ABSTRACT
This chapter introduces the analytical characters of a finite quasigroup Q with element e, as almost-periodic functions on the stabilizer of e in the universal multiplication group U(Q;Q) of Q. Although the finite-dimensional complex representations of a finite group are determined up to equivalence by its ordinary characters, the corresponding combinatorial characters of Q, as treated in Chapters 6 and 7, are inadequate for the task of classifying all the so-called ordinary Q-modules, the finite-dimensional complex vector spaces in the slice category Q/Q. As shown by Theorem 12.4, this classification is achieved by the analytical characters. The chapter is organized around various spaces of complex-valued functions.
Section 12.1 looks at the space L!(Q) of functions f : Q→ C. The combinatorial character theory of Q defines so-called “generalized Laplace operators” on this space. If Q has s conjugacy classes and basic characters, then it has s generalized Laplace operators ∆1, . . . ,∆s. The ordinary representation theory of Q furnishes coefficient functions fm in L1(Q) for each element m of M = p−1{e} in a Q-module p : E → Q. One thus studies the behavior of the coefficient functions under the generalized Laplace operators. The intimate connection between modules and characters in the group case is interpreted in this theory as Theorem 12.1. The limitations of the approach are readily seen in the ordinary representation theory of the singleton quasigroup. There the unique generalized Laplace operator ∆1 on the 1-dimensional space L1({e}) cannot hope to classify the rich supply of modules. As an intermediate step towards the analytical character theory which does
classify modules, Section 12.2 looks at periodic functions on groups. These are complex-valued functions that remain invariant under translation by a subgroup of finite index. The space L1(Q) is embedded into the space P (G˜) of periodic functions on the universal multiplication group G˜ = U(Q;Q) of Q, and the generalized Laplace operators are extended from L1(Q) to P (G˜). The Laplace operator ∆1 is related to the Laplace operator used in harmonic analysis on free groups (Theorem 12.2). The actual analytical character theory for classifying modules is given in
Section 12.3, in terms of almost-periodic functions on the stabilizer G˜e of e in the universal multiplication group G˜. Almost-periodic functions on the universal stabilizer G˜e correspond to continuous complex-valued functions on a
