Functor morphing and representations of automorphism groups of modules

ABSTRACTThe unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is a commutative variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in the case where the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids.

Symplectic graphs over finite commutative rings

Monoidal indeterminates and categories of possible worlds

We introduce the notion of <em>strong concatenable process</em> for Petri nets as the least refinement of non-sequential (concatenable) processes which can be expressed abstractly by means of a <em> functor</em> Q[_] from the category of Petri nets to an appropriate category of symmetric strict monoidal categories with free non-commutative monoids of objects, in the precise sense that, for each net N, the strong concatenable processes of N are isomorphic to the arrows of Q[N]. This yields an axiomatization of the causal behaviour of Petri nets in terms of symmetric strict monoidal categories.<br /> <br />In addition, we identify a <em>coreflection</em> right adjoint to Q[_] and we characterize its replete image in the category of symmetric monoidal categories, thus yielding an abstract description of the category of net computations.

Kan spectra provide a combinatorial model for the stable homotopy category. They were introduced by Dan Kan in 1963 under the name semisimplicial spectra. A Kan spectrum is similar to a pointed simplicial set, but it has simplices in negative degrees as well and all its simplices have infinitely many faces and degeneracies. In the first part of this thesis, we define a functor from the category of Gamma-spaces to the category of Kan spectra without passing through any other category of spectra. We show that the resulting Kan spectrum of a Gamma-space A agrees with the usual object associated to A in the stable homotopy category by comparing it to Bousfield-Friedlander's spectrum construction. In particular, applying our construction to the Gamma-space associated to a symmetric monoidal category provides a combinatorial model of its algebraic K-theory spectrum. For the Gamma-space associated to an abelian group, this yields via the stable Dold-Kan correspondence the unbounded chain complex with the abelian group concentrated in degree zero. The second part of this work concerns group spectra and twisting structures. Group spectra are the group objects in the category of Kan spectra. They provide an algebraic, combinatorial model for the stable homotopy category. We transfer Brown's model structure from the category of Kan spectra to a Quillen equivalent model structure on the category of group spectra. We then construct the analogues of Kan's loop group functor and its right adjoint Wbar together with corresponding classifying bundles, so that the category of Kan spectra becomes a twisted homotopical category in the sense of Farjoun and Hess.

The main object of this chapter is to construct and study the irreducible polynomial representations of the general linear group GL m ℂ = GL ( E ), where E is a complex vector space of dimension m . These can be formed by a basic construction in linear algebra that generalizes a well known construction of symmetric and exterior products; they make sense for any module over a commutative ring. These representations are parametrized by Young diagrams λ with at most m rows, and have bases corresponding to Young tableaux on λ with entries from [ m ]. They can also be constructed from representations of symmetric groups. Like the latter, these have useful realizations both as subspaces and as quotient spaces of naturally occurring representations, with relations given by quadratic equations. The characters of the representations are given in §8.3. To prove that these give all the irreducible representations we use a bit of the Lie group–Lie algebra story, which is sketched in this setting in §8.2. In the last section we describe some variations on the quadratic equations. In particular, we identify the sum of all polynomial representations with a ring constructed by Deruyts a century ago. A construction in linear algebra For any commutative ring R and any R -module E , and any partition λ, we will construct an R -module denoted E λ . (For applications in these notes, the case where R = ℂ, so E is a complex vector space, will suffice.)

This third chapter presents a brief and somewhat sketchy introduction to the theory of unitary representations of reductive and semisimple Lie groups G. The basic fact for an irreducible unitary representation π of G on a Hilbert space ℋ, is that every irreducible representation к of a maximal compact subgroup K ⊂ G has multiplicity m(к, π| к) ≤ dim к. This yields up the infinitesimal character χπ: L(g)→ ℂ and the distribution character θπ: C c ∞ (G) → ℂ, and consequently the differential equations $$z({\theta _\pi }) = \,{\chi _\pi }(z){\theta _\pi }$$ for $$z \in L{\text{(g)}}$$ which are the starting point for serious harmonic analysis on G.

The classification of irreducible square integrable representations of classical p p -adic groups is completed in this paper, under a natural local assumption. Further, this classification gives a parameterization of irreducible tempered representations of these groups. Therefore, it implies a classification of the non-unitary duals of these groups (modulo cuspidal data). The classification of irreducible square integrable representations is directly related to the parameterization of irreducible square integrable representations in terms of dual objects, which is predicted by Langlands program.

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.

The results of the note are inspired by the theory of representations of the infinite-dimensional classical groups. A new series of irreducible unitary representations of the group U(p,q) is described. These representations are constructed in the Gel'fand-Tsetlin basis and also as induced by nonunitary finite-dimensional representations of a maximal parabolic subgroup. As q → ∞ they approximate irreducible unitary representations of the group U(p,∞).

MacMahon's Master Theorem, Double Tableau Polynomials, and Representations of Groups

A systematic method is presented whereby any compact Lie group of n-real parameters is dealt with from an infinitesimal approach with the representative matrix method based on a group of inner automorphisms suggested in a previous paper. The group manifold, defined in terms of a metric of group parameters, is identified as a Riemannian one in which these parameters play a role of n curvilinear coordinates. Riemannian geometry is thus valid in the group manifold, and geometric quantities are explicitly calculated in terms of the symmetric or (0) connection by a straightforward application of the ordinary procedure of tensor analysis. A new and simpler method of computing the invariant volume element is presented within this framework. Furthermore, we discuss in detail the group of inner automorphisms for the calculation of the matrix element of finite rotations for any irreducible representation (abbreviated MEFRIR). It is found that our method works very well and yields right and left vector fields together with a set of 2n equations to be satisfied by the MEFRIR. The global properties of the group may, therefore, be obtained as a solution to these equations. Moreover, it provides not only the generalized Maurer-Cartan equations, the Lie structure formulas, two parameter groups of point transformations and the adjoint group, but also two additional nonsymmetric (+) and (−) connections with zero curvature, which do not possess any preassigned metric but possess two absolute parallelisms. Thus, our results on differential geometry completely agree with Cartan and Schouten's. A link between differential geometry and representations is presented by the right and left vector fields which are explicitly calculable in terms of the n parameters and through which geometric quantities, e.g., the Riemann tensor, the Ricci tensor, and the scalar curvature, of any connection are explicitly displayed. A theorem relating both the vector fields to the metric tensors is also included. Finally, the l (rank of the group) invariant differential equations to be satisfied by the MEFRIR are cast in the covariant (or Lie derivative) forms in any connection. Examples of the invariant equations are given for SU(2), SO(3), and SU(3). The two invariant equations of the latter can be cast in terms of the eigenvalues of isospin and hypercharge upon carrying out charge and hypercharge quantizations; in this connection, a new nonrelativistic wave equation to be satisfied by SU(3) multiplets as a generalization of the Schrödinger equation of the symmetric top is also proposed.

As E. Formanek has characterized low dimensional representations of the braid group B n , we extend these representations to the group of conjugating automorphisms C n , when n 5. We then give a classification for irreducible representations of C n in dimensions of at most n -3.Next, we determine representations of C n in dimension n -1 when each of the restrictions to the symmetric group S n and the braid group B n are irreducible.

We construct, for any finite commutative ring R, a family of representations of the general linear group whose intertwining properties mirror those of the principal series for over a finite field.