Subgroups of depth three

Burciu, Sebastian; Kadison, Lars

doi:10.4310/sdg.2010.v15.n1.a2

Cited by 17 publications

(47 citation statements)

References 15 publications

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“…For example, for the permutation groups Σ n < Σ n+1 and their corresponding group algebras B ⊆ A over any commutative ring K, one has depth d(B, A) = 2n − 1 [6,4]. Depths of subgroups in P GL(2, q), twisted group algebras and Young subgroups of Σ n are computed in [10,9,11].…”

Section: 2mentioning

confidence: 99%

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Uniquely separable extensions

Kadison¹

2020

Expositiones Mathematicae

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The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only the H-depth is one. Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1. The depth one Hopf algebroids are derived explicitly. If the separable idempotent is unique for some reason, then the separable extension is called uniquely separable. For example, a Frobenius extension with invertible E-index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier results for subalgebra pairs of group algebras are recovered, with corollaries on depth 1 over fields of characteristic zero and more general ground rings.1991 Mathematics Subject Classification. 12F10, 13B02, 16D20, 16H05, 16S34.

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Section: 2mentioning

confidence: 99%

“…(We have seen in Section 1 that ⇒ is more generally true.) It is known that a semisimple complex algebra-subalgebra pair form a split, separable Frobenius algebra extension [3,6,17]. From [35,37] and group theory, the following is a common setup.…”

Section: Uniquely Separable Frobenius Extensionsmentioning

confidence: 99%

Uniquely separable extensions

Kadison¹

2020

Expositiones Mathematicae

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“…Let G be a finite group, and Cl(G) denote the set of conjugacy classes of G. For group algebras, we recall where B has minimum polynomial (X − 1)(X − 3) and C has minimal polynomial X(X − 1)(X − 3). The depth is computed to be d 0 (S 2 , S 3 ) = 3 in [10] (and d 0 (S n−1 , S n ) = 2n − 1 in [11], d h (S n−1 , S n ) = 2n + 1 [35]). Proof.…”

Section: Minimal Polynomials Of Q In A(r) and A(h)mentioning

confidence: 99%

An In-Depth Look at Quotient Modules

Kadison

2018

Algebr Represent Theor

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The coset G-space of a finite group and a subgroup is a fundamental module of study of Schur and others around 1930; for example, its endomorphism algebra is a Hecke algebra of double cosets. We study and review its generalization Q to Hopf subalgebras, especially the tensor powers and similarity as modules over a Hopf algebra, or what's the same, Morita equivalence of the endomorphism algebras. We prove that Q has a nonzero integral if and only if the modular function restricts to the modular function of the Hopf subalgebra. We also study and organize knowledge of Q and its tensor powers in terms of annihilator ideals, sigma categories, trace ideals, Burnside ring formulas, and when dealing with semisimple Hopf algebras, the depth of Q in terms of the McKay quiver and the Green ring.1991 Mathematics Subject Classification. 16D20, 16D90, 16T05, 18D10, 20C05.

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“…Although the depth two subgroups are precisely the normal subgroups [8], a similar description is not yet known for any higher depth. For example, in [4] a sufficient condition for depth three was given but it is not yet known if that condition is necessary. More precisely if H ⊂ G is an extension of finite groups, it was shown that H is a depth three subgroup of G provided that there is x ∈ G such that H…”

Section: Introductionmentioning

confidence: 99%