The Depth of Subgroups of Suzuki Groups

Héthelyi, László; Horváth, Erzsébet; Petényi, Franciska

doi:10.1080/00927872.2014.949731

Cited by 12 publications

(12 citation statements)

References 10 publications

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“…The subgroup depth of the alternating group series A n ⊂ A n+1 is 2 (n − ⌈ √ n ⌉) + 1 over the complex numbers [6] (a precise value is not known in positive characteristic, although it is between the value just given and 2n−3 [2]). Other results on subgroup depth can be seen in [13,14,15,20,21].…”

Section: Introductionmentioning

confidence: 89%

A quantum subgroup depth

2017

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The Green ring of the half quantum group H = U n (q) is computed in [9]. The tensor product formulas between indecomposables may be used for a generalized subgroup depth computation in the setting of quantum groups -to compute depth of the Hopf subalgebra H in its Drinfeld double D(H). In this paper the Hopf subalgebra quotient module Q (a generalization of the permutation module of cosets for a group extension) is computed and, as H-modules, Q and its second tensor power are decomposed into a direct sum of indecomposables. We note that the least power n, referred to as depth, for which Q ⊗(n) has the same indecomposable constituents as Q ⊗(n+1) is n = 2, since Q ⊗(2) contains all H-module indecomposables, which determines the minimum even depth d ev (H, D(H)) = 6. √ 8n+1−1 2 + 1, as computed in 1991 Mathematics Subject Classification. 16D20, 16D90.

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Section: Introductionmentioning

confidence: 89%

A quantum subgroup depth

2017

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“…Recall from Section 1 that for a Hopf algebra-Hopf subalgebra pair Combinatorial depth is first defined in [5]. A certain simplification in the definition of minimum even combinatorial depth of a subgroup pair H ≤ G, denoted by d ev c (H, G),were highlighted in [26]as follows. Let F 0 = {H} and for each i ∈ N ,…”

Section: 4mentioning

confidence: 99%

“…; the precise determination is explained in [5,25,26]. A particularly easy characterization is d c (H, G) = 1 if and only if G = HC G (H) [5].…”

Section: 4mentioning

confidence: 99%

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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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“…For example, for the permutation groups Σ n < Σ n+1 and their corresponding group algebras over any commutative ring K, one has depth d K (Σ n , Σ n+1 ) = 2n − 1 [3]. Depths of subgroups in P GL(2, q), Suzuki groups, twisted group algebra extensions and Young subgroups of Σ n are computed in [14,17,11,15]. If B and A are semisimple complex algebras, the minimum odd depth is computed from powers of an order r symmetric matrix with nonnegative entries S := MM T where M is the inclusion matrix K 0 (B) → K 0 (A) and r is the number of irreducible representations of B in a basic set of K 0 (B); the depth is 2n + 1 if S n and S n+1 have an equal number of zero entries [8].…”

Section: Introductionmentioning

confidence: 99%

Subgroup Depth and Twisted Coefficients

Hernández

Kadison

Szamotulski

2016

Communications in Algebra

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Abstract. Danz computes the depth of certain twisted group algebra extensions in [11], which are less than the values of the depths of the corresponding untwisted group algebra extensions in [8]. In this paper, we show that the closely related h-depth of any group crossed product algebra extension is less than or equal to the h-depth of the corresponding (finite rank) group algebra extension. A convenient theoretical underpinning to do so is provided by the entwining structure of a right H-comodule algebra A and a right H-module coalgebra C for a Hopf algebra H. Then A ⊗ C is an A-coring, where corings have a notion of depth extending hdepth. This coring is Galois in certain cases where C is the quotient module Q of a coideal subalgebra R ⊆ H. We note that this applies for the group crossed product algebra extension, so that the depth of this Galois coring is less than the h-depth of H in G. Along the way, we show that subgroup depth behaves exactly like combinatorial depth with respect to the core of a subgroup, and extend results in [23] to coideal subalgebras of finite dimension.

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The Depth of Subgroups of Suzuki Groups

Cited by 12 publications

References 10 publications

A quantum subgroup depth

A quantum subgroup depth

An In-Depth Look at Quotient Modules

Subgroup Depth and Twisted Coefficients

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