The partition function and Hecke operators

The theory of congruences for the partition function p(n) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P(z)|T(ℓ2), where P(z) is the relevant modular generating function. We obtain such formulas using Eulerʼs Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujanʼs Delta-function.