Unitary designs and codes

A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code—a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct absolute values—and give an upper bound for the size of a code with s inner product values in U(d), for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary t-design in U(d), and we catalogue some t-designs that arise from finite groups.      

On the normalizers of the unitary subgroup in an integral group ring

In this paper, we investigate the properties of the normalizers of the unitary subgroup u_f(ZG) in an integral group rings. One of our main results is Theorem 2.6 which character¬izes the second normalizer of the unitary subgroup. As a conse¬quence of this theorem, we prove that the second normalizer of u_f(ZG) coincides with the first normalizer when G is a periodic group. Among other results, we give necessary and sufficient conditions for which the unitary subgroup is normal in the unit group when G is periodic and also characterize when all bicyclic units are nontrivial and elements of the normalizer of the unitary subgroup.     

Tensor products of singular representations and an extension of the $ \theta $-correspondence

In this paper we consider the problem of decomposing tensor products of certain singular unitary representations of a semisimple Lie group G. Using explicit models for these representations (constructed earlier by one of us) we show that the decomposition is controlled by a reductive homogeneous space . Our procedure establishes a correspondence between certain unitary representations of G and those of . This extends the usual -correspondence for dual reductive pairs. As a special case we obtain a correspondence between certain representations of real forms of E ₇ and F ₄.     

CertainC*-algebras with real rank zero and their corona and multiplier algebras: II

We first determine the homotopy classes of nontrivial projections in a purely infinite simpleC*-algebraA, in the associated multiplier algebraM(A) and the corona algebraM A/A in terms ofK _*(A). Then we describe the generalized Fredholm indices as the group of homotopy classes of non-trivial projections ofA; consequently, we determine theK _*-groups of all hereditaryC*-subalgebras of certain corona algebras. Secondly, we consider a group structure of *-isomorphism classes of hereditaryC*-subalgebras of purely infinite simpleC*-algebras. In addition, we prove that ifA is aC*-algebra of real rank zero, then each unitary ofA, in caseA it unital, each unitary ofM(A) and ofM(A)/A, in caseA is nonunital but -unital, can be factored into a product of a unitary homotopic to the identity and a unitary matrix whose entries are all partial isometries (with respect to a decomposition of the identity).Partially supported by a grant from the National Science Foundation.     

Unitary spherical spectrum for p-adic classical groups

Let G be a split reductive p-adic group. Then the determination of the unitary representations with nontrivial Iwahori fixed vectors can be reduced to the determination of the unitary dual of the corresponding Iwahori-Hecke algebra. In this paper we study the unitary dual of the Iwahori-Hecke algebras corresponding to the classical groups. We determine all the unitary spherical representations.     

Repeated Quantum Interactions and Unitary Random Walks

Among the discrete evolution equations describing a quantum system ℋ _S undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in ℝ ^N . The characterization we obtain is entirely algebraical in terms of the unitary operator driving the elementary interaction. We show that the solutions of these equations are then random walks on the group U(ℋ₀) of unitary operators on ℋ₀.     

On periods of CUSP forms and algebraic cycles forU(3)

In this paper we discuss relations between the following types of conditions on a representationπ in a cuspidalL-packet ofU(3): (1)L(s, π×ξ) has a pole ats=1 for someξ; (2) aperiod ofπ over some algebraic cycle inU(3) (coming from a unitary group in two variables) is non-zero; and (3) π is atheta-series lifting from some unitary group in two variables. As an application of our analysis, we show that the algebraic cycles on theU(3) Shimura variety arenot spanned (over the Hecke algebra) by the modular and Shimura curves coming from unitary subgroups. All three authors are supported by a grant from the U.S.-Israel Binational Science Foundation; the second author is also supported by an NSF Grant.     

Irreducible decompositions of unitary representations of infinite- dimensional groups

This paper concerns the problem of irreducible decompositions of unitary representations of topological groups G, including the group Diff₀(M) of diffeomorphisms with compact support on smooth manifolds M. It is well known that the problem is affirmative, when G is a locally compact, separable group (cf. [3, 4]). We extend this result to infinite-dimensional groups with appropriate quasi-invariant measures, and, in particular, we show that every continuous unitary representation of Diff₀(M) has an irreducible decomposition under a fairly mild condition. This research was partially supported by a Grant-in-Aid for Scientific Research (No.14540167), Japan Socieity of the Promotion of Science.     

Towards the Eaton-Moretó conjecture on the minimal height of characters

Abstract

In this note we prove that the Eaton-Moretó conjecture holds for all blocks of finite general linear and unitary groups for all primes. Also, we show that no block of a finite quasi-simple group of classical Lie type provides a minimal counterexample to the conjecture, and so for ? > 5 no ?-block of any quasi-simple group can be a minimal counterexample.     

Singular unitary dilations

We study the problem of determining which bounded linear operator on a Hilbert space can be dilated to a singular unitary operator. Some of the partial results we obtained are (1) every strict contraction has a diagonal unitary dilation, (2) everyC ₀ contraction has a singular unitary dilation, and (3) a contraction with one of its defect indices finite has a singular unitary dilation if and only if it is the direct sum of a singular unitary operator and aC ₀(N) contraction. Such results display a scenario which is in marked contrast to that of the classical case where we have the absolute continuity of the minimal unitary power dilation of any completely nonunitary contraction.     

Kazhdan's Property (T) for the unitary group of aseparable Hilbert space

We show that the unitary group of a separable Hilbert space has Kazhdan's Property (T), when it is equipped with the strong operator topology. More precisely, for every integer m 2, we give an explicit Kazhdan set consisting of m unitary operators and determine an optimal Kazhdan constant for this set. Moreover, we show that a locally compact group with Kazhdan's Property (T) has a finite Kazhdan set if and only if its Bohr compactification has a finite Kazhdan set. As a consequence, if a locally compact group with Property (T) is minimally almost periodic, then it has a finite Kazhdan set.     

Some permanence properties of C-unique groups

We will study some permanence properties of C^*-unique groups in details. In particular, normal subgroups and extensions will be considered. Among other interesting results, we prove that every second countable amenable group with an injective finite-dimensional representation (not necessarily unitary) is a retract of a C^*-unique group. Moreover, any amenable discrete group is a retract of a discrete C^*-unique group.     

Canonical forms for unitary congruence and *congruence

We use methods of the general theory of congruence and *congruence for complex matrices – regularization and cosquares – to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices A such that āA (respectively, A ²) is normal. As special cases of our canonical forms, we obtain – in a coherent and systematic way – known canonical forms for conjugate normal, congruence normal, coninvolutory, involutory, projection, λ-projection, and unitary matrices. But we also obtain canonical forms for matrices whose squares are Hermitian or normal, and other cases that do not seem to have been investigated previously. We show that the classification problems under (a) unitary *congruence when A ³ is normal, and (b) unitary congruence when AāA is normal, are both unitarily wild, so these classification problems are hopeless.     

On the Homotopy Type of the Unitary Group and the Grassmann Space of Purely Infinite Simple C*-Algebras

We completely determine the homotopy groups _n (.) of the unitary group and the space of projections of purely infinite simple C ^*-algebras in terms of K-theory. We also prove that the unitary group of a purely infinite simple C ^*-algebra A is a contractible topological space if and only if K0(A) = K1(A) = {0}, and again if and only if the unitary group of the associated generalized Calkin algebra L(HA) / K(HA) is contractible. The well-known Kuiper's theorem is extended to a new class of C ^*-algebras.     

Analysis on the minimal representation of O(p,q): III. Ultrahyperbolic equations on

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q=2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Möbius group of conformal transformations on , and preserves a space of solutions of the ultrahyperbolic Laplace equation on . We construct in an intrinsic and natural way a Hilbert space of solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on L²(C), where C is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of O(p,q).     

The fundamental group of a unitary orbit

The unitary orbit of a complex n × n matrix A is simply connected if and only if the portion of the commutant {A} which resides in the special unitary group is path connected.     

Association Schemes and Fusion Algebras (An Introduction)

We introduce the concept of fusion algebras at algebraic level, as a purely algebraic concept for the fusion algebras which appear in conformal field theory in mathematical physics. We first discuss the connection between fusion algebras at algebraic level and character algebras, a purely algebraic concept for Bose-Mesner algebras of association schemes. Through this correspondence, we establish the condition when the matrix S of a fusion algebra at algebraic level is unitary or symmetric. We construct integral fusion algebras at algebraic level, from association schemes, in particular from group association schemes, whose matrix S is unitary and symmetric. Finally, we consider whether the modular invariance property is satisfied or not, namely whether there exists a diagonal matrix T satisfying the condition (ST)³ = S ². We prove that this property does not hold for some integral fusion algebras at algebraic level coming from the group association scheme of certain groups of order 64, and we also prove that the (nonintegral) fusion algebra at algebraic level obtained from the Hamming association scheme H(d, q) has the modular invariance property.