For prime levels 5 ≤ p ≤ 19, sets of Γ 0 (p)-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for Γ 1 (p). An explicit formulation of the permutation representation and several applications are given, including a new representation for the number of t-core partitions. The Γ 0 (p)-action induces coefficient symmetries within representations for modular forms and invariance subgroups for coupled systems of differential equations. The symmetry for levels p = 5, 7, 11 is linked to the Kleinian automorphism groups.
Abstract. We demonstrate that quotients of septic theta functions appearing in S. Ramanujan's Notebooks and in F. Klein's work satisfy a new coupled system of nonlinear differential equations with interesting symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends a technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein's quartic relation induces new symmetric representations for low weight Eisenstein series in terms of weight one modular forms of level seven.
We introduce the notion of "finite general representation type" for a finitedimensional algebra, a property related to the "dense orbit property" introduced by Chindris-Kinser-Weyman. We use an interplay of geometric, combinatorial, and algebraic methods to produce a family of algebras of wild representation type but finite general representation type. For completeness, we also give a short proof that the only local algebras of finite general representation type are already of finite representation type. We end with a Brauer-Thrall style conjecture for general representations of algebras.
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