Clifford theory for infinite dimensional modules

Szechtman, Fernando

doi:10.1080/00927872.2016.1172604

Cited by 2 publications

(8 citation statements)

References 34 publications

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“…What seems to be unknown is when the action of H on W is extendible to I. Theorem 8.14 answers this question: when D has no special triples, in the sense of Definition 8.12. Combining Theorems 8.8 with Theorem 8.14 and the generalization of Gallagher's theorem found in [Sz,Theorem 3.11], we obtain a much sharper decomposition of V (D, f ), as described in Theorem 8.15. This decomposition becomes even sharper if I/H is abelian, and this is stated in Corollary 8.16.…”

Section: Introductionmentioning

confidence: 65%

“…Proof. See [Sz,Theorem 3.5] for the first assertion. As for the second, by Frobenius reciprocity (cf.…”

Section: Clifford Theory For Infinite Dimensional Modulesmentioning

confidence: 99%

“…No calculations of any kind are required. However, for full generality, we must resort to the extension of Gallagher's theorem found in [Sz,Theorem 3.11] (this extension is not required for those interested only in U n (q), or even U n (R) with R finite). As in the case of U n (q), associated to each closed subset Γ of Φ there is a corresponding pattern subgroup M (Γ) of M .…”

Section: Introductionmentioning

confidence: 99%

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Representations of McLain groups

2017

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Abstract. Basic modules of McLain groups M = M (Λ, ≤, R) are defined and investigated. These are (possibly infinite dimensional) analogues of André's supercharacters of Un(q). The ring R need not be finite or commutative and the field underlying our representations is essentially arbitrary: we deal with all characteristics, prime or zero, on an equal basis. The set Λ, totally ordered by ≤, is allowed to be infinite. We show that distinct basic modules are disjoint, determine the dimension of the endomorphism algebra of a basic module, find when a basic module is irreducible, and exhibit a full decomposition of a basic module as direct sum of irreducible submodules, including their multiplicities. Several examples of this decomposition are presented, and a criterion for a basic module to be multiplicity-free is given. In general, not every irreducible module of a McLain group is a constituent of a basic module.

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Section: Introductionmentioning

confidence: 65%

“…Proof. See [Sz,Theorem 3.5] for the first assertion. As for the second, by Frobenius reciprocity (cf.…”

Section: Clifford Theory For Infinite Dimensional Modulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Representations of McLain groups

2017

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“…Such tools are expounded in detail in [14], a brief summary of which appears in Section 2. Our use of Gallagher's theorem makes the irreducibility of V .D; / conceptually transparent and free of calculations.…”

Section: Introductionmentioning

confidence: 99%

“…However, this fact will play no role whatsoever in our arguments. The flexibility of being able to deal with finite and infinite-dimensional modules on equal terms is based on [14], which develops a Clifford theory for modules of arbitrary dimensionality.…”

Section: Introductionmentioning

confidence: 99%

Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

Szechtman

2016

Journal of Group Theory

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Abstract. Let A be a ring with 1 ¤ 0, not necessarily finite, endowed with an involution , that is, an anti-automorphism of order Ä 2. Let H n .A/ be the additive group of all n n hermitian matrices over A relative to . Let U n .A/ be the subgroup of GL n .A/ of all upper triangular matrices with ones along the main diagonal. Let P D H n .A/ Ì U n .A/, where U n .A/ acts on H n .A/ by -congruence transformations. We may view P as a unipotent subgroup of either a symplectic group Sp 2n .A/, if D 1 A (in which case A is commutative), or a unitary group U 2n .A/ if ¤ 1 A . In this paper we construct and classify a family of irreducible representations of P over a field F that is essentially arbitrary. In particular, when A is finite and F D C, we obtain irreducible representations of P of the highest possible degree.

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Clifford theory for infinite dimensional modules

Abstract: Abstract. Clifford theory of possibly infinite dimensional modules is studied.

Cited by 2 publications

References 34 publications

Representations of McLain groups

Representations of McLain groups

Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings

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