History of the notion of irreducible representation I am looking for the earliest references where the study of irreducible representations appears. There has been many articles and books on the history of representation theory. A fundamental feature of this theory, is that in good situations where one is dealing with a semisimple category, one can decompose objects into simple ones, or here, irreducible representations. My understanding is that the introduction of this circle of ideas is usually credited to Frobenius around the end of the 19th century.
However, the decomposition of tensor products of irreducible representations of $SL_2$ can be found in the article by Paul Gordan "Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist." in J. reine angew. Math. 323 (1868), 323-354.
It is written in old fashioned language that can be difficult to decipher, but the Clebsch-Gordan decomposition for $SL_2$ is basically there in Section 2 of that article. One could also ask: when was it realized that it was important and very useful to decompose general representations in terms of irreducibles? Reading the proof in that article, one can only conclude that Gordan was very well aware of that.
Also note that for the irreducible representations for $SL_n$, one can find a description already in the article by Alfred Clebsch "Ueber eine Fundamenthalaufgabe der Invariantentheorie", in
Abhandlungen der Königlichen Gesellschaft der Wissenschaften in Göttingen 17 (1872), 3-62, and its shorter follow-up "Ueber eine Fundamentalaufgabe der Invariantentheorie", Math. Ann. 5 (1872), 427-434. 
Are there earlier references about irreducible representations?
 A: I convert my comments to an answer per Abdelmalek’s request:


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*Dieudonné attributes the classification of irreducible $sl_2$-modules to Cayley (1856).

*Also the theory of spherical and cylindrical harmonics should qualify as prehistory — told in e.g. Heine (1878, pp. 1–10), Burkhardt (1902–1903, Chap. V).

*The words “irreducible” and “degree” hint at another root: if $G$ is finite, decomposing its regular representation $L$ on $\mathbf C[G]$ (elements $x=\smash{\sum x_g\delta^g}$, product $\smash{\delta^g\cdot\delta^h}=\smash{\delta^{gh}}$, $L(x)y=x\cdot y$) amounts to factoring the “group determinant” $\det(L(x))$ into irreducible polynomials in the $x_g$.
(“When was it realized that it was important and very useful to decompose general representations in terms of irreducibles?” inadvertently evokes a whole other question involving the origin of Fourier analysis, going back to at least D. Bernoulli, not to mention celestial epicycles, Pythagorean ideas on musical harmony; or Lang’s Algebra’s casting of Jordan form as “Representation Theory of One Endomorphism” (or the monoid algebra $k[\mathrm X]$ of $(\mathbf N,+)$, with irreducibles etc.) — but I understand the intent was to keep it non-commutative.) 
