Local duality for representations of finite group schemes

A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the $\mathfrak{p}$-local and $\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal $\mathfrak{p}$ in the cohomology ring of the group scheme.

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Finite group schemes of p-rank ≤ 1

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000834 ◽

2017 ◽

Vol 166 (2) ◽

pp. 297-323

Author(s):

HAO CHANG ◽

ROLF FARNSTEINER

Keyword(s):

Finite Group ◽

Lie Algebras ◽

Group Scheme ◽

Closed Field ◽

Difficult Case ◽

Modular Representation Theory ◽

Group Schemes ◽

Restricted Lie Algebras ◽

A Finite Group ◽

Finite Group Schemes

AbstractLet be a finite group scheme over an algebraically closed field k of characteristic char(k) = p ≥ 3. In generalisation of the familiar notion from the modular representation theory of finite groups, we define the p-rank rkp() of and determine the structure of those group schemes of p-rank 1, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height 1, which correspond to restricted Lie algebras. Our results show that group schemes of p-rank ≤ 1 are closely related to those being of finite or domestic representation type.

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CONTINUOUS TIME QUANTUM WALKS AND QUOTIENT GRAPHS

International Journal of Quantum Information ◽

10.1142/s0219749911007435 ◽

2011 ◽

Vol 09 (03) ◽

pp. 1005-1017

Author(s):

R. SUFIANI ◽

S. NAMI ◽

M. GOLMOHAMMADI ◽

M. A. JAFARIZADEH

Keyword(s):

Finite Group ◽

Continuous Time ◽

Group Scheme ◽

Quantum Walks ◽

Normal Subgroups ◽

Arbitrary Vertex ◽

Group Schemes ◽

Time Quantum ◽

A Finite Group ◽

Finite Group Schemes

Continuous-time quantum walks (CTQW) over finite group schemes is investigated, where it is shown that some properties of a CTQW over a group scheme defined on a finite group G induces a CTQW over group scheme defined on G/H, where H is a normal subgroup of G with prime index. This reduction can be helpful in analyzing CTQW on underlying graphs of group schemes. Even though this claim is proved for normal subgroups with prime index (using the Clifford's theorem from representation theory), it is checked in some examples that for other normal subgroups or even non-normal subgroups, the result is also true! It means that CTQW over the graph on G, starting from any arbitrary vertex, is isomorphic to the CTQW over the quotient graph on G/H if we take the sum of the amplitudes corresponding to the vertices belonging to the same cosets.

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Cohomology and projectivity of modules for finite group schemes

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005266 ◽

2001 ◽

Vol 131 (3) ◽

pp. 405-425 ◽

Cited By ~ 4

Author(s):

CHRISTOPHER P. BENDEL

Keyword(s):

Finite Group ◽

Representation Theory ◽

Hopf Algebra ◽

Group Scheme ◽

Steenrod Algebra ◽

Group Schemes ◽

Finite Dimensional ◽

A Finite Group ◽

Finite Group Schemes ◽

Cocommutative Hopf Algebra

Let G be a finite group scheme over a field k, that is, an affine group scheme whose coordinate ring k[G] is finite dimensional. The dual algebra k[G]* ≡ Homk(k[G], k) is then a finite dimensional cocommutative Hopf algebra. Indeed, there is an equivalence of categories between finite group schemes and finite dimensional cocommutative Hopf algebras (cf. [19]). Further the representation theory of G is equivalent to that of k[G]*. Many familiar objects can be considered in this context. For example, any finite group G can be considered as a finite group scheme. In this case, the algebra k[G]* is simply the group algebra kG. Over a field of characteristic p > 0, the restricted enveloping algebra u([gfr ]) of a p-restricted Lie algebra [gfr ] is a finite dimensional cocommutative Hopf algebra. Also, the mod-p Steenrod algebra is graded cocommutative so that some finite dimensional Hopf subalgebras are such algebras.Over the past thirty years, there has been extensive study of the modular representation theory (i.e. over a field of positive characteristic p > 0) of such algebras, particularly in regards to understanding cohomology and determining projectivity of modules. This paper is primarily interested in the following two questions:Questions1·1. Let G be a finite group scheme G over a field k of characteristic p > 0, and let M be a rational G-module.(a) Does there exist a family of subgroup schemes of G which detects whether M is projective?(b) Does there exist a family of subgroup schemes of G which detects whether a cohomology class z ∈ ExtnG(M, M) (for M finite dimensional) is nilpotent?

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Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes

Forum Mathematicum ◽

10.1515/forum-2017-0007 ◽

2018 ◽

Vol 30 (2) ◽

pp. 479-495

Author(s):

Yang Pan

Keyword(s):

Finite Group ◽

Lie Algebras ◽

Finite Groups ◽

Positive Characteristic ◽

Group Scheme ◽

Closed Field ◽

Group Schemes ◽

Frobenius Kernel ◽

A Finite Group ◽

Finite Group Schemes

AbstractWe investigate the saturation rank of a finite group scheme defined over an algebraically closed field{\Bbbk}of positive characteristicp. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group{\operatorname{SL}_{n}}.

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The cohomology ring of a finite group scheme

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1970-0266928-1 ◽

1970 ◽

Vol 26 (4) ◽

pp. 567-567

Author(s):

Gustave Efroymson

Keyword(s):

Finite Group ◽

Cohomology Ring ◽

Group Scheme ◽

A Finite Group

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Freyd's Generating Hypothesis for Groups with Periodic Cohomology

Canadian Mathematical Bulletin ◽

10.4153/cmb-2011-090-5 ◽

2012 ◽

Vol 55 (1) ◽

pp. 48-59 ◽

Cited By ~ 3

Author(s):

Sunil K. Chebolu ◽

J. Daniel Christensen ◽

Ján Mináč

Keyword(s):

Finite Group ◽

Module Category ◽

Tate Cohomology ◽

Stable Module Category ◽

Finite Dimensional ◽

Stable Module ◽

A Finite Group ◽

Thick Subcategory ◽

Generating Hypothesis

AbstractLet G be a finite group, and let k be a field whose characteristic p divides the order of G. Freyd's generating hypothesis for the stable module category of G is the statement that a map between finite-dimensional kG-modules in the thick subcategory generated by k factors through a projective if the induced map on Tate cohomology is trivial. We show that if G has periodic cohomology, then the generating hypothesis holds if and only if the Sylow p-subgroup of G is C2 or C3. We also give some other conditions that are equivalent to the GH for groups with periodic cohomology.

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