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 
 $$\mathcal {T}$$
 consisting of symmetric polynomials in the part sizes and multiplicities.

/pdf/a-symmetric-bloch-okounkov-theorem-286wbjcm56.pdf

A symmetric Bloch-Okounkov theorem

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.

/pdf/a-symmetric-bloch-okounkov-theorem-1c4672tmnz.pdf

We identify a class of “semi-modular” forms invariant on special subgroups of $$GL_2({\mathbb {Z}})$$
 , which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.

Partition Eisenstein series and semi-modular forms

There are many families of functions on partitions, such as the shifted symmetric functions, for which the corresponding q-brackets are quasimodular forms. We extend these families so that the corresponding q-brackets are quasimodular for a congruence subgroup. Moreover, we find subspaces of these families for which the q-bracket is a modular form. These results follow from the properties of Taylor coefficients of strictly meromorphic quasi-Jacobi forms around rational lattice points.

The Bloch-Okounkov theorem for congruence subgroups and Taylor coefficients of quasi-Jacobi forms.

. We study the area Siegel–Veech constants of components of strata of abelian diﬀerentials with even or odd spin parity. We prove that these con- stants may be computed using either: (I) quasimodular forms, or (II) intersection theory. These results reﬁne the main theorems of [CMZ18] and [CMSZ20] which described the area Siegel–Veech constants of the full strata. Along the proof of (II), we establish a new identity for Siegel–Veech constants of cylin- ders.

pdf/cylinder-counts-and-spin-refinement-of-area-siegel-veech-2lkadtt4.pdf

Cylinder counts and spin refinement of area Siegel-Veech constants

We obtain a condition describing when the quasimodular forms given by the Bloch–Okounkov theorem as q-brackets of certain functions on partitions are actually modular. This condition involves the kernel of an operator 
$$\Delta $$

. We describe an explicit basis for this kernel, which is very similar to the space of classical harmonic polynomials.

/pdf/when-is-the-bloch-okounkov-q-bracket-modular-qh5wwhec7n.pdf

When is the Bloch–Okounkov q-bracket modular?

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A symmetric Bloch-Okounkov theorem

When is the Bloch–Okounkov q-bracket modular?