Issai Schur determined the polynomial replesentations of the complex general linear group GLn(¢) in his doctoral dissertation [S], published in 1901. This remarkable work contained many very original ideas, developed with superb algebraic skill. Schur showed that these representations are completely reducible, that each irreducible one is "homogeneous" of some degree r ~ 0 (see 2.2), and that the equivalence types of irreducible polynomial representations of GLn(~) , of fixed homogeneous degree r, are in one-one correspondence with the partitions k = (~i'''" kn) of r into not more than n parts. Moreover Schur showed that the character of an irreducible representation of type k is given by a certain symmetric function ~I in n variables (since described as a "Schur function"; see 3.5). An essential part of Schur's technique was to set up a correspondence between representations of GL (~) of fixed homogeneous degree r, n and representations of the finite symmetric group G(r) on r symbols, and through this correspondence to apply C. Frobenius's discovery of the characters of G(r) [F, 1900].

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