Laurence Barker

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Certain solutions to Harpers equation are discrete analogues of (and approximations to) the Hermite-Gaussian functions. They are the energy eigenfunctions of a discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the discrete harmonic oscillator.

The discrete fractional Fourier transform and Harper's equation

It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.

Continuum quantum systems as limits of discrete quantum systems. III. Operators

Convergence of a “discrete” operator to a “continuum” operator is defined. As examples, the circular rotor, the one-dimensional box, the harmonic oscillator, and the fractional Fourier transform are realized as limits of finite-dimensional quantum systems. Limits, thus defined, preserve algebraic structure. The results prepare for a sequel in which some affine canonical transforms will be “discretized.”

Tornehave Morphisms I: Resurrecting the Virtual Permutation Sets Annihilated by Linearization

Tom Dieck introduced a commutative triangle whereby the exponential morphism from the Burnside functor to the unit functor is factorized through the real representation functor. Tornehave introduced a p-adic variant of the exponential morphism. His construction involves real representations that are not well defined up to isomorphism. To obtain a well defined commutative triangle, we introduce the orientation functor, a quotient of the real representation functor.

Continuum quantum systems as limits of discrete quantum systems. IV. Affine canonical transforms

Affine canonical transforms, complex-order Fourier transforms, and their associated coherent states appear in two scenarios: finite-discrete and continuum. We examine the relationship between the two scenarios, making systematic use of inductive limits, which were developed in the preceding articles in this series.

A General Approach to Green Functors Using Bisets

We introduce a biset-theoretic notion of a Green functor which accommodates the functorial and ring-theoretic structural features of the modular character functor. For any Green functor A in that sense, we introduce an algebra ΛA. Any ΛA-module has the same kind of functorial structure as A and is also a module for the algebra , where G runs over the underlying family of finite groups. In a globally defined scenario and also in a scenario localized to the subquotients of a fixed finite group, we take A to be the modular character functor, and we classify the simple ΛA-modules.

THE DIMENSION OF A PRIMITIVE INTERIOR G-ALGEBRA

We give the residue class, modulo a certain power of p, for the dimension of a primitive interior G-algebra in terms of the dimension of the source algebra. To illustrate, we improve a theorem of Brauer on the dimension of a block algebra. Almost always, the G-algebras arising in group representation theory have been interior. Both in applications and in the general theory, it often suAces to consider primitive interior G-algebras. One of the themes of the theory is the characterisation of a primitive interior G-algebra in terms of its source algebra S. Stories revolving around this theme are told in the two books devoted to G-algebra theory, namely Kulshammer (8), Thevenaz (15) and in the papers listed in their bibliographies. We mention particularly Puig (11), (12). These stories focus on rich algebraic relation- ships between A and S; for a start, (11, 3.5) tells us that A and S are Morita equivalent. However, many outstanding conjectures, some old and some new, hark back to Brauers more arithmetical approach to group representation theory. See, for instance, conjectures in Alperin (1), Dade (4), Feit (6, Section 4.6) and Robinson (13). In this note, we point out an arithmetical relationship between A and S.A s an illustration, we shall discuss a theorem of Knorr on the dimension of a simply defective module, and shall improve a theorem of Brauer on the dimension of a block algebra. See also Ellers (5). Our notation is as in Thevenaz (15); we repeat a little of it to set the scene, and extend it slightly. LetO be a complete local noetherian ring with an algebraically closed residue field k of prime characteristic p. Let G be a finite group, and let A be an interior G-algebra; as usual, we assume that A is finitely generated overO, and either free over O or annihilated by J(O). Given a pointed group H on A ,w e choose an element j2, and define A :a jAj as an interior H-algebra. Now let X be an A-module; again we assume that X is finitely generated overO, and either free overO or annihilated by J(O). We define X :a jX as an A-module. It is easy to extend the use of embeddings in Puig (12, 2.13.1) to show that X is unique up to a natural isomorphism of A-modules. Henceforth, let us assume that A is primitive. Let Pg be a defect pointed group on A. The source algebra A associated with Pg is an interior P-algebra. The multi- plicity module V() associated with Pg is a projective indecomposable k ^ NOPU- module. By the construction of V(), if 1Aa P t2T t as a sum of mutually orthogo-

Local representation theory and möbius inversion

Various representation-theoretic parameters of a finite group are shown to satisfy formulas similar to formulas appearing in reformulations by Kolshammer, Robinson, and Thevenaz of Alperins Conjecture. The conjecture itself is reformulated again, now as a statement not mentioning characters or conjugacy classes.

MÖBIUS INVERSION AND THE LEFSCHETZ INVARIANTS OF SOME p-SUBGROUP COMPLEXES

ABSTRACT: By generalising Mobius inversion to finite acyclic directed multigraphs, we show that the coefficients for the reduced Lefschetz invariant of a suitable G-poset of p-subgroups are, essentially, values of a Mobius function. This allows us to present the Knorr–Robinson reformulation of Alperins conjecture as a case of Mobius inversion.