The origins of the theory of group characters

Relations between polynomial roots

“…The following lemma provides a natural generalization of the well-known computation of the cyclic determinant and was originally due to Dedekind (see [4]). The computation of the rank is an analogue of a theorem by A. Schinzel concerning the rank of a cyclic matrix ( [3]).…”

The concept of “character” in Dirichlet’s theorem on primes in an arithmetic progression

“…Similarly, abstracting proofs of properties of characters on (Z/mZ) * that rely on features specific to the integers modulo m paves the way to extending these properties to group characters more generally. Characters and their properties form the basis for representation theory, which has been an essential part of group theory since the turn of the twentieth century [48,60]. Authors like Hadamard, de la Vallée-Poussin, and Landau were also interested in extending Dirichlet's methods to other kinds of Dirichlet series, which now play a core role in analytic number theory.…”

“…These facts are now quite familiar in constructive and computable analysis. 48 Kronecker's treatment of characters provides an interesting combination of Dirichlet's approach and modern ones. Fixing a modulus m, Kronecker, like Dirichlet, provided a fully explicit description of the characters modulo m in terms of primitive elements of the powers of primes giving m and primitive roots of unity.…”

Frobenius, Schur, and the Berlin Algebraic Tradition

“…(9) Frobenius's most famous mathematical achievement is to be found in the theory of group characters (for the history of representation theory see [32], [33], and [37]). In 1896 he initiated representation theory of finite groups (over the field C of complex numbers) and established what we now conceive as its most basic vocabulary and results [17], [18]' [19].…”