Character theory
The character carries the essential information about the representation in a more condensed form.Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves.This is possible because a complex representation of a finite group is determined (up to isomorphism) by its character.The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well.Many deep theorems on the structure of finite groups use characters of modular representations.Characters of irreducible representations encode many important properties of a group and can thus be used to study its structure.Character theory is an essential tool in the classification of finite simple groups.Close to half of the proof of the Feit–Thompson theorem involves intricate calculations with character values.Easier, but still essential, results that use character theory include Burnside's theorem (a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem of Richard Brauer and Michio Suzuki stating that a finite simple group cannot have a generalized quaternion group as its Sylow 2-subgroup.Let V be a finite-dimensional vector space over a field F and let ρ : G → GL(V) be a representation of a group G on V. The character of ρ is the function χρ : G → F given by where Tr is the trace.The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a compact form.Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class of G. The columns are labelled by (representatives of) the conjugacy classes of G. It is customary to label the first row by the character of the trivial representation, which is the trivial action of G on a 1-dimensional vector space byHere is the character table of the cyclic group with three elements and generator u: where ω is a primitive third root of unity.Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism.This group is connected to Dirichlet characters and Fourier analysis.Since the irreducible characters of G form an orthonormal basis for the space of complex-valued class functions of G, there is a unique class function θG of G with the property that for each irreducible character χ of G (the leftmost inner product is for class functions of G and the rightmost inner product is for class functions of H).The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions.This led to an alternative description of the induced character θG.This induced character vanishes on all elements of G which are not conjugate to any element of H. Since the induced character is a class function of G, it is only now necessary to describe its values on elements of H. If one writes G as a disjoint union of right cosets of H, say then, given an element h of H, we have: Because θ is a class function of H, this value does not depend on the particular choice of coset representatives.The general technique of character induction and later refinements found numerous applications in finite group theory and elsewhere in mathematics, in the hands of mathematicians such as Emil Artin, Richard Brauer, Walter Feit and Michio Suzuki, as well as Frobenius himself.The Mackey decomposition was defined and explored by George Mackey in the context of Lie groups, but is a powerful tool in the character theory and representation theory of finite groups.Its basic form concerns the way a character (or module) induced from a subgroup H of a finite group G behaves on restriction back to a (possibly different) subgroup K of G, and makes use of the decomposition of G into (H, K)-double cosets.is a disjoint union, and θ is a complex class function of H, then Mackey's formula states that where θt is the class function of t−1Ht defined by θt(t−1ht) = θ(h) for all h in H. There is a similar formula for the restriction of an induced module to a subgroup, which holds for representations over any ring, and has applications in a wide variety of algebraic and topological contexts.Mackey decomposition, in conjunction with Frobenius reciprocity, yields a well-known and useful formula for the inner product of two class functions θ and ψ induced from respective subgroups H and K, whose utility lies in the fact that it only depends on how conjugates of H and K intersect each other.The formula (with its derivation) is: (where T is a full set of (H, K)-double coset representatives, as before).This formula is often used when θ and ψ are linear characters, in which case all the inner products appearing in the right hand sum are either 1 or 0, depending on whether or not the linear characters θt and ψ have the same restriction to t−1Ht ∩ K. If θ and ψ are both trivial characters, then the inner product simplifies to |T|.One may interpret the character of a representation as the "twisted" dimension of a vector space.[3] Treating the character as a function of the elements of the group χ(g), its value at the identity is the dimension of the space, since χ(1) = Tr(ρ(1)) = Tr(IV) = dim(V).A sophisticated example of this occurs in the theory of monstrous moonshine: the j-invariant is the graded dimension of an infinite-dimensional graded representation of the Monster group, and replacing the dimension with the character gives the McKay–Thompson series for each element of the Monster group.is a complex semisimple Lie algebra with Cartan subalgebra