a full solution in level one and a partial answer in higher levels for Young multidiagrams, using Fayers’s notions of core block and weight of a multipartition. We apply the result in level one to study a shift operation on partitions.
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DOBA, EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, IZUM, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UILJ, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
We prove that cyclotomic Yokonuma–Hecke algebras of type A are cyclotomic quiver Hecke algebras and we give an explicit isomorphism with its inverse, using a similar result of Brundan and Kleshchev ...on cyclotomic Hecke algebras. The quiver we use is given by disjoint copies of cyclic quivers. We relate this work to an isomorphism of Lusztig.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We study a shift action defined on multipartitions and on residue multisets of their Young diagrams. We prove that the minimal orbit cardinality among all multipartitions associated to a given ...multiset depends only on the orbit cardinality of the multiset. Using abaci, this problem reduces to a convex optimisation problem over the integers with linear constraints. We solve it by proving an existence theorem for binary matrices with prescribed row, column and block sums. Finally, we give some applications to the representation theory of the Hecke algebra of the complex reflection group G(r,p,n).
Nous étudions une action de décalage définie sur les multipartitions et sur les multi-ensembles de résidus de leurs diagrammes de Young. Nous montrons que la taille minimale de l'orbite parmi toutes les multi-partitions associées à un multi-ensemble donné ne dépend que le la taille de l'orbite du multi-ensemble. Via les abaques, ce problème se réduit à un problème d'optimisation convexe à variables entières sous contraintes linéaires. Nous le résolvons en montrant un théorème d'existence de matrices binaires de sommes prescrites sur les lignes, colonnes et blocs. Finalement, nous donnons des applications à la théorie des représentations de l'algèbre de Hecke du groupe de réflexions complexes G(r,p,n).
This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew ...cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras.
As an application of this general theory, the main result of this paper proves that the Hecke algebras of type
G
(
ℓ
,
p
,
n
)
G(\ell ,p,n)
are graded skew cellular algebras. In the special case when
p
=
2
p = 2
this implies that the Hecke algebras of type
G
(
ℓ
,
2
,
n
)
G(\ell ,2,n)
are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting.
As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type
G
(
ℓ
,
p
,
n
)
G(\ell ,p,n)
are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.
We give a Morita equivalence theorem for so-called cyclotomic quotients of affine Hecke algebras of type B and D, in the spirit of a classical result of Dipper-Mathas in type A for Ariki-Koike ...algebras. As a consequence, the representation theory of affine Hecke algebras of type B and D reduces to the study of their cyclotomic quotients with eigenvalues in a single orbit under multiplication by $q^2$ and inversion. The main step in the proof consists in a decomposition theorem for generalisations of quiver Hecke algebras that appeared recently in the study of affine Hecke algebras of type B and D. This theorem reduces the general situation of a disconnected quiver with involution to a simpler setting. To be able to treat types B and D at the same time we unify the different definitions of generalisations of quiver Hecke algebra for type B that exist in the literature.
Nous énonçons un théorème d'équivalence de Morita pour les quotients cyclotomiques des algèbres de Hecke affines de type B et D, suivant un résultat classique de Dipper-Mathas en type A pour les algèbres d'Ariki-Koike. Ainsi, la théorie des représentations des algèbres de Hecke affines de type B et D se réduit à l'étude de leurs quotients cyclotomiques où les valeurs propres sont dans une unique orbite pour la multiplication par q² et l'inversion. La preuve consiste notamment en un théorème de décomposition pour des généralisations d'algèbres de Hecke carquois introduites récemment dans l'étude des algèbres de Hecke affines de type B et D, ramenant la situation générale d'un carquois non connexe avec involution à un cadre plus simple. Pour traiter simultanément les deux types, nous unifions les différentes définitions d'algèbres de Hecke carquois pour le type B déjà existantes.
A celebrated result of Kerov-Vershik and Logan-Shepp gives an asymptotic shape for large partitions under the Plancherel measure. We prove that when we consider \(e\)-regularisations of such ...partitions we still have a convex limit shape, which is given by a shaking of the Kerov-Vershik-Logan-Shepp curve. We deduce an explicit form for the first asymptotics of the length of the first rows and the first columns for the \(e\)-regularisation.
To any Young diagram we can associate the multiset of residues of all its nodes. This paper is concerned with the inverse problem: given a multiset of elements of Z/eZ, does it comes from a Young ...diagram? We give a full solution in level one and a partial answer in higher levels for Young multidiagrams, using Fayers's notions of core block and weight of a multipartition. We apply the result in level one to study a shift operation on partitions.
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