History and variation on the theme of the frobenius reciprocity theorem

Abstract

The German mathematician Georg Ferdinand Frobenius (1849-1917) was born in Berlin on 26 October 1849, the son of Christian Ferdinand Frobenius. After at tending the Joachimthal Gymnasium, Frobenius began his mathematical studies at Gottingen in 1867. Under the direction of the inimitable Karl Weierstrass he earned a doctorate three years later in Berlin. In ensuing years, Frobenius made fundamental discoveries in such diverse areas as matrix theory mod p and 0-functions in several variables. His particularly productive work during the 1870-75 period is chronicled in Crelle's Journal. The major achievements of Frobenius are undoubtedly in the theory of group representat ions--a theory he initiated in 1896 at the age of 47. Along with its three historic roots--Galois theory, Lie theory, and number theory-i t stands as a cornerstone of modern mathematics. group, so we begin with the latter notion. It appears, implicitly, as early as 1801 in the work of Gauss--viz . , in his Dzsquisitiones Arithmetzcae--and m later years, beginning in 1878, Richard Dedekind also considered the notion (again implicitly). An extensive correspondence be tween Dedekind and Frobenius, spanning periods from 1882-83 and from 1895-98, provides valuable insight into the development of Frobenius's creation. A fascinating and learned account of this cor-