Irreducible Pair Research Articles

A Cauchy Harish-Chandra integral for a dual pair over a p-adic field, the definition and a conjecture

For a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the group with the larger or equal rank.The purpose of this article is to transfer this construction to the p-adic case. We provide the precise definition of the integral kernel operator and formulate a conjecture.

Sounds of Unridden Waves and the Contemporary Sublime

Abstract This article offers a speculative model intended to demonstrate that the sublime remains a viable construct in contemporary artworld contexts. Specifically, it proposes that sublimity and its neglected Other “profundity”—conceptualized as an irreducible pair—are well-suited to thinking about how to take seriously the still vital legacies bequeathed by postmodernism, while attempting to move beyond its more debilitating nihilistic and relativistic tendencies. First, somewhat neglected aspects of the Late Classical rhetorical heritage of the sublime are reviewed. Second, a model is progressively developed, building upon a formulation of human discourse articulated by Paul Ricoeur, and then introducing ideas relating to irresolvable self-contradiction and dialogical processes, as found in the thought of Western philosophers, such as Hegel, and in a number of Eastern philosophies, particularly Daoism. Finally, this model is applied to a brief discussion of an original moving image artwork by the authors, Sounds of Unridden Waves, published in a previous issue of the Journal of Asia-Pacific Popular Culture (2021).

Projective cases for the restriction of the oscillator representation to dual pairs of type I

For all the irreducible dual pairs of type I $(G,G')$, we analyze the restriction of the oscillator representation as a $(\mathfrak{g}', K')$-module, when $G'$ is the smaller group. For all $(G,G')$ in the stable range, as well as one more case, the modules obtained are projective. We use the duality correspondence introduced by Howe to analyze these restrictions.

THE SLOT LENGTH OF A FAMILY OF MATRICES

AbstractWe introduce the notion of the slot length of a family of matrices over an arbitrary field ${\mathbb {F}}$ . Using this definition it is shown that, if $n\ge 5$ and A and B are $n\times n$ complex matrices with A unicellular and the pair $\{A,B\}$ irreducible, the slot length s of $\{A,B\}$ satisfies $2\le s\le n-1$ , where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of $n\times n$ matrices, where $n\ge 5$ , is at most $n-1$ . The slot length of a family of rank-one complex matrices can be equal to n.

Extremal unipotent representations for the finite Howe correspondence

We study the Howe correspondence for unipotent representations of irreducible dual pairs (G,G′)=(Um(Fq),Un(Fq)) and (G,G′)=(Sp2m(Fq),O2nϵ(Fq)), where Fq denotes the finite field with q elements (q odd) and ϵ=±1. We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image Θ(π′) of a unipotent representation π′ of G′.

Quaternionic hyperbolic Fenchel–Nielsen coordinates

Let $$\mathrm {Sp}(2,1)$$ be the isometry group of the quaternionic hyperbolic plane $$\mathbf{H}_{\mathbb {H}}^2$$ . An element g in $$\mathrm {Sp}(2,1)$$ is hyperbolic if it fixes exactly two points on the boundary of $$\mathbf{H}_{\mathbb {H}}^2$$ . We classify pairs of hyperbolic elements in $$\mathrm {Sp}(2,1)$$ up to conjugation. A hyperbolic element of $$\mathrm {Sp}(2,1)$$ is called loxodromic if it has no real eigenvalue. We show that the set of $$\mathrm {Sp}(2,1)$$ conjugation orbits of irreducible loxodromic pairs is a $$(\mathbb {C}\mathbb {P}^1)^4$$ bundle over a topological space that is locally a semi-analytic subspace of $$\mathbb {R}^{13}$$ . We use the above classification to show that conjugation orbits of ‘geometric’ representations of a closed surface group (of genus $$g \ge 2$$ ) into $$\mathrm {Sp}(2,1)$$ can be determined by a system of $$42g-42$$ real parameters. Further, we consider the groups $$\mathrm {Sp}(1,1)$$ and $$\mathrm {GL}(2, \mathbb {H})$$ . These groups also act by the orientation-preserving isometries of the four and five dimensional real hyperbolic spaces respectively. We classify conjugation orbits of pairs of hyperbolic elements in these groups. These classifications determine conjugation orbits of ‘geometric’ surface group representations into these groups.

Isospectral drums and simple groups

Nearly every known pair of isospectral but nonisometric manifolds — with as most famous members isospectral bounded [Formula: see text]-planar domains which makes one “not hear the shape of a drum” [M. Kac, Can one hear the shape of a drum? Amer. Math. Monthly 73(4 part 2) (1966) 1–23] — arise from the (group theoretical) Gassmann–Sunada method. Moreover, all the known [Formula: see text]-planar examples (so counter examples to Kac’s question) are constructed through a famous specialization of this method, called transplantation. We first describe a number of very general classes of length equivalent manifolds, with as particular cases isospectral manifolds, in each of the constructions starting from a given example that arises itself from the Gassmann–Sunada method. The constructions include the examples arising from the transplantation technique (and thus in particular the known planar examples). To that end, we introduce four properties — called FF, MAX, PAIR and INV — inspired by natural physical properties (which rule out trivial constructions), that are satisfied for each of the known planar examples. Vice versa, we show that length equivalent manifolds with FF, MAX, PAIR and INV which arise from the Gassmann–Sunada method, must fall under one of our prior constructions, thus describing a precise classification of these objects. Due to the nature of our constructions and properties, a deep connection with finite simple groups occurs which seems, perhaps, rather surprising in the context of this paper. On the other hand, our properties define in some sense physically irreducible pairs of length equivalent manifolds — “atoms” of general pairs of length equivalent manifolds, in that such a general pair of manifolds is patched up out of irreducible pairs — and that is precisely what simple groups are for general groups.

Coherent state transforms and the Mackey-Stone-Von Neumann theorem

Mackey showed that for a compact Lie group K, the pair (K, C 0(K)) has a unique non-trivial irreducible covariant pair of representations. We study the relevance of this result to the unitary equivalence of quantizations for an infinite-dimensional family of K × K invariant polarizations on T*K. The Kähler polarizations in the family are generated by (complex) time-τ Hamiltonian flows applied to the (Schrödinger) vertical real polarization. The unitary equivalence of the corresponding quantizations of T*K is then studied by considering covariant pairs of representations of K defined by geometric prequantization and of representations of C 0(K) defined via Heisenberg time-(−τ) evolution followed by time-(+τ) geometric-quantization-induced evolution. We show that in the semiclassical and large imaginary time limits, the unitary transform whose existence is guaranteed by Mackey's theorem can be approximated by composition of the time-(+τ) geometric-quantization-induced evolution with the time-(−τ) evolution associated with the momentum space [W. D. Kirwin and S. Wu, “Momentum space for compact Lie groups and the Peter-Weyl theorem” (unpublished)] quantization of the Hamiltonian function generating the flow. In the case of quadratic Hamiltonians, this asymptotic result is exact and unitary equivalence between quantizations is achieved by identifying the Heisenberg imaginary time evolution with heat operator evolution, in accordance with the coherent state transform of Hall.

Water dimer dipole moment

The peculiar properties of the behavior of effective polarizability of water molecules in its saturated vapor are discussed on the basis of the experimental data on the static dielectric constant. It is taken into account that in the region 273 K < T < 485 K there is a mixture of monomers and dimers in the water vapor. The comparison of the theoretical and experimental expressions for the effective polarizability shows that the dipole moment of water dimer is connected with the dipole moment of monomer via relation: dD = √2dm. The relative value of the irreducible pair contributions to the polarizability of the dimer is determined.

Conservation relations for local theta correspondence

We prove Kudla-Rallis conjecture on first occurrences of local theta correspondence, for all irreducible dual pairs of type I and all local fields of characteristic zero.

The structure of ν-isologic pairs of groups

In [2], the first two authors generalized the concept of n-isoclinism to the class of all pairs of groups. In this paper, we extend that notion to ?-isologism and study the details of this notion. In addition, it is shown that every pair of groups is ?-isologic with a quotient irreducible pair of groups. Finally, as an application, we drive some inequalities for the Baer-invariant of a pair of groups.

On the lengths of irreducible pairs of complex matrices

The length of a pair of matrices is the smallest integer l l such that words in the matrices with at most l l factors span the unital algebra generated by the pair. Upper bounds for lengths have been much studied. If B B is a rank one n × n n\times n (complex) matrix, the length of the irreducible pair { A , B } \{A,B\} is 2 n − 2 2n-2 and the subwords of A n − 1 B A n − 2 A^{n-1}BA^{n-2} form a basis for M n ( C ) M_n(\mathbb {C}) . New examples are given of irreducible pairs of n × n n\times n matrices of length n n . There exists an irreducible pair of 5 × 5 5\times 5 matrices of length 4 4 . We begin the study of determining lower bounds for lengths.

Researching Living/Lived Mathematical Work

In qualitative research, accounts of experience are often taken for the experience itself despite ample phenomenological research that has articulated the difference between the living presence and the presence of the present, which requires representation. In this contribution, I provide practical examples that exhibit the difference between two aspects of mathematics that form an irreducible pair: living/lived mathematical work and accounts of mathematical work. Directions for the future practice of research are provided.

Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the total degree and the vector degree, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.

Study of irreducible balanced pairs for substitutive languages

Let be a language. A balanced pair (u,v) consists of two words u and v in which have the same number of occurrences of each letter. It is irreducible if the pairs of strict prefixes of u and v of the same length do not form balanced pairs. In this article, we are interested in computing the set of irreducible balanced pairs on several cases of languages. We make connections with the balanced pairs algorithm and discrete geometrical constructions related to substitutive languages. We characterize substitutive languages which have infinitely many irreducible balanced pairs of a given form.

Quantum Symmetric Pairs and Their Zonal Spherical Functions

We study the space of biinvariants and zonal spherical functions associated to quantum symmetric pairs in the maximally split case. Under the obvious restriction map, the space of biinvariants is proved isomorphic to the Weyl group invariants of the character group ring associated to the restricted roots. As a consequence, there is either a unique set, or an (almost) unique two-parameter set of Weyl group invariant quantum zonal spherical functions associated to an irreducible symmetric pair. Included is a complete and explicit list of the generators and relations for the left coideal subalgebras of the quantized enveloping algebra used to form quantum symmetric pairs.

Directed graphs, 2D state models, and characteristic polynomials of irreducible matrix pairs

The definition and main properties of a 2D digraph, namely a directed graph with two kinds of arcs, are introduced. Under the assumption of strong connectedness, the analysis of its paths and cycles is performed, based on an integer matrix whose rows represent the compositions of all circuits, and on the corresponding row module. Natural constraints on the composition of the paths connecting each pair of vertices lead to the definition of a 2D strongly connected digraph. For a 2D digraph of this kind the set of vertices can be partitioned into disjoint 2D-imprimitivity classes, whose number and composition are strictly related to the structure of the row module. Irreducible matrix pairs, i.e. pairs endowed with a 2D strongly connected digraph, are subsequently discussed. Equivalent descriptions of irreducibility, naturally extending those available for a single irreducible matrix, are obtained. These refer to the free evolution of the 2D state models described by the pairs and to their characteristic polynomials. Finally, primitivity is viewed as a special case of irreducibility, and completely characterized in terms of 2D-digraphs, characteristic polynomials, and 2D system dynamics.

On pairs of unbounded self-adjoint operators satisfying an algebraic relation

Unbounded pairs of self-adjoint operatorsA andB satisfying the algebraic relationF1(A)B=BF2(A) are studied. For these relations, various definitions of “integrable” pairs of operators are presented and the class of “tame” relations is indicated; for the “tame” relations, the irreducible pairs are described and a structure theorem is presented.

Positive commutators [formula omitted

We consider commutators of the form R ƒ, g = i[ƒ(P), g(Q)] , where P, Q are an irreducible canonical pair of selfadjoint operators in Hilbert space ([ P, Q] = − i), and where ƒ, g are real-valued functions with ƒ′, g′ ϵ L 1( R) . A necessary and sufficient condition for R ƒ, g to be of rank one [zero] is given. Some necessary and some sufficient conditions are given for R ƒ, g ⩾ 0 . For example, it is necessary that both ƒ, g are strictly monotone (unless R ƒ, g = 0 ). If K a is the set of bounded functions ƒ such that ƒ has an analytic continuation to a strip ¦ Im z¦ < a with ( Im z)( Im ƒ(z)) ⩾ 0 , then it is sufficient that ƒ ϵ K a and g ϵ K b with ab ⩾ π 2 .

High-frequency asymptotic behavior of the spectrum of the velocity autocorrelation function in a simple liquid

The effect of short range forces of interparticle repulsion on the formation of high-frequency asymptotic behavior of the velocity autocorrelation function in a simple liquid is studied. For this purpose, a method is developed of a group expansion of the evolution operator of a many-particle system. It is shown that only irreducible pair collisions make a contribution to the high-frequency asymptotic behavior. As an illustration, calculations are performed for interatomic potentials modeled by a power function. Application of the developed methods to the case of classical high-temperature plasma leads to the well-known Ginzburg formula for the high frequency conductivity.