Feb 25, 2020 namely, we construct of a super version of the frobenius schur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the eight types of irreducible real super representations described in wall1964 it belongs to. We previously calculated the character table of s 4. Introduction to representation theory pavel etingof, oleg golberg, sebastian hensel, tiankai liu, alex schwendner, dmitry vaintrob, and elena yudovina. Introduction to representation theory of nite groups. This book is written for students who are studying nite group representation theory beyond the level of a rst course in abstract algebra. Representation theory was created by frobenius about 100 years ago. The representation theory of symmetric groups is a special case of the representation. It provides a new viewpoint from which one can examine various aspects of representation theory and areas of application. The proof of the peterweyl theorem in chapter v was strongly inspired by taos online notes 34 and 33. The other possible and worse definition is a group that admits an action so that the pair forms a frobenius group. The super frobeniusschur indicator and finite group gauge. This was the first example of frobenius group with nonabelian kernel it was.
The representation theory of symmetric groups james, g. Representation theory university of california, berkeley. A group is an abstract mathematical object, a set with elements and an operation satisfying certain axioms. A matrix representation of gover f is a group homomorphism. Their work was inspired in part by two largely unrelated developments which occurred earlier in the nineteenth century. Glv, where v is an ndimensional vector space over kand glv denotes the group of invertible linear maps v. Lecture 17 perronfrobenius theory stanford university. Reph that gives a representation of hfrom a representation of g, by simply restricting the group action to h.
Group representations and harmonic analysis from euler to. Construction of hadamard matrices from certain frobenius. In this letter dedekind made the following observation. If g is a 2concos group and g is a frobenius group with kernel g0 and complement h, then h is cyclic. A representation of a group gis a homomorphism from gto the group glv of invertible linear operators on v, where v.
This result which we call the frobenius theorem has stimulated widespread inter. On primitive permutation groups inducing a frobenius group on a suborbit, ii, the quarterly journal of mathematics, volume 31, issue 3, 1 septe we use cookies to enhance your experience on our website. I shouldve paid more attention to how the physicists teach general relativity. Mackey received december 9, 1952 introduction in the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by frobeniuss notion of induced representation. Finally, the representation theory of the symmetric group s n is explored, using both methods from both combinatorics. In the rst problem we explore how often they vanish in artintype representations. Frobenius twists in the representation theory of the. Frobenius algebras include matrix algebras over any field where the bilinear form is given by the trace of the product of two matrices, semisimple algebras, group algebras of finite groups even if they are not semisimple, which includes the crucial case for modular representation theory, when the characteristic of the field divides the order. Group matrices, group determinants and representation theory. Frobenius twists in the representation theory of the symmetric group david j. Lam recapitulation the origin of the representation theory of finite groups can be traced back to a correspondence between r. For the general linear group glnk over an algebraicallyclosed. A frobenius group is a transitive permutation group that is not regular and such that only the identity fixes more than one point.
If there is torsion in the homology these representations require something other than ordinary character theory to be understood. This article defines a property that can be evaluated for finite groups and hence, a particular kind of group property view other properties of finite groups or view all group properties definition. Right division in groups, dedekindfrobenius group matrices, and ward quasigroups. Lecture 17 perronfrobenius theory positive and nonnegative matrices and vectors. I am certain that this talk will go horribly there are too many indices ying around. The present article is based on several lectures given by the author in 1996 in. Prior to this there was some use of the ideas which we can now identify as representation theory characters of cyclic groups as used by. Construction of hadamard matrices from certain frobenius groups. Here is an overview of the course quoted from the course page.
Namely, we construct of a super version of the frobeniusschur indicator whose value for a real irreducible super representation is an eighth root of unity, distinguishing which of the. The representation theory of symmetric groups is a special case of the. In this theory, one considers representations of the group algebra a cg. The creation of the theory of group characters by thomas hawkins the creation of the theory of group characters and representations by georg frobenius is one of those relatively rare episodes in the history of mathematics for which we have much more information at our disposal than that conveyed through the published papers themselves. Mackey received december 9, 1952 introduction in the study of the relationship between the representation theory of a group and those of its various subgroups an important role is played by frobenius s notion of induced representation. A representation of a group realizes the elements of the group concretely as geometric symmetries. Mat 4451196 introduction to representation theory chapter 1 representation theory of groups algebraic foundations. A representation of a group gis a homomorphism from gto the group glv of invertible. A representation of a group gis a homomorphism from gto the group glv of invertible linear operators on v, where v is a nonzero. Frobenius reciprocity, and frobenius character formula chapter 3 representations of sl 2f q. Representations of frobenius kernels these are notes for a talk that i was coerced into giving for the langlands support group on modular representation theory. A graphical frobenius representation gfr of a frobenius group g is a graph whose automorphism group, as a group of permutations of the vertex set, is isomorphic to g. G is called a frobenius group, and k is called its.
Frobenius algebra, symmetric algebra, group algebra, hopf algebra, separable algebra, character, grothendieck ring, integrality, irreducible representation, adjoint representation, rank. Representation theory was born in 1896 in the work of the german mathematician f. In this theory, one considers representations of the group algebra a cg of a. If k is the nonabelian group of order 7 3 with exponent 7, and h is the cyclic group of order 3, then there is a frobenius group g that is an extension k. This shows that g is a frobenius group with kernel g0 and complement h. Group matrices, group determinants and representation. If d is a divisor of the order of a finite group g, then the number of solutions of xd 1 in g is a multiple of d. The representation theory of finite groups began with the pioneering research of frobenius, burnside, and schur at the turn of the century. Parts of it may be used to introduce undergraduates to representation theory by studying the appealing pattern structure of group matrices. Frobenius algebra, symmetric algebra, group algebra, hopf algebra, separable algebra. This book sets out an account of the tools which frobenius used to discover representation theory for nonabelian groups and describes its modern applications. The same group will generally have many di erent such representations. Since it is often easier to find representations of the smaller group h than of g, the operation of forming induced representations is an important tool to construct new representations.
The book is designed to appeal to several audiences, primarily mathematicians working either in group representation theory or in areas of mathematics where representation theory is involved. A representation of a group gon a complex vector space vis a group action of gon vby linear transformations, i. Finally, chapter vi was my attempt to specialize highest weight theory to the lie group sun and the complex lie algebra sl n. The problem of classifying which frobenius groups admit a gfr is a natural extension of the. We give an upper bound for the density of the set of vanishing frobenius traces in terms of the multiplicities of the irreducible components of the adjoint representation. Introduction to representation theory mit opencourseware. Representations of the symmetric group via young tableaux jeremy booher as a concrete example of the representation theory we have been learning, let us look at the symmetric groups s n and attempt to understand their representations.
Representation theory of groups algebraic foundations 1. Research of the author supported in part by nsa grant h982300910026. An ndimensional representation of gover k n 1 is a group homomorphism g. If a group contains a proper non trivial subgroup such that for all, then there exists a normal subgroup such that is the semidirect product of and. The representation theory of nite groups has a long history, going back to the 19th century and earlier. This thesis studies frobenius traces in galois representations from two di erent directions. Induced representations and frobenius reciprocity math g4344, spring 2012 1 generalities about induced representations for any group gand subgroup h, we get a restriction functor resg h. Given a representation of h, the induced representation is, in a sense, the most general representation of g that extends the given one. The rst results about representation theory in characteristic p are also discussed. This gives an example of a frobenius group with nonabelian kernel. A theorem of frobenius says that if g is a finite frobenius group given as a permutation group, as above, the set consisting of the identity of g and those elements with no fixed point forms a normal subgroup n. This work was triggered by a letter to frobenius by r. Johnson this book sets out an account of the tools which frobenius used to discover representation theory for nonabelian groups and describes its modern applications.