scholarly journals CLASSICAL AND QUANTUM FERMIONS LINKED BY AN ALGEBRAIC DEFORMATION

2005 ◽  
Vol 02 (05) ◽  
pp. 777-782 ◽  
Author(s):  
ALI MOSTAFAZADEH
Keyword(s):  
Irreducible Representation ◽  
Regular Representation ◽  
Faithful Representation ◽  
Dirac Fermion ◽  
Reducible Representation ◽  
Fermion Algebra ◽  
Algebraic Deformation

We study the regular representation ρζ of the single-fermion algebra [Formula: see text], i.e., c2 = c+2 = 0, cc+ + c+c = ζ1, for ζ ∈ [0,1]. We show that ρ0 is a four-dimensional nonunitary representation of [Formula: see text] which is faithfully irreducible (it does not admit a proper faithful subrepresentation). Moreover, ρ0 is the minimal faithfully irreducible representation of [Formula: see text] in the sense that every faithful representation of [Formula: see text] has a subrepresentation that is equivalent to ρ0. We therefore identify a classical fermion with ρ0 and view its quantization as the deformation: ζ : 0 → 1 of ρζ. The latter has the effect of mapping ρ0 into the four-dimensional, unitary, (faithfully) reducible representation ρ1 of [Formula: see text] that is reminiscent of a Dirac fermion.

scholarly journals GAUSSIAN FLUCTUATIONS OF REPRESENTATIONS OF WREATH PRODUCTS

2006 ◽  
Vol 09 (04) ◽  
pp. 529-546
Author(s):  
PIOTR ŚNIADY
Keyword(s):  
Finite Group ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Regular Representation ◽  
Wreath Products ◽  
Young Diagrams ◽  
Reducible Representation ◽  
Irreducible Components ◽  
Left Regular Representation ◽  
Left Regular

We study the asymptotics of the reducible representations of the wreath products G≀Sq = Gq ⋊ Sq for large q, where G is a fixed finite group and Sq is the symmetric group in q elements; in particular for G = ℤ/2ℤ we recover the hyperoctahedral groups. We decompose such a reducible representation of G≀Sq as a sum of irreducible components (or, equivalently, as a collection of tuples of Young diagrams) and we ask what is the character of a randomly chosen component (or, what are the shapes of Young diagrams in a randomly chosen tuple). Our main result is that for a large class of representations, the fluctuations of characters (and fluctuations of the shape of the Young diagrams) are asymptotically Gaussian. The considered class consists of the representations for which the characters asymptotically almost factorize and it includes, among others, the left regular representation therefore we prove the analogue of Kerov's central limit theorem for wreath products.

Polynomial Invariant Theory and Taylor Series

1991 ◽  
Vol 43 (6) ◽  
pp. 1243-1262 ◽  
Author(s):  
John E. Gilbert
Keyword(s):  
Irreducible Representation ◽  
Taylor Series ◽  
Invariant Theory ◽  
Regular Representation ◽  
Polynomial Functions ◽  
Polynomial Invariant ◽  
Finite Dimensional ◽  
Isotypic Component ◽  
Image Position ◽  
The Right

For any group K and finite-dimensional (right) K-module V let be the right regular representation of K on the algebra of polynomial functions on V. An Isotypic Component of is the sum of all k-submodules of on which π restricts to an irreducible representation can then be written as f = ΣƬ ƒƬ with ƒƬ in .

scholarly journals On the Structure of Finite Groupoids and Their Representations

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 414 ◽  
Author(s):  
Alberto Ibort ◽  
Miguel Rodríguez
Keyword(s):  
Irreducible Representation ◽  
Regular Representation ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Linear Representations ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Quantum Mechanical Systems ◽  
Maschke’S Theorem ◽  
Totally Disconnected

In this paper, both the structure and the theory of representations of finite groupoids are discussed. A finite connected groupoid turns out to be an extension of the groupoids of pairs of its set of units by its canonical totally disconnected isotropy subgroupoid. An extension of Maschke’s theorem for groups is proved showing that the algebra of a finite groupoid is semisimple and all finite-dimensional linear representations of finite groupoids are completely reducible. The theory of characters for finite-dimensional representations of finite groupoids is developed and it is shown that irreducible representations of the groupoid are in one-to-one correspondence with irreducible representation of its isotropy groups, with an extension of Burnside’s theorem describing the decomposition of the regular representation of a finite groupoid. Some simple examples illustrating these results are exhibited with emphasis on the groupoids interpretation of Schwinger’s description of quantum mechanical systems.

Representations of Compact Right Topological Groups

1993 ◽  
Vol 36 (3) ◽  
pp. 314-323 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Topological Group ◽  
Irreducible Representation ◽  
Regular Representation ◽  
List Type ◽  
Topological Groups ◽  
Regular Representations ◽  
Left Regular Representation ◽  
Left Regular ◽  
Enveloping Semigroups ◽  
The Right

AbstractCompact right topological groups arise naturally as the enveloping semigroups of distal flows. Recently, John Pym and the author established the existence of Haar measure μ on such groups, which invites the consideration of the regular representations. We start here by characterizing the continuous representations of a compact right topological group G, and are led to the conclusion that the right regular representation r is not continuous (unless G is topological). The domain of the left regular representation l is generally taken to be the topological centreor a tractable subgroup of it, furnished with a topology stronger than the relative topology from G (the goals being to have l both defined and continuous). An analysis of l and r on H = L2(G) for some non-topological compact right topological groups G shows, among other things, that: (i)for the simplest (perhaps) G generated by ℤ, (l, H) decomposes into one copy of each irreducible representation of ℤ and c copies of the regular representation.(ii)for the simplest (perhaps) G generated by the euclidean group of the plane , (l, H) decomposes into one copy of each of the continuous one-dimensional representations of and c copies of each continuous irreducible representation Ua,a > 0.(iii)when Λ(G) is not dense in G, it can seem very reasonable to regard r as a continuous representation of a related compact topological group, and also, G can be almost completely "lost" in the measure space (G, μ).

Some numerical results on the characters of exceptional Weyl groups

1978 ◽  
Vol 84 (3) ◽  
pp. 417-426 ◽  
Author(s):  
W. M. Beynon ◽  
G. Lusztig
Keyword(s):  
Irreducible Representation ◽  
Weyl Group ◽  
Irreducible Character ◽  
Regular Representation ◽  
Finite Subgroup ◽  
Weyl Groups ◽  
Real Vector ◽  
Invariant Vectors ◽  
Image Position ◽  
The Ideal

1. Let V be an l-dimensional real vector space and let W be a finite subgroup of GL(V) generated by reflexions such that the space of W-invariant vectors in V is zero. Then W acts naturally on the symmetric algebra S of V preserving the natural grading . LetI be the ideal in S generated by the w-invariant elements in . The quotient algebra inherits the W-action and also a grading is the image of Sk under S → S̅. It is well known that the W-module S̅ is isomorphic to the regular representation of W (see (3), ch. v, 5·2); in particular, S̅k = 0 for large k. (More precisely, S̅k = 0 for k > ν, where ν is the number of reflexions in W.)If ρ is an irreducible character of W, we denote by nk(ρ) the multiplicity of ρ in the W- module S̅k. The sequence n(ρ) = (n0(ρ), n1(ρ), n2(ρ), …) is an interesting invariant of the character ρ For example, in the study of unipotent classes in semisimple groups, one encounters the following question: what is the smallest k for which nk(ρ) 4= 0 (with ρ as above) and, then, what is nk(ρ) Also, the polynomial in qcan sometimes be interpreted as the dimension of an irreducible representation of a Chevalley group over the field with q elements. For these reasons it seems desirable to describe explicitly the sequence n(ρ) (or, equivalently, the polynomial Pρ(q)) for the various irreducible characters ρ of W. When W is a Weyl group of type Al, this is contained in the work of Steinberg (8); in the case where W is a Weyl group of type Bl or Dl this is done in (7),§2.

Vérités et Mensonges

2020 ◽  
Vol 39 (2) ◽  
pp. 188-224
Author(s):  
Erik Gunderson
Keyword(s):  
Theory And Practice ◽  
The Self ◽  
Faithful Representation ◽  
Political Subject ◽  
Concrete Reality ◽  
Hermeneutic Process ◽  
The Way

This is a survey of some of the problems surrounding imperial panegyric. It includes discussions of both the theory and practice of imperial praise. The evidence is derived from readings of Cicero, Quintilian, Pliny, the Panegyrici Latini, Menander Rhetor, and Julian the Apostate. Of particular interest is insincere speech that would be appreciated as insincere. What sort of hermeneutic process is best suited to texts that are politically consequential and yet relatively disconnected from any obligation to offer a faithful representation of concrete reality? We first look at epideictic as a genre. The next topic is imperial praise and its situation “beyond belief” as well as the self-positioning of a political subject who delivers such praise. This leads to a meditation on the exculpatory fictions that these speakers might tell themselves about their act. A cynical philosophy of Caesarism, its arbitrariness, and its constructedness abets these fictions. Julian the Apostate receives the most attention: he wrote about Caesars, he delivered extant panegyrics, and he is also the man addressed by still another panegyric. And in the end we find ourselves to be in a position to appreciate the way that power feeds off of insincerity and grows stronger in its presence.

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

scholarly journals Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

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