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The modern definition of group representation is a homomorphism between a group $G$ and the group $GL(V, K)$ of some vector space over the field $K$.

But as far as I know, when Frobenius developed his character theory, he did not have this picture in his mind.

So, who invented (or discovered) this definition?

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It is true that Frobenius did not have the notion of a group representation in his celebrated "trilogy" of 1896 papers on factorization of the group determinant problem, posed to him by Dedekind in their correspondence the same year. Instead, he defined characters by working with a commutative complex algebra (which he later identified as the center of the group algebra), and taking traces of some projection maps. These are the papers where he proved the orthogonality relations and related characters to the structure constants, among other things. Some inspiration came from the work of Weierstrass, Dedekind, and Study on commutative hypercomplex systems.

Frobenius did however give a more or less modern definition of a group representation shortly thereafter, also on Dedekind's suggestion:"Let $H$ be an abstract group, $A, B, C\dots$ be its elements. One associates to the element $A$ the matrix $(A)$, to the element $B$ the matrix $(B)$, etc., in such away that the group $\mathfrak{H}'$ is isomorphic to the group $\mathfrak{H}$, that is, $(A)(B)=(AB)$. Then I say that the substitutions or the matrices $(A),(B),(C)\dots$ represent the group $\mathfrak{H}$" ("isomorphic" did not yet mean one-to-one, so it corresponds to modern "homomorphic"). He then pointed out that the characters are given by the traces of matrices in "primitive" representations, defined as those with irreducible determinants.

Within two years of his initial trilogy Frobenius introduced the “composition” (now tensor product) of characters, established relations between the characters of a group and its subgroups, introduced induced representations, and proved the reciprocity law for them. He did acknowledge as inspiration two works of Molien (1893, 1897), who analyzed group algebras as hypercomplex systems. Molien anticipated methods of Maschke, Wedderburn, and Noether in modern group representation theory.

Frobenius's work is described in detail in Lam's Representations of Finite Groups: a Hundred Years in AMS Notices, which references the original papers.

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    $\begingroup$ See also "The mathematics of Frobenius in context" by Hawkins. $\endgroup$ – user2255 Dec 23 '15 at 5:51

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