Linear Representations Of Finite Groups Research Articles

Frobenius Manifolds on Orbits Spaces

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group \({\mathcal {W}}\) acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.

RepnDecomp: A GAP package for decomposing linear representations of finite groups

RepnDecomp: A GAP package for decomposing linear representations of finite groups

Convergence and limits of linear representations of finite groups

Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in continuous algebras. We show that under a certain integrality condition, the algebras above are skew fields. Our main result is the extension of Schramm's characterization of hyperfiniteness to linear representations.

A numerical invariant for linear representations of finite groups

We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi–Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that in the modular setting ed($\rho$) can be arbitrary large (under a mild assumption on $G$). Here $G$ is fixed, and $\rho$ is allowed to range over the finite-dimensional representations of $G$. The appendix gives a constructive version of this result.

An introduction to the linear representations of finite groups

A few elements of the formalism of finite group representations are recalled. As to avoid a too mathematically oriented approach the discussed items are limited to the most essential aspects of the linear and matrix representations of standard use in chemistry and physics.

Algorithms for highly symmetric linear and integer programs

This paper deals with exploiting symmetry for solving linear and integer programming problems. Basic properties of linear representations of finite groups can be used to reduce symmetric linear programming to solving linear programs of lower dimension. Combining this approach with knowledge of the geometry of feasible integer solutions yields an algorithm for solving highly symmetric integer linear programs which only takes time which is linear in the number of constraints and quadratic in the dimension.

Linear Representations and Isospectrality with Boundary Conditions

We present a method for constructing families of isospectral systems, using linear representations of finite groups. We focus on quantum graphs, for which we give a complete treatment. However, the method presented can be applied to other systems such as manifolds and two-dimensional drums. This is demonstrated by reproducing some known isospectral drums, and new examples are obtained as well. In particular, Sunada’s method (Ann. Math. 121, 169–186, 1985) is a special case of the one presented.

On Primitive Linear Representations of Finite Groups

Let F be a field, let G be a finite group, and let π be a linear representation of G over F; that is, π is a group homomorphism π: G → GL(V) of G into the general linear group on a finite-dimensional vector space V over F. We say π is AI if π is completely reducible and for each normal subgroup H of G, each irreducible FH-submodule of V is absolutely irreducible. For example, if F is algebraically closed then all completely reducible representations over F are AI. In particular, all of our theorems hold over the complex numbers without the hypothesis that the representation is AI.

On fuzzy linear representations of finite groups

A linear representation ϱ of a finite group G in a finite-dimensional vector space V induces, through Zadeh's extension principle, a function, 9≈, from I G to into I GL( V) where GL( V) is the group of all linear automorphisms of V. If W is a fuzzy subspace of V, the group of all fuzzy linear automorphisms of W GL( W ), is a subgroup of GL( V). W is said to be stable under the action of a fuzzy subgroup A of G if 9≈( A ) is a subset of GL( W ), i.e. 9≈( A ) is zero at every f inside GL( V) and outside GL( q ). If W is stable under the action of A then its support subspace is stable under the action of the support subgroup of A in the crisp sense. Finally, we show that; if there are two stable fuzzy subspaces one of them is contained in the other, then the smaller one has a fuzzy direct summand in the bigger which is also a stable fuzzy subspace.

Methods that combine finite group theory with component mode synthesis in the analysis of repetitive structures

The static and dynamic calculations of symmetric mechanical structures are considered. This type of configuration is frequently found in rotary machines, turbomachines, parabolic antennas, spatial structures, etc. The objective of this study is to present two methods and a calculation code which takes this repetitivity into account in order to minimize the time and cost of predictive calculations. Both proposed methods are applicable to bidimensional and tridimensional problems with arbitrary loadings. They are based on the theory of linear representations of finite groups and a technique of component mode synthesis.

Book Review: Linear representations of finite groups

Book Review: Linear representations of finite groups