scholarly journals Branching law for finite subgroups of 𝐒𝐋 3 ℂ$\mathbf {SL}_3\mathbb {C}$ and McKay correspondence

2014 ◽  
Vol 17 (2) ◽  
Author(s):  
Frédéric Butin ◽  
Gadi S. Perets
Keyword(s):  
Finite Subgroup ◽  
Mckay Correspondence ◽  
Branching Law ◽  
Finite Subgroups

Abstract.Given a finite subgroup

Coinvariant Algebras of Finite Subgroups of SL(3;C)

2004 ◽  
Vol 56 (3) ◽  
pp. 495-528 ◽  
Author(s):  
Yasushi Gomi ◽  
Iku Nakamura ◽  
Ken-ichi Shinoda
Keyword(s):  
Hilbert Scheme ◽  
Dual Graph ◽  
Orbit Space ◽  
Rational Curves ◽  
Finite Subgroup ◽  
Crepant Resolution ◽  
Mckay Correspondence ◽  
Finite Subgroups ◽  
Explicit Formulae ◽  
Molien Series

AbstractFor most of the finite subgroups of SL(3; C) we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae [McKay99] for subgroups of SU(2). We also study the G-orbit Hilbert scheme HilbG(C3) for any finite subgroup G of SO(3), which is known to be a minimal (crepant) resolution of the orbit space C3/G. In this case the fiber over the origin of the Hilbert-Chow morphism from HilbG(C3) to C3/G consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of G. This is an SO(3) version of the McKay correspondence in the SU(2) case.

scholarly journals RESTRICTIONS OF ALGEBRAIC GROUP REPRESENTATIONS TO FINITE SUBGROUPS

2002 ◽  
Vol 34 (2) ◽  
pp. 165-173
Author(s):  
HARM DERKSEN
Keyword(s):  
Algebraic Group ◽  
Short Proof ◽  
Linear Algebraic Group ◽  
Finite Subgroup ◽  
Group Representations ◽  
Rational Representation ◽  
Finite Dimensional ◽  
Finite Subgroups

Suppose that H is a finite subgroup of a linear algebraic group, G. It was proved by Donkin that there exists a finite-dimensional rational representation of G whose restriction to H is free. This paper gives a short proof of this in characteristic 0. The author also studies more closely which representations of H can appear as a restriction of G.

scholarly journals G-GRAPHS AND SPECIAL REPRESENTATIONS FOR BINARY DIHEDRAL GROUPS IN GL(2,ℂ)

2012 ◽  
Vol 55 (1) ◽  
pp. 23-57
Author(s):  
ALVARO NOLLA DE CELIS
Keyword(s):  
Moduli Space ◽  
Cyclic Subgroup ◽  
Finite Subgroup ◽  
Explicit Description ◽  
Vector Spaces ◽  
Minimal Resolution ◽  
Mckay Correspondence ◽  
Dihedral Groups ◽  
Resolution Of Singularity ◽  
Distinguished Basis

AbstractGiven a finite subgroup G⊂GL(2,ℂ), it is known that the minimal resolution of singularity ℂ2/G is the moduli space Y=G-Hilb(ℂ2) of G-clusters ⊂ℂ2. The explicit description of Y can be obtained by calculating every possible distinguished basis for as vector spaces. These basis are the so-called G-graphs. In this paper we classify G-graphs for any small binary dihedral subgroup G in GL(2,ℂ), and in the context of the special McKay correspondence we use this classification to give a combinatorial description of special representations of G appearing in Y in terms of its maximal normal cyclic subgroup H ⊴ G.

scholarly journals Solvable-by-finite subgroups of GL(2, F)

1978 ◽  
Vol 19 (1) ◽  
pp. 45-48 ◽  
Author(s):  
Abdul Majeed ◽  
A. W. Mason
Keyword(s):  
Linear Group ◽  
Characteristic Zero ◽  
Finite Subgroup ◽  
Free Subgroup ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Irreducible Subgroup ◽  
Finite Subgroups

In a recent paper [5] Tits proves that a linear group over a field of characteristic zero is either solvable-by-finite or else contains a non-cyclic free subgroup. In this note we determine all the infinite irreducible solvable-by-finite subgroups of GL(2, F), where F is an algebraically closed field of characteristic zero. (Every reducible subgroup of GL(2, F) is metabelian.) In addition, we prove that an irreducible subgroup of GL(2, F) has an irreducible solvable-by-finite subgroup if and only if it contains an element of zero trace.

FINITE SUBGROUPS OF HYPERBOLIC GROUPS

2000 ◽  
Vol 10 (04) ◽  
pp. 399-405 ◽  
Author(s):  
NOEL BRADY
Keyword(s):  
Upper Bound ◽  
Identity Element ◽  
Hyperbolic Group ◽  
Finite Subgroup ◽  
Hyperbolic Groups ◽  
Generating Set ◽  
Number Of Generators ◽  
Finite Subgroups ◽  
Hyperbolicity Constant

We prove that every finite subgroup of a hyperbolic group G can be conjugated to a 2δ+1 neighborhood of the identity element, where δ is the hyperbolicity constant for G with respect to a given generating set. This gives an upper bound for the size of such finite subgroups in terms of δ and the number of generators for G.

ON THE DIMENSION OF GROUPS THAT SATISFY CERTAIN CONDITIONS ON THEIR FINITE SUBGROUPS

2020 ◽  
pp. 1-6
Author(s):  
LUIS JORGE SÁNCHEZ SALDAÑA
Keyword(s):  
Maximal Subgroup ◽  
Modular Group ◽  
Cohomological Dimension ◽  
Finite Subgroup ◽  
Fuchsian Groups ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Finite Subgroups ◽  
One Relator Groups ◽  
Virtual Cohomological Dimension

Abstract We say a group G satisfies properties (M) and (NM) if every nontrivial finite subgroup of G is contained in a unique maximal finite subgroup, and every nontrivial finite maximal subgroup is self-normalizing. We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for EG and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one-relator groups, the Hilbert modular group, and 3-manifold groups.

scholarly journals Sums of multivariate polynomials in finite subgroups

2018 ◽  
Vol 14 (02) ◽  
pp. 301-311
Author(s):  
Paolo Leonetti ◽  
Andrea Marino
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Subgroup ◽  
Multivariate Polynomials ◽  
Multivariate Polynomial ◽  
Group Of Units ◽  
Finite Subgroups

Let [Formula: see text] be a commutative ring, [Formula: see text] a multivariate polynomial, and [Formula: see text] a finite subgroup of the group of units of [Formula: see text] satisfying a certain constraint, which always holds if [Formula: see text] is a field. Then, we evaluate [Formula: see text], where the summation is taken over all pairwise distinct [Formula: see text]. In particular, let [Formula: see text] be a power of an odd prime, [Formula: see text] a positive integer coprime with [Formula: see text], and [Formula: see text] integers such that [Formula: see text] divides [Formula: see text] and [Formula: see text] does not divide [Formula: see text] for all non-empty proper subsets [Formula: see text]; then [Formula: see text] where the summation is taken over all pairwise distinct [Formula: see text]th residues [Formula: see text] modulo [Formula: see text] coprime with [Formula: see text].

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