Character theory

Character theory studies representations of groups through their characters: functions that assign each group element the trace of its representing operator.

Group representation: A homomorphism from a group to the invertible linear transformations of a vector space. Character theory extracts trace-valued functions from these homomorphisms.
Character table: A table listing irreducible character values against the conjugacy classes of a finite group. It organizes the principal data character theory computes for a finite group.
Frobenius character formula: A formula expressing irreducible characters of symmetric groups using partitions and cycle data. It makes character calculations for symmetric groups explicit.
Ferdinand Georg Frobenius: A German mathematician who developed foundational methods in group representation theory. His work established character theory as a systematic study of finite groups.
Matrix coefficient: A scalar-valued function on a group obtained by pairing a vector with its image under a representation. Unlike a character, a matrix coefficient can retain information beyond the operator's trace.
Group character: The function assigning each group element the trace of its representing linear operator. This trace function is the central object studied by character theory.
Orthogonality relations for characters: Identities expressing orthogonality among irreducible characters and among rows or columns of character tables. They enable character tables to identify and separate irreducible representations.
Murnaghan–Nakayama rule: A recursive rule for computing symmetric-group character values from cycle types and partitions. It provides an efficient combinatorial method for entries in symmetric-group character tables.
Richard Dedekind: A German mathematician whose work shaped modern algebra and number theory. His investigations of group determinants helped motivate Frobenius's character methods.
Representation equivalence: A relation identifying representations connected by an invertible intertwining linear map. For finite groups over suitable fields, characters can test whether representations are equivalent.

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