Filtrations on the representation ring of a finite group

1983 ◽  
pp. 397-405 ◽  
Author(s):  
C. B. Thomas
Keyword(s):  
Finite Group ◽  
Representation Ring ◽  
A Finite Group

J-Equivalence of group representations

1971 ◽  
Vol 70 (1) ◽  
pp. 9-14 ◽  
Author(s):  
V. P. Snaith
Keyword(s):  
Finite Group ◽  
Galois Group ◽  
Complex Representation ◽  
Group Representations ◽  
Root Of Unity ◽  
Representation Ring ◽  
A Finite Group

If G is a finite group of order N and ΓN is the Galois group of Q(w) over Q, where w is a primitive Nth root of unity then ΓN acts on the complex representation ring, R(G), of G. The group of co-invariants is denoted by R(G)ΓN = R(G)/W(G).

scholarly journals Regulator constants of integral representations of finite groups

2018 ◽  
Vol 168 (1) ◽  
pp. 75-117 ◽  
Author(s):  
ALEX TORZEWSKI
Keyword(s):  
Finite Group ◽  
Elliptic Curves ◽  
Finite Groups ◽  
Isomorphism Class ◽  
Integral Representations ◽  
Dihedral Groups ◽  
Representation Ring ◽  
Representations Of Finite Groups ◽  
Cohomological Invariant ◽  
A Finite Group

AbstractLet G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

The Representation Ring and the Centre of a Hopf Algebra

1999 ◽  
Vol 51 (4) ◽  
pp. 881-896 ◽  
Author(s):  
Sarah J. Witherspoon
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Character Table ◽  
Quantum Double ◽  
Representation Ring ◽  
Finite Dimensional ◽  
Dual Character ◽  
Structure Constants ◽  
A Finite Group ◽  
Cocommutative Hopf Algebra

AbstractWhen H is a finite dimensional, semisimple, almost cocommutative Hopf algebra, we examine a table of characters which extends the notion of the character table for a finite group. We obtain a formula for the structure constants of the representation ring in terms of values in the character table, and give the example of the quantum double of a finite group. We give a basis of the centre of H which generalizes the conjugacy class sums of a finite group, and express the class equation of H in terms of this basis. We show that the representation ring and the centre of H are dual character algebras (or signed hypergroups).

The Representation Ring of the Twisted Quantum Double of a Finite Group

1996 ◽  
Vol 48 (6) ◽  
pp. 1324-1338 ◽  
Author(s):  
S. J. Witherspoon
Keyword(s):  
Finite Group ◽  
Group Algebras ◽  
Quantum Double ◽  
Grothendieck Ring ◽  
Representation Ring ◽  
Twisted Group ◽  
A Finite Group

AbstractWe provide an isomorphism between the Grothendieck ring of modules of the twisted quantum double of a finite group, and a product of centres of twisted group algebras of centralizer subgroups. It follows that this Grothendieck ring is semisimple. Another consequence is a formula for the characters of this ring in terms of representations of twisted group algebras of centralizer subgroups.

scholarly journals Brauer relations in finite groups II: Quasi-elementary groups of order paq

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Alex Bartel ◽  
Tim Dokchitser
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Burnside Ring ◽  
Rational Representation ◽  
Generators And Relations ◽  
Representation Ring ◽  
A Finite Group

Abstract.This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we provide such a classification in terms of generators and relations for quasi-elementary groups of order

Some computations in the modular representation ring of a finite group

1971 ◽  
Vol 69 (1) ◽  
pp. 163-166 ◽  
Author(s):  
John Santa Pietro
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Modular Representation ◽  
Multiplicative Structure ◽  
Characteristic P ◽  
Isomorphism Classes ◽  
Direct Sum ◽  
Representation Ring ◽  
Ring Isomorphism ◽  
A Finite Group

Let p be an odd prime and G = HB be a semi-direct product where H is a cyclic, p-Sylow subgroup and B is finite Abelian. If K is a field of characteristic p the isomorphism classes of KG-modules relative to direct sum and tensor product generate a ring a(G) called the representation ring of G over K. If K is algebraically closed it is shown in (4) that there is a ring isomorphism a(G) ≃ a(HB2)⊗a(B1) where B1 is the kernel of the action of B on H and B2 = B/B1.> 2, Aut (H) is cyclic thus HB2 is metacyclic. The study of the multiplicative structure of a(G) is thus reduced to that of the known rings a(B1) and a(HB2) (see (3)).

On the cohomology of certain groups

1961 ◽  
Vol 57 (4) ◽  
pp. 731-733 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Finite Group ◽  
Differential Operator ◽  
Spectral Sequence ◽  
Large Class ◽  
Recent Work ◽  
Finite Groups ◽  
Classifying Spaces ◽  
Representation Ring ◽  
Grothendieck Rings ◽  
A Finite Group

Recent work of Atiyah (1) on the Grothendieck rings of classifying spaces of finite groups has yielded, among many other interesting results, a spectral sequence relating the simple integer cohomology of a finite group to its representation ring. He has determined the first differential operator of the spectral sequence which affects the p-part of the cohomology, and the question naturally arises, when is it the only non-zero one. My object in this note is to show that this is so for a large class of groups.

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