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This is a survey article related to:linear representation theory
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This article describes the duality between conjugacy classes and representations, as well as puts important results of character theory in the correct perspective.

Contents 1 Conjugacy classes, representations, and automorphisms of the group 1.1 The conjugacy classes 1.2 The representations 1.3 Automorphism group 2 Two notions of evaluation and equivalence 2.1 Locally conjugate 2.2 Character-conjugate 2.3 A tabular description of both 3 Notions of collapse 3.1 For locally conjugate 3.2 For character-conjugate 3.3 In the tabular description 4 What happens in special cases 4.1 Maschke's averaging lemma 4.2 For finite groups 4.3 Automorphisms acting on these 4.4 Induction rules for subgroups 5 Further exploiting the duality 5.1 Brauer's permutation lemma 5.2 Supercharacter theory

Conjugacy classes, representations, and automorphisms of the group

The conjugacy classes

Let C(G) denote the set of conjugacy classes of G. Clearly, C(G) is just a plain set.

The representations

Let I(G) denote the space of indecomposable linear representations of G. Again, I(G) is just a plain set. However, we can view the -algebra freely generated by these, and in fact, the elements with all coefficients positive can be identified precisely with the representations that can be expressed a finite linear combinations of indecomposable representations. We denote the set of all representations as R(G).

Automorphism group

Let G be a group. Then, the automorphism group acts naturally on the elements of G.

We now have two induced actions of A:

• A acts on C(G). That is, every element permutes the conjugacy classes in G.
• A acts on the collection of all linear representations of G, as follows . This action preserves indecomposability (that is, an indecomposable representation is mapped to an indecomposable representation under the action of A). Thus, A acts on I(G).

Observe some further things:

• The class automorphisms of G act trivially on C(G), in fact, these are precisely the automorhpisms that act trivially on C(G). Hence, the group that effectively acts is the quotient of the whole group of automorphisms by the group of class automorphisms.
• The inner automorphisms of G act trivially on I(G). However, it is not immediately clear whether these are the only automorphisms that act trivially on I(G).

Two notions of evaluation and equivalence

Locally conjugate

Let C(GL) denotes the collection of all linear automorphisms of any vector space Vover k, upto the following equivalence: a linear automorphism σ₁ of V₁ is equivalent to a linear automorhpism σ₂ of V₂ if there is an isomorphism such that .

C(GL) can be viewed as the union over all isomorphism classes of vector spaces V of the set of conjugacy classes in GL(V).

Consider the following map:

given by 's equivalence class for any .

We are now in a position to define a notion of locally conjugate:

• Two conjugacy classes c₁ and c₂ are termed locally conjugate if for any , L(c₁,ρ) = L(c₂,ρ). In other words, for any indecomposable representation, the image of elements in the two conjugacy classes are conjugate in the general linear group.
• Two representations ρ₁ and ρ₂ are termed locally conjugate if for any conjugacy class c, L(c,ρ₁) = L(c,ρ₂). In other words, two representations are locally conjugate if for any conjugacy class, the images under the two representations are similar linear transformations.

Note that being locally conjugate, both for elements and for representations, are a priori weaker conditions than being conjugate (or equivalent). The reason is that the choice of conjugating element may differ for every instance.

Character-conjugate

Consider the map:

where χ(c,ρ) = Tr(ρ(g)) where . This is the famed character.

First note that χ factors through L, in the sense that if χ(c,ρ) is completely determined from L(c,ρ).

We have two notions:

• Two conjugacy classes c₁ and c₂ are termed character-conjugate if χ(c₁,ρ) = χ(c₂,ρ) for any . In other words, the character takes the same value for every representation.
• Two representations ρ₁ and ρ₂ are termed character-conjugate representations if χ(c,ρ₁) = χ(c,ρ₂) for any conjugacy class c. In other words, the character takes the same value on every conjugacy class.

A tabular description of both

We can visualize the notion of locally conjugate as follows. Consider a table or grid in which the rows are labelled by indecomposable representations ρ (alliterated for convenience) and the columns are labelled by conjugacy classes c. In row ρ and column c fill in the value L(c,ρ).

Then two representations are locally conjugate if the corresponding rows are identical, and two conjugacy classes are locally conjugate if the corresponding columns are equal.

If instead we fill, in the entries of the table, the value of the character of ρ evaluated at c, then two representations are character-conjugate if the corresponding rows are identical, and two conjugacy classes are character-conjugate if the corresponding oclumns are identical.

Notions of collapse

For locally conjugate

• A field is termed class-separating for a group if the following holds: . In other words, any two locally conjugate conjugacy classes are the same. In other words, a conjugacy class is completley determined by the conjugacy types of its images under all representations.
• A field is termed class-determining for a group if the following holds: . In other words, any two locally conjugate representations are equivalent. In other words, a representation is completely determined by the conjugacy type of the image of every conjugacy class.

For character-conjugate

• A field is termed character-separating for a group if the following holds: . In other words, a conjugacy class is completely determined by the value of the character of all representations. In other words, any two character-conjugate conjugacy classes are the same.
• A field is termed character-determining for a group if the following holds: . In other words, any two character-conjugate representations are equivalent. In other words, a representation is completely determined by its character.

Note that since χ factors through L, any character-separating field is class-separating, and any character-determining field is class-determining.

In the tabular description

In the description where we make a table with rows indexed by representations ρ and columns indexed by conjugacy classes c, and where the entry in row ρ and column c is L(c,ρ), we say that the field is class-determining if no two rows are identical, and class-separating if no two columns are identical.

A similar observation golds if we replace the L with the character values.

What happens in special cases

Maschke's averaging lemma

Further information: Maschke's averaging lemma

Maschke's averaging lemma tells us that for a finite group over a field whose characteristic does not divide the order of the group, every finite-dimensional representation is completely reducible, and in particular, all the indecomposable representations are irreducible and finite-dimensional.

This in particular means that I(G) = Irr(G).

For finite groups

Further information: Non-modular implies character-determining, Sufficiently large implies splitting, Splitting implies character-separating

The orthonormality theorems tell us the following:

• For a finite group, if the field has characteristic zero or relatively prime to the order of the group, the field is character-determining. In other words, every representation of the group over the field is completely determined by its character.
• For a finite group, if the field is sufficiently large for the group, viz it contains all the m^th roots of unity where m is the exponent of G, then the field is a splitting field for the group, in the sense that all representations of the group realized over some extension of the field are also realized over the field. Thus, it is also a character-separating field. That is, a conjugacy class is uniquely determined by the characters of various representations, evaluated at it.

Combining both of these, we conclude that for a finite group, a sufficiently large field is both character-determining and character-separating.

This can be seen directly from the fact that the character table of a finite group over a sufficiently large field, when scaled by a suitable factor, becomes an orthogonal matrix.

Automorphisms acting on these

Further information: Linearly pushforwardable implies class-preserving for class-separating field, class-preserving implies linearly pushforwardable

We saw earlier that A acts both on C(G) and on R(G). Note also that under the A-action, the value of the function L remains unchanged. In other words, if , we have:

.

Thus, we see that:

• If the field is class-separating, then any linearly pushforwardable automorphism must be a class-preserving automorphism. A linearly pushforwardable automorphism is an automorphism that can be pushed forward to an inner automorphism for any linear representation. In terms of the tabular description, this is simply saying that if no two columns are identical, any automorphism that preserves the rows must preserve the columns.
• If the field is class-determining, then any class-preserving automorphism must be a linearly pushforwardable automorphism. In terms of the tabular description, this is simply saying that if no two rows are identical, any automorphism that if no two rows are identical, any automorphism that preserves the columns must preserve the rows.

For finite groups and sufficiently large fields both the above assumptions are valid, and hence, the class-preserving automorphisms are the same as the linearly pushforwardable automorphisms.

Induction rules for subgroups

We want to correlate the representation-theoretic properties related to induction and restriction of representations, with the properties of restricting and extending equivalence relations. To correlate these, again we need hypotheses on the field, such as class-determining, class-separating.

Call a subgroup of a group induction-isotypical if every irreducible representation of the subgroup inducts to an isotypical representation of the whole group. Here are some easy observations:

• If the subgroup is a central factor, then it is induction-isotypical for any field. This follows from the fact that all the conjugates of the representation in the group are equivalent to it in the subgroup.

Further exploiting the duality

Brauer's permutation lemma

Further information: Brauer's permutation lemma

This is a special result that works for irreducible representations. It essentially states that under any Galois automorphism, the decomposition of the set of conjugacy classes into orbits, is the same as the decomposition of the set of irreducible representations into orbits.

Supercharacter theory

Further information: Supercharacter theory

Supercharacter theories are a way of partitioning the characters into equivalence classes, and simultaneously partitioning the conjugacy classes into equivalence classes, in such a way that a duality of sorts continues to hold.