Klara Zimmerman: MacMahons teorem

Abstract:

In this essay we explore symmetric plane partitions, which can be visualized as three dimensional arrangements of unit cubes that satisfy certain symmetry conditions. I have studied David M. Bressoud’s book Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, and focused on the parts leading up to MacMahon’s theorem. Bressoud takes the reader through important combinatorial theorems and their proofs, such as the Vandermonde determinant theorem, the Weyl denominator formula and the Jacobi–Trudi Identity.

I will present these theorems step by step, focusing on details and illustrating through basic examples. In order to understand the theorems presented, fundamental combinatorial concepts such as inversion numbers, semi standard Young tableaux and Schur functions are introduced. Finally, I will use these concepts and theorems to prove Percy MacMahon’s generating function for symmetric plane partitions.