Browsing by Author "Barker, L."
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Item Open Access Alperin's fusion theorem and G-posets(Walter de Gruyter GmbH, 1998) Barker, L.Some G-posets comprising Brauer pairs or local pointed groups belong to a class of G-posets which satisfy a version of Alperin's fusion theorem, and as a consequence, have simply connected orbit spaces.Item Open Access Blocks of Mackey categories(Elsevier, 2016) Barker, L.For a suitable small category F of homomorphisms between finite groups, we introduce two subcategories of the biset category, namely, the deflation Mackey category MF← and the inflation Mackey category MF→. Let G be the subcategory of F consisting of the injective homomorphisms. We shall show that, for a field K of characteristic zero, the K-linear category KMG=KMG←=KMG→ has a semisimplicity property and, in particular, every block of KMG owns a unique simple functor up to isomorphism. On the other hand, we shall show that, when F is equivalent to the category of finite groups, the K-linear categories KMF← and KMF→ each have a unique block. © 2015 Elsevier Inc.Item Open Access Continuum quantum systems as limits of discrete quantum systems, I: State vectors(Academic Press, 2001) Barker, L.Dynamical systems on "continuum" Hilbert spaces may be realized as limits of dynamical systems on "discrete" (possibly finite-dimensional) Hilbert spaces. In this first of four papers on the topic, the "continuum" and "discrete" spaces are interfaced to one another algebraically, convergence of vectors is defined in such a way as to preserve inner products, and a necessary and sufficient coordinate-wise criterion for convergence is proved. © 2001 Academic Press.Item Open Access Continuum quantum systems as limits of discrete quantum systems. III. Operators(A I P Publishing LLC, 2001) Barker, L.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." © 2001 American Institute of Physics.Item Open Access Continuum quantum systems as limits of discrete quantum systems. IV. Affine canonical transforms(American Institute of Physics, 2003) Barker, L.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. © 2003 American Institute of Physics.Item Open Access Continuum quantum systems as limits of discrete quantum systems: II. State functions(Institute of Physics Publishing Ltd., 2001) Barker, L.In this second of four papers on the eponymous topic, pointwise convergence of a 'discrete' state function to a 'continuum' state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel. As examples (and as a prerequisite for the sequels), the normal approximation theorem and the convergence of the Kravchuk functions to the Hermite-Gaussians are expressed in terms of the algebraic notion of convergence.Item Open Access Counting positive defect irreducible characters of a finite group(University of Auckland, Department of Mathematics, 1998) Barker, L.Let z+ (G) be the number of ordinary irreducible characters of a finite group G which have positive defect with respect to a prime p. We express z+(G) as the p- adic limit of a sequence of enumerative parameters of G and p. When p = 2, and under a suitable hypothesis on the Sylow 2- subgroups of G, we give two local characterisations of the parity of z+(G), one of them compatible with Alperin’s Weight Conjecture, the other apparently independent.Item Open Access Defects of irreducible characters of p-Soluble groups(Academic Press, 1998) Barker, L.We prove a refinement of thep-soluble case of Robinson's conjectural local characterization of the defect of an irreducible character. © 1998 Academic Press.Item Open Access The dimension of a primitive interior G-Algebra(Cambridge University Press, 1999) Barker, L.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. © Glasgow Mathematical Journal Trust 1999.Item Open Access The discrete fractional Fourier transform and Harper's equation(London Mathematical Society, 2000) Barker, L.It is shown that the discrete fractional Fourier transform recovers the continuum fractional Fourier transform via a limiting process whereby inner products are preserved.Item Open Access The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform(Institute of Physics Publishing, 2000) Barker, L.; Candan, C.; Hakioğlu, T.; Kutay, M. A.; Özaktaş, Haldun M.Certain solutions to Harper's 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.Item Open Access Fibred permutation sets and the idempotents and units of monomial Burnside rings(Elsevier, 2004-11-15) Barker, L.We study the units of monomial Burnside rings and the idempotents of monomial Burnside algebras. Introducing a tenduction map, we realise the unit group and the torsion unit group as a Mackey functor. © 2004 Elsevier Inc. All rights reserved.Item Open Access A general approach to green functors using bisets(Taylor & Francis, 2016) Barker, L.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 ⊕G A(G) , 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. © 2016, Copyright © Taylor & Francis Group, LLC.Item Open Access Genotypes of irreducible representations of finite p-groups(Academic Press, 2006-12-15) Barker, L.For any characteristic zero coefficient field, an irreducible representation of a finite p-group can be assigned a Roquette p-group, called the genotype. This has already been done by Bouc and Kronstein in the special cases Q and C. A genetic invariant of an irrep is invariant under group isomorphism, change of coefficient field, Galois conjugation, and under suitable inductions from subquotients. It turns out that the genetic invariants are precisely the invariants of the genotype. We shall examine relationships between some genetic invariants and the genotype. As an application, we shall count Galois conjugacy classes of certain kinds of irreps. © 2006 Elsevier Inc. All rights reserved.Item Open Access Local representation theory and Möbius inversion(Taylor & Francis Inc., 1999) Barker, L.Various representation-theoretic parameters of a finite group are shown to satisfy formulas similar to formulas appearing in reformulations by Külshammer, Robinson, and Thévenaz of Alperin's Conjecture. The conjecture itself is reformulated again, now as a statement not mentioning characters or conjugacy classes.Item Open Access A new notion of rank for finite supersolvable groups and free linear actions on products of spheres(Walter de Gruyter GmbH, 2003) Barker, L.; Yalçın, E.For a finite supersolvable group G, we define the saw rank of G to be the minimum number of sections Gk/Gk-1 of a cyclic normal series G* such that Gk -Gk-1 contains an element of prime order. The axe rank of G, studied by Ray [10], is the minimum number of spheres in a product of spheres admitting a free linear action of G. Extending a question of Ray, we conjecture that the two ranks are equal. We prove the conjecture in some special cases, including that where the axe rank is 1 or 2. We also discuss some relations between our conjecture and some questions about Bieberbach groups and free actions on tori.Item Open Access The number of blocks with a given defect group(London Mathematical Society, 1997) Barker, L.Given a P-subgroup P of a finite group G, we express the number of P-blocks of G with defect group P as the /?-rank of a symmetric integer matrix indexed by the N (P) / P-conjugacy classes in PC(P)/P. We obtain a combinatorial criterion for P to be a defect group in G.Item Open Access On contractibility of th orbit space of a G-poset of Brauer pairs(Elsevier, 1999-02-15) Barker, L.Given ap-blockbof a finite groupG, we show that theG-poset of Brauer pairs strictly containing (1,b) has contractibleG-orbit space. A similar result is proved for certainG-posets ofp-subgroups. Both results generalise P. Symonds' verification of a conjecture of P. Webb. © 1999 Academic Press.Item Open Access On p-soluble groups and the number of simple modules associated with a given Brauer pair(Oxford University Press, 1997) Barker, L.Item Open Access Real representation spheres and the real monomial Burnside ring(Elsevior, 2012) Barker, L.; Tuvay, İ.We introduce a restriction morphism, called the Boltje morphism, from a given ordinary representation functor to a given monomial Burnside functor. In the case of a sufficiently large fibre group, this is Robert Boltje's splitting of the linearization morphism. By considering a monomial Lefschetz invariant associated with real representation spheres, we show that, in the case of the real representation ring and the fibre group {±1}, the image of a modulo 2 reduction of the Boltje morphism is contained in a group of units associated with the idempotents of the 2-local Burnside ring. We deduce a relation on the dimensions of the subgroup-fixed subspaces of a real representation. © 2011 Elsevier Inc.