Introduction

Summary

Between the years 1896 and 1900, G. Frobenius [1896], [1900] invented and mapped out the theory of complex representations of finite groups, paying special attention to representations of the symmetric group Σ_r. Thus, given a finite group we want to determine all group homomorphisms from our group into GL_m(C) for arbitrary positive integers m. Around this time Frobenius suggested to his pupil Issai Schur that he might examine the representation theory of the infinite group Γ_c = GL_n(C). The subsequent investigations appeared in Schur's [1901] beautiful doctoral thesis of 1901. Schur studied Γ_c by means of the C-space A_c(n, r) of r-homogeneous polynomial functions in the n² coordinate functions on Γ_c. In particular he showed that the isomorphism types irreducible representations of CΓ_c with a given degree of homogeneity r are in one-to-one correspondence with the partitions of r into at most n parts; he also showed that the character of an irreducible module indexed by such a partition λ is a certain symmetric function, now called ‘Schur function’, s_λ. In order to exploit Frobenius' work on Σ_r he set up (in modern parlance) an equivalence between the category of polynomial representations and the module category of Σ_r.

In another paper, Schur [1927] to analysed the actions of both Σ_r and Γ_c on the rth tensor power of the natural module for Γ_c (the space of n-columns over C).