Generalized partitions and eta functions

Question

Let $\sigma$ be an element of $SL_{24}(\mathbb{Z})$ with its Jordan normal form is diagonal and the eigen values are $\epsilon_j$ for $1 \le j \le 24$ are n th root of unity where $n|N$ and $N$ is the finite order of $\sigma$. Equivalently we are describing $\sigma$ through its cycle shape $(a_1)^{b_1}\cdots(a_s)^{b_s}$.

We associate $\sigma$ to the following modular form:

$$\eta_{\sigma}(q) := \eta(\epsilon_1q)\cdots\eta (\epsilon_{24} q)=\eta(q^{a_1})^{b_1}\cdots \eta(q^{a_s})^{b_s}$$

Here $\eta$ stands for the Dedekind eta-function. Using the above defined $\eta_{\sigma}$ we define: $$\sum_{j>0}p_{\sigma}(1+j)q^{1+j} = \frac{q}{\eta_{\sigma}(q)}$$ This is a generalized partition function.

We assume the cycle type of $\sigma$ is $1^{1}23^{1}$ and hence $N = 23$. In this case I have the following questions :

1. What is this generalized partition function $p_{\sigma}$ and how to find $p_{\sigma}(n)$ for some natural number $n$?

2. There are many generalisations of partitions functions and hence what is the reference for this particular type of generalized partition function?

Thanks for your time.

Have a good day.

Answer 1

What you have is positive integers such that $\;a_1b_1+\dots+a_sb_s=24.$ Your $q/\eta_\sigma(q)$ is an example of an eta-quotient and is a modular function of negative weight. As just one example, if $\;a_1=1,b_1=24\;$ then $\eta_\sigma(q)=\Delta(q)$ is the generating function of the Ramanujan tau function. You may find some similar kinds of partitions in the OEIS. For example, sequence A005758 is partitions into parts of 12 kinds. For your first question, just expand $\;\eta(q^a)=\prod_{n>0}1-q^{an}\;$ and multiply the $q$-series together. For example, in your case where $b_1=b_2=a_1=1, a_2=23,$ $$f(q):=\frac1{\prod_{k>0}(1-q^k)(1-q^{23k})}=1+q+2q^2+3q^3+5q^4+\dots$$ which is the generating function of partitions of $n$ into positive integers where the integers come in two kinds and the second kind has weight $23$ times the weight of the first kind.

Answer 2

It is not an answer, but just some comments where such $\eta$ products appear:

• A version of Moonshine for the Mathieu group M24 proposed by G. Mason around 1985, the coefficients $a_i, b_i$ are related with the cycle structre of M24 elements (see here), in particular (1,23) appears there. The result is that such products are Hecke eigen-forms

• Related theorem by D. Dummit, H. Kisilevsky, and J. McKay, Multiplicative products of η functions, Finite Groups - Coming of Age, American Mathematical Society, 1985 which describes all such multiplicative products - (most come form M24). Nice introduction to that and previous results can be found in Jeremy Booher The Spirit of Moonshine: Connections between the Mathieu Groups and Modular Forms

• McKay conjecture proved by Koike 1984 (ON McKAY'S CONJECTURE) gives conditions on $a_i, b_i$ when such $\eta$-product is primitive cusp form, again is quite related to the above mentioned results

• S. Galkin found (see here) that Mason's $\eta$-products are related with Gromov-Witten invariants of certain Fano manifolds

• K.Saito considered $\eta$-products in his papers e.g. Extended Affine Root System V (Elliptic Eta-product and their Dirichlet series). Some of his conjectures has been proved in arXiv:math/0702027. See also MO270241 Are the Fourier coefficients of η(qm)m/η(q)η(qm)m/η(q) non-negative?

• MO33058: Eta-products and modular elliptic curves where elliptic curves whose modular are such $\eta$-products are described. See also MO238646 On η(6z)η(18z)η(6z)η(18z) and the splitting / modularity of x3−2

• MO91672 Linear eta product identities - how many are there?, MO92119 Duality of eta product identities: a new idea?