scholarly journals Description of unitary representations of the group of infinite 𝑝-adic integer matrices

2021 ◽  
Vol 25 (21) ◽  
pp. 606-643
Author(s):  
Yury Neretin
Keyword(s):  
Irreducible Representation ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Infinite Matrices ◽  
Natural Topology ◽  
Content Type ◽  
Residue Ring ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

We classify irreducible unitary representations of the group of all infinite matrices over a p p -adic field ( p ≠ 2 p\ne 2 ) with integer elements equipped with a natural topology. Any irreducible representation passes through a group G L GL of infinite matrices over a residue ring modulo p k p^k . Irreducible representations of the latter group are induced from finite-dimensional representations of certain open subgroups.

Note on Weight Spaces of Irreducible Linear Representations

1968 ◽  
Vol 11 (3) ◽  
pp. 399-403 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Weight Function ◽  
Wide Class ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
One Dimensional ◽  
Highest Weight ◽  
Linear Representations ◽  
Finite Dimensional

Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.

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.

scholarly journals Multiplicities in the Tensor Product of Finite-Dimensional Representations of Discrete Groups

1978 ◽  
Vol 21 (1) ◽  
pp. 17-19
Author(s):  
Dragomir Ž. Djoković
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Unitary Representations ◽  
Vector Spaces ◽  
Closed Field ◽  
Discrete Groups ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

Groups with Representations of Bounded Degree

1949 ◽  
Vol 1 (1) ◽  
pp. 105-112 ◽  
Author(s):  
Irving Kaplansky
Keyword(s):  
Compact Group ◽  
Group Element ◽  
Unitary Representations ◽  
Bounded Degree ◽  
One Dimensional ◽  
Unit Matrix ◽  
Finite Dimensional ◽  
Weyl Theorem ◽  
Irreducible Unitary Representations ◽  
Complete Set

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.

scholarly journals One more method to construct the irreducible unitary representations of nipotent Lie groups

1986 ◽  
Vol 40 (1) ◽  
pp. 89-94
Author(s):  
A. A. Astaneh
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Unitary Representations ◽  
Irreducible Representations ◽  
Nilpotent Lie Group ◽  
Irreducible Unitary Representations ◽  
Simply Connected

AbstractIn this paper one more canonical method to construct the irreducible unitary representations of a connected, simply connected nilpotent Lie group is introduced. Although we used Kirillov' analysis to deduce this procedure, the method obtained differs from that of Kirillov's, in that one does not need to consider the codjoint representation of the group in the dual of its Lie algebra (in fact, neither does one need to consider the Lie algebra of the group, provided one knows certain connected subgroups and their characters). The method also differs from that of Mackey's as one only needs to induce characters to obtain all irreducible representations of the group.

Irreducible representations of finitely generated nilpotent groups

1977 ◽  
Vol 81 (2) ◽  
pp. 201-208 ◽  
Author(s):  
Daniel Segal
Keyword(s):  
Irreducible Representation ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Irreducible Representations ◽  
Closed Field ◽  
Finitely Generated ◽  
Dimensional Representation ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Irreducible Representation

1. Introduction. It is well known that every finite-dimensional irreducible representation of a nilpotent group over an algebraically closed field is monomial, that is induced from a 1-dimensional representation of some subgroup. However, even a finitely generated nilpotent group in general has infinite-dimensional irreducible representations, and as a first step towards an understanding of these one wants to discover whether they too are necessarily monomial. The main point of this note is to show how far they can fail to be so.

Existence of Weight Space Decompositions for Irreducible Representations of Simple Lie Algebras

1971 ◽  
Vol 14 (1) ◽  
pp. 113-115 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Characteristic Zero ◽  
Finite Dimension ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
Space Decomposition ◽  
Finite Dimensional

Let L denote a finite-dimensional simple Lie algebra over an algebraically closed field K of characteristic zero. It is well known that every finite-dimension 1, irreducible representation of L admits a weight space decomposition; moreover every irreducible representation of L having at least one weight space admits a weight space decomposition.

scholarly journals Splitting fields of real irreducible representations of finite groups

2021 ◽  
Vol 25 (31) ◽  
pp. 897-902
Author(s):  
Dmitrii Pasechnik
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Finite Groups ◽  
Irreducible Representations ◽  
Content Type ◽  
Cyclotomic Numbers ◽  
Representations Of Finite Groups ◽  
A Finite Group ◽  
Algorithmic Procedure

We show that any irreducible representation ρ \rho of a finite group G G of exponent n n , realisable over R \mathbb {R} , is realisable over the field E ≔ Q ( ζ n ) ∩ R E≔\mathbb {Q}(\zeta _n)\cap \mathbb {R} of real cyclotomic numbers of order n n , and describe an algorithmic procedure transforming a realisation of ρ \rho over Q ( ζ n ) \mathbb {Q}(\zeta _n) to one over E E .

Family of measures on a space of curves that are quasi-invariant with respect to some action of diffeomorphisms group

2016 ◽  
Vol 19 (03) ◽  
pp. 1650019
Author(s):  
Evgenii Dmitrievich Romanov
Keyword(s):  
Invariant Measures ◽  
Functional Space ◽  
Geometric Interpretation ◽  
Unitary Representations ◽  
Multidimensional Case ◽  
Dimensional Euclidean Space ◽  
Future Research ◽  
Finite Dimensional ◽  
Irreducible Unitary Representations ◽  
Nikodym Derivative

A family of quasi-invariant measures on the special functional space of curves in a finite-dimensional Euclidean space with respect to the action of diffeomorphisms is constructed. The main result is an explicit expression for the Radon–Nikodym derivative of the transformed measure relative to the original one. The stochastic Ito integral allows to express the result in an invariant form for a wider class of diffeomorphisms. These measures can be used to obtain irreducible unitary representations of the diffeomorphisms group which will be studied in future research. A geometric interpretation of the action considered together with a generalization to the multidimensional case makes such representations applicable to problems of quantum mechanics.

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