8.2 Character Theory of Finite Groups

Character Theory of Finite Groups is a powerful tool in algebraic combinatorics. It uses group representations to study group structure through character tables, which capture essential information about a group's irreducible representations and conjugacy classes.

This theory connects to Combinatorial Representation Theory by providing methods to analyze group actions and count solutions to equations in groups. It showcases how abstract algebra concepts can solve concrete combinatorial problems, bridging pure and applied mathematics.

Character tables of finite groups

Definition and properties of character tables

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• A character of a representation is the trace of the corresponding matrix representation
  • The character is a class function on the group, meaning it is constant on conjugacy classes
• The character table of a group is a table whose rows correspond to the irreducible characters and whose columns correspond to the conjugacy classes of the group
  • The entry in the ith row and jth column is the value of the ith irreducible character on an element of the jth conjugacy class
• The character table of a finite group determines the group up to isomorphism
  • Two groups with the same character table are isomorphic (e.g., $D_8$ and $Q_8$)

Computing character tables and their applications

• The sum of the squares of the dimensions of the irreducible representations equals the order of the group
  • This follows from the orthogonality relations for irreducible characters
• The number of irreducible representations of a group equals the number of conjugacy classes of the group
  • This is because the character table is a square matrix
• Character tables can be used to study the structure of a group, such as determining its center, commutator subgroup, and normal subgroups
  • For example, the center of a group consists of elements $g$ for which $|\chi(g)| = \chi(1)$ for all irreducible characters $\chi$

Orthogonality relations for characters

First and second orthogonality relations

• The first orthogonality relation states that the sum of the products of the values of two different irreducible characters over all group elements is zero
  • In other words, distinct irreducible characters are orthogonal with respect to the inner product defined by summing over the group
• The second orthogonality relation states that for an irreducible character $\chi$, the sum of $|\chi(g)|^2$ over all group elements $g$ equals the order of the group
  • This implies that an irreducible character has norm equal to the square root of the order of the group

Applications of orthogonality relations

• Orthogonality relations can be used to decompose a given character into a sum of irreducible characters
  • The multiplicity of an irreducible character in this decomposition can be computed using the inner product of characters
• The orthogonality relations imply that the irreducible characters form an orthonormal basis for the space of class functions on the group
  • This is with respect to the inner product defined by summing over the group
• Orthogonality relations are crucial in proving character-theoretic formulas for counting solutions to equations in groups
  • For example, they are used in the proof of Burnside's lemma (also known as the Cauchy-Frobenius lemma)

Character-theoretic formulas for counting solutions

Burnside's lemma and its applications

• Burnside's lemma is a formula for counting the number of orbits of a group action
  • It states that the number of orbits equals the average number of fixed points over all group elements
• The number of fixed points of a group element $g$ acting on a set $X$ can be computed using the character of the permutation representation associated to the action
  • Specifically, the number of fixed points equals the value of this character at $g$
• Applying Burnside's lemma to the conjugation action of a group on itself yields a formula for the number of conjugacy classes
  • The number of conjugacy classes equals the average value of an irreducible character over the group

Counting solutions to equations in groups

• Character-theoretic methods can be used to count the number of solutions to certain equations in finite groups
  • For example, the number of solutions to the equation $x^n = 1$ in a group $G$ equals the sum of the values of the irreducible characters of $G$ at an element of order $n$
• Similar techniques can be applied to count the number of solutions to other equations, such as $x^2 = 1$ or $xy = yx$
  • These methods often involve expressing the number of solutions in terms of character sums and then using orthogonality relations to simplify the expressions

Conjugacy classes and normal subgroups using characters

Determining conjugacy classes from character tables

• Two elements of a group are conjugate if and only if they have the same character values for all irreducible characters
  • Thus, the conjugacy classes can be determined from the character table
• The kernel of a character $\chi$ is the set of group elements $g$ for which $\chi(g) = \chi(1)$
  • This is a normal subgroup of the group
• If $\chi$ is a faithful character (i.e., its kernel is trivial), then the center of the group consists precisely of the elements $g$ for which $|\chi(g)| = \chi(1)$

Characterizing normal subgroups using characters

• A subgroup $H$ of a group $G$ is normal if and only if every irreducible character of $G$ restricts to a sum of irreducible characters of $H$
  • This criterion can be used to test for normality using the character tables of $G$ and $H$
• The commutator subgroup of a group (i.e., the subgroup generated by all commutators) is the intersection of the kernels of all linear characters (i.e., one-dimensional characters) of the group
  • This characterization can be used to compute the commutator subgroup from the character table
• Characters can also be used to prove that certain subgroups are characteristic (i.e., invariant under all automorphisms of the group)
  • For example, the center and the commutator subgroup are always characteristic subgroups