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Plethysm of Schur Functions and Irreducible Polynomial Representations of the Complex General Linear Group

We show the intimate connection Schur functions have with the irreducible polynomial representations of GL(n,C). Following Schur’s thesis, in Section 1, we prove that the characters of these representations are given by Schur functions. In the second section, we elaborate on how a specific combinatorial operation on symmetric functions called plethysm gives the characters of the composition of irreducible representations. There is no general combinatorial description such as the Littlewood-Richardson rule, which is used to multiply Schur functions, that is known for the plethysm of Schur functions. There are, however, numerous algorithms and a few combinatorial formulas for special cases. In hopes of inspiring future work, we give a survey of the most notable results.