MacMahon's Master Theorem, Double Tableau Polynomials, and Representations of Groups

Abstract

It is shown that MacMahon's master theorem gives the diagonal elements of a class of irreducible representations of the general linear group,Gl(n,C), and hence the trace of these representations, or the group characters. These representations are unitary under restriction to the unitary subgroup and constitute the so-called totally symmetric representations. A generalization of MacMahon's master theorem that generates the elements of the representation matrix themselves is given. These relations for groups are, in fact, more general in that they apply to arbitrary matrices of indeterminates. These results are proved by using the properties of a class of Young–Weyl tableau polynomials defined overn2indeterminates. Forn=2, two sets of basic polynomials are defined in terms of Young–Weyl tableaux. The first set corresponds to the unitary irreducible representations of U(2), as defined in the physics literature and whose properties comprise a substantive part of what is known as the quantum theory of angular momentum; the second set is Rota's double standard tableau polynomials. These two sets of polynomials span the same space, and a comprehensive set of relations between them is given. A further generalization of MacMahon's master theorem pertaining to the general polynomial representations ofGL(n,C), as enumerated by partitions, is conjectured and is shown to contain the classical Schur function generating function as a special case.