Irreducible representations of finite abelian groups pdf

The first part was originally written for quantum chemists. Any irreducible complex representation of an abelian group is 1dimensional. Representation theory for finite groups contents 1. Representations of elementary abelian pgroups and vector. Finite groups and character theory this semester well be studying representations of lie groups, mostly compact lie groups. The irreducible complex representations of a finite group g can be characterized using results from character theory. Phase retrievable projective representation frames for. Abelian topological groups g with prototype g r or c are something of an ex treme in the opposite direction. Now, suppose that is an irreducible representation of so, by definition, is a simple module and hence by the second part of diximierschurs lemma.

Representation theory for finite groups shaun tan abstract. Pdf representation theory of finite groups researchgate. Irreducible representations of finite groups sciencedirect. Representation theory of nite abelian groups october 4, 2014 1. We brie y discuss some consequences of this theorem, including the classi cation of nite. Finite abelian groups amin witno abstract we detail the proof of the fundamental theorem of nite abelian groups, which states that every nite abelian group is isomorphic to the direct product of a unique collection of cyclic groups of prime power orders. For an abelian group, any irreducible representation over a splitting field is onedimensional in particular, for a finite abelian group of order, the degrees of irreducible representations. Here we know that unitary irreducible representations are all onedimensional we proved this as a consequence of the unitary schur in 1. Determination of a representation by its character. If the locally compact abelian group g has a finite dimensional unitary irreducible projective representation with factor system.

The following theorem will be presented in a more general way, as it provides a very beautiful result about representations of compact and therefore also of finite groups. Linear representations of finite groups jeanpierre. Chapter 5 characters and character tables in great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy. Every linear representation of a compact group over a field of characteristic zero is a direct sum of irreducible representations. While it is comparatively simple to do so for nite groups and there are. If g is abelian, then gi and nj i implying that all irreducible representations will be one dimensional.

We consider character theory, constructions of representations, and conjugacy classes. Pdf induced representations of abelian groups of finite rank. Representation of finite abelian groups 561 nj 2 igi jl where gi is the order of the finite group 6. Introduction to representation theory of finite groups. Classify all representations of a given group g, up to isomorphism. I have freely used the language of abelian categories projective modules, grothendieck groups, which is well suited to this sort of question. Some of the general structure theory in the compact case is quite similar to that of the case of.

Every complex representation of a finite abelian group is completely re ducible, and every irreducible representation is 1dimensional. Irreducible representations of finite groups 491 appendix the remaining degeneracy discussed in section 3 can completely be removed by a linear combination of the irreducible matrix representatives, as shown by the following matrix generating theorem, which is actually a special case of burnsides theorem. It will further be shown that a locally compact group has all of its irreducible representations of finite dimension if and only if it is a projective limit of lie groups with the same property, and finally that a lie group has this. We are now in a position to describe the irreducible representations of. Irreducible characters of finite abelian groups mathoverflow. Linear representations of finite groups jeanpierre serre auth. The characterization of abelian groups, which have faithful irreducible projective representations over algebraic number fields andpadic number fields can be found in barannik l. A finite abelian group has a faithful irreducible representation if and only if it is cyclic. Browse other questions tagged finitegroups representationtheory abeliangroups characters or ask your own question. Reducible representations of abelian groups by aharon atzmon 1.

One side of the theorem was already proved in remark 2. Representation theory of finite abelian groups applied to a. Let t be an irreducible representation of g over f and d. On irreducible characters of dihedral groups of degree. Which finite groups have faithful complex irreducible. Faithful irreducible projective representations of metabelian. Linear representations of finite groups springerlink. We prove that any irreducible faithful representation of an almost torsionfree abelian group g of finite rank over a finitely generated field of characteristic zero is induced from an irreducible. That is, we claim that v is a direct sum of simultaneous eigenspaces for all operators in g. Request pdf phase retrievable projective representation frames for finite abelian groups we consider the problem of characterizing projective representations that admit frame vectors with the. Also, we have already described all the representations of a finite cyclic group in example 1. Let a be a finite abelian group and let v be an irreducible representation of a.

Questions about modular representation theory of finite groups can often be reduced to elementary abelian subgroups. It was shown that the existence of such representations strongly depends on construction of the socle of the group g. Abelian implies every irreducible representation is one. On the projective representations of finite abelian groups. This is the first book to offer a detailed study of the representation theory of elementary abelian groups, bringing together information from many papers and journals, as well as unpublished research. This book consists of three parts, rather different in level and purpose. Pdf on irreducible partial representations of groups. Irreducible representations of finite abelian groups. Representations of finite groups ubc math university of british. Irreducible representations of finite groups jason fulman abstract.

Representation theory authorstitles recent submissions. Irreducible representations of nilpotent groups generate. We refer the readers to curtis and reiner 3 for relevant definitions on projective representations of finite groups. Let h be a finite group acting by group automorphisms on a cyclic. Random walk on the set of irreducible representations of a. As we have explained above, a representation of a group g over k is the same thing as a representation of its group algebra kg. On the degrees of irreducible representations of finite groups atumi watanabe received november 29, 1979 1. Introduction let g be a finite group of order g \ and f be an algebraically closed field of characteristic 0. Complete sets of inequivalent irreducible projective representations of cnm w1,wn. Representation theory for finite groups 5 proposition 3. We find that the irreducible representations of finite abelian groups are, in gen.

Gcas and projective representations of finite abelian groups, alladi ramakrishnans. Groups with an irreducible character of large degree are solvable frank demeyer abstract. Representation theory university of california, berkeley. Irreducible representations of nilpotent groups generate classi able c algebras caleb eckhardt and elizabeth gillaspy miami university of ohio and university of montana uct masterclass k. The characterization of abelian groups, which have faithful irreducible projective representations over algebraic number fields andadic number fields can be found in barannik 1. Dihedral groups are good example of finite groups and have a series of applications in chemistry. There are an infinite number of equivalent representations but it is possible to find one. As related results, an asymptotic description of plancherel. The degree of an irreducible complex character afforded by a finite group is bounded above by the index of an abelian normal subgroup and by the square root of the index of the center. Feb 08, 2011 explain how to find all irreducible representations of a finite abelian group.

Show that every irreducible representation of g over a. Representation theory of finite groups anupam singh iiser pune. For the symmetric and general linear groups, a sharp convergence rate bound is obtained and a cuto. Irreducible representations of the groups z and r in a nuclear frechet space are constructed in 3 and 4, respectively. Read 23 implies that there exists an irreducible representation. Irreducible representations of finite abelian groups 14 4. We are now in a position to describe the irreducible representations of an abelian group. In particular, all such representations decompose as a direct sum of irreps, and the number of irreps of is equal to the number of conjugacy classes of. First recall that, by the fundamental theorem of finite abelian groups, every finite abelian group is a finite direct product of cyclic groups.

We obtained some sufficient and necessary conditions of existence of faithful irreducible representations of a soluble group g of finite rank over a field k. P i z c i is a finitely generated abelian group, each c i is. Whenever a finite group affords an irreducible character. Connected compact lie groups are reductive essentially semsimple plus torus, so gg, g is a torus, which misses most irreducible unitary representations of g if g has a simple component. Irreducible representations of the product of two groups 4. The case of finite groups was solved by gaschutz in. A representation of a finite abelian group is irreducible if and only if proof. Gaschutz, endliche gruppen mit treuen absolutirreduziblen darstellungen.