A symmetric Bloch–Okounkov theorem

Abstract

The algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

Quantum KdV hierarchy and quasimodular forms

“…The proof is given in Section 2 and builds on the previous work [18] of the first author. In the special case c = 0, the theorem states that {g(0)} q is quasimodular of homogeneous weight if B −1 g is homogeneous.…”