Finite group schemes of p-rank ≤ 1

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.

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

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}}.

CONTINUOUS TIME QUANTUM WALKS AND QUOTIENT GRAPHS

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.

scholarly journals Local duality for representations of finite group schemes

2019 ◽  
Vol 155 (2) ◽  
pp. 424-453 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause ◽  
Julia Pevtsova
Keyword(s):  
Finite Group ◽  
Cohomology Ring ◽  
Group Scheme ◽  
Stable Category ◽  
Group Schemes ◽  
Stable Module Category ◽  
Serre Duality ◽  
Stable Module ◽  
A Finite Group ◽  
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.

scholarly journals Cohomology and projectivity of modules for finite group schemes

2001 ◽  
Vol 131 (3) ◽  
pp. 405-425 ◽  
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?

scholarly journals Vector bundles trivialized by proper morphisms and the fundamental group scheme

2010 ◽  
Vol 10 (2) ◽  
pp. 225-234 ◽  
Author(s):  
Indranil Biswas ◽  
João Pedro P. Dos Santos
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Characteristic Zero ◽  
Vector Bundles ◽  
Projective Variety ◽  
Group Scheme ◽  
Smooth Projective Variety ◽  
Closed Field ◽  
Surjective Morphism ◽  
Finite Vector

AbstractLet X be a smooth projective variety defined over an algebraically closed field k. Nori constructed a category of vector bundles on X, called essentially finite vector bundles, which is reminiscent of the category of representations of the fundamental group (in characteristic zero). In fact, this category is equivalent to the category of representations of a pro-finite group scheme which controls all finite torsors. We show that essentially finite vector bundles coincide with those which become trivial after being pulled back by some proper and surjective morphism to X.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

scholarly journals Stratification for module categories of finite group schemes

2017 ◽  
Vol 31 (1) ◽  
pp. 265-302 ◽  
Author(s):  
Dave Benson ◽  
Srikanth B. Iyengar ◽  
Henning Krause ◽  
Julia Pevtsova
Keyword(s):  
Finite Group ◽  
Group Schemes ◽  
Finite Group Schemes

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

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