A symmetric Bloch-Okounkov theorem

TL;DR

This paper introduces a new quasimodular algebra of symmetric polynomials on partitions, extending the algebraic framework related to the Bloch-Okounkov theorem and quasimodular forms.

Contribution

It constructs a novel quasimodular algebra of symmetric polynomials, broadening the understanding of algebraic structures associated with partitions and quasimodular forms.

Findings

The algebra of symmetric polynomials in part sizes and multiplicities is quasimodular.

This algebra generalizes the Bloch-Okounkov theorem.

The $q$-bracket of elements in this algebra yields quasimodular forms.

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 consisting of symmetric polynomials in the part sizes and multiplicities.