Correspondence among congruence families for generalized Frobenius partitions via modular permutations
Rong Chen, Xiao-Jie Zhu
TL;DR
This work develops a unified, representation-theoretic framework for congruences of generalized Frobenius partitions by constructing vector-valued modular forms $f_{k,\beta}$ linked to the generating functions $C\Psi_{k,\beta}$. It establishes explicit vector-valued SL$_2(\mathbb{Z})$ transformation laws and a Γ$_0(k)$-permutation mechanism that relates $\beta$-families, and it shows how linear combinations of modular transformations can bridge different congruence classes. The approach yields concrete Ramanujan-type congruences, notably for $k=3$, by exploiting $U_p$-sequences and modular-function identities, and it provides a pathway to extend Garvan–Sellers–Smoot results to general $k$ and $\beta$. The results highlight a deep interplay between vector-valued modular forms, Weil representations, and Jacobi-theoretic decompositions, with potential broad impact on producing new congruences and understanding the structure of partition-related generating functions.
Abstract
In 2024, Garvan, Sellers and Smoot discovered a remarkable symmetry in the families of congruences for generalized Frobenius partitions $cψ_{2,0}$ and $cψ_{2,1}$. They also emphasized that the considerations for the general case of $cψ_{k,β}$ are important for future work. In this paper, for each $k$ we construct a vector-valued modular form for the generating functions of $cψ_{k,β}$, and determine an equivalence relation among all $β$. Within each equivalence class, we can identify modular transformations relating the congruences of one $cψ_{k,β}$ to that of another $cψ_{k,β'}$. Furthermore, correspondences between different equivalence classes can also be obtained through linear combinations of modular transformations. As an example, with the aid of these correspondences, we prove a family of congruences of $cφ_{3}$, the Andrews' $3$-colored Frobenius partition.