# Earliest known usage of letter gamma “Γ” for reducible representation in group theory

Does any know the earliest known usage of the Greek letter gamma for showing a reducible representation of a group? This symbolism is commonly used in character tables in chemical applications of group theory, for example as shown here. Thank you in advance. • Yes, some authors use an index i in Γ to show that it is irreducible. – M. Farooq Apr 9 '18 at 14:34
• I agree. The convention at least in the chemical literature is to use capital gamma for reducible and capital gamma with i as superscript or subscript as irreducible representation. – M. Farooq Apr 10 '18 at 14:06
• Citation needed. This excerpt only involves $\color{red}{\textrm{ir}}$reducibles (for which $\color{blue}i$ is an enumeration $\color{blue}{\textrm{i}}$ndex). – Francois Ziegler Apr 10 '18 at 21:39

## 1 Answer

Early examples are Burnside (1910, pp. 324-325; 1911, p. 271) where $\color{red}{\textrm{ir}}$reducible representations are called $\Gamma$, $\Gamma_1$, $\Gamma_2$, etc. (Earlier in (1901) he had called them $G_1$, $G_2$, etc.)

Speiser (1923, p. 104; 1927, p. 151) uses the same convention, but allows $\Gamma$ to be reducible.

Note that for them a representation is not a homomorphism to $\mathrm{GL}(n)$ (a map) but rather its image (a group). So your question morphs into: who first wrote $\Gamma$ for a group of linear substitutions? Now that goes back to at least Jordan (1870, pp. 221 sq).

• Thank you. I checked the unabridged Oxford dictionary, it also traces the earliest use of the term representation, by Burnside in 1897. "W. Burnside Theory Groups of Finite Order ii. 22 As long as we are dealing with the properties of a group per se, and not with properties which depend on the form of representation, the group may, if convenient, be replaced by any group which is simply isomorphic with it." Does anyone know if Frobenius papers on groups have been translated? I could not find them. Thanks. – M. Farooq Apr 9 '18 at 14:38
• @M.Farooq Frobenius introduced the German word, Darstellung, nearly simultaneously (18 November 1897). I don’t think he’s been translated, but Curtis (1999) has a good exposition. Anyway, right: almost as soon as someone was in a position to call a representation $\Gamma$, they did. But it was really a group — hence, probably, the letter. – Francois Ziegler Apr 11 '18 at 17:04
• @ Francois, thank you once again. I am attaching picture from a 1933 paper by Tisza (Zeitschrift fur Physik). It seems like that Γ was not that universal symbol for representation. He uses D instead of gamma for Darstellung. – M. Farooq Apr 11 '18 at 20:28
• Here is one example of using D instead of Γ in the early literature Tisza (vide supra). – M. Farooq Apr 11 '18 at 20:30