Imprimitivity theorem for groupoid representations

Pysiak, Leszek

doi:10.1515/dema-2013-0294

Cited by 8 publications

(18 citation statements)

References 21 publications

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“…There is a further refinement of the decomposition in invariant subspaces (15). We have seen that each subspace W +…”

Section: Propositionmentioning

confidence: 98%

“…However, it is clear that there are other elements in the groupoid which do not change the state (in the sense that the empty box is still in its place) but the pieces around it have been exchanged. For instance, if the original state is s 16 , consider the sequence of moves (taking into account the numbered boxes) (12,16), (11,12), (15,11), (16,15), where the pair (j, i) means that the move takes the box in position j (in the original configuration of boxes) to position i, that is, the move (15,16) will move the box sitting in position 15 down to occupy position 16, and so on. This would imply that the state s i is transformed in the state s j as the empty box changes from position i to j.…”

Section: Loyd's Puzzlesmentioning

confidence: 99%

“…Then the source of the move (j, i) would be i and target j. We could denote the composition of the sequence of moves as (16,15) • (15,11) • (11,12) • (12, 16) (where the circle • will stand for the composition rule of the groupoid that consists in performing one move after the other). The final state would be s 16 again, but the pieces have been moved around (we have changed the configuration of the square, see Figure 3, right, displaying the final configuration).…”

Section: Loyd's Puzzlesmentioning

confidence: 99%

“…Hence, the theory of linear representations of finite groupoids provides a different perspective to the standard algebra of matrices. There is a number of relevant contributions to the theory of linear representations of groupoids describing the notion of measurable and continuous representations (see for instance the aforementioned work by Landsman [3] or [13] and references therein), unitary representations of groupoids and the extension of Mackey's imprimitivity theorem [14][15][16] or the representation theory of Lie groupoids [17].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the Structure of Finite Groupoids and Their Representations

Ibort

Rodrı́guez

2019

Symmetry

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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.

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“…There is a further refinement of the decomposition in invariant subspaces (15). We have seen that each subspace W +…”

Section: Propositionmentioning

confidence: 98%

Section: Loyd's Puzzlesmentioning

confidence: 99%

Section: Loyd's Puzzlesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Structure of Finite Groupoids and Their Representations

Ibort

Rodrı́guez

2019

Symmetry

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“…In 1981 the notion of differential group and the notion of group differential structure (based on the notion of Sikorski's differential space -see [9]) was introduced and investigated by the second author in his PhD thesis [4]. Independently, in the same time, an analogous notions was investigated by P. Multarzyński in his PhD thesis (prepared in the Jagiellonian University in Krakov).…”

mentioning

confidence: 99%

Differential Groupoids

Burzyńska¹,

Pasternak-Winiarski²

2015

JMSS

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Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

Drewnik¹,

Miller

Pasternak-Winiarski

2021

Complex Anal. Oper. Theory

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The aim of the paper is to create a link between the theory of reproducing kernel Hilbert spaces (RKHS) and the notion of a unitary representation of a group or of a groupoid. More specifically, it is demonstrated on one hand how to construct a positive definite kernel and an RKHS for a given unitary representation of a group(oid), and on the other hand how to retrieve the unitary representation of a group or a groupoid from a positive definite kernel defined on that group(oid) with the help of the Moore–Aronszajn theorem. The kernel constructed from the group(oid) representation is inspired by the kernel defined in terms of the convolution of functions on a locally compact group. Several illustrative examples of reproducing kernels related with unitary representations of groupoids are discussed in detail. The paper is concluded with the brief overview of the possible applications of the proposed constructions.

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Imprimitivity theorem for groupoid representations

Cited by 8 publications

References 21 publications

On the Structure of Finite Groupoids and Their Representations

On the Structure of Finite Groupoids and Their Representations

Differential Groupoids

Reproducing Kernel Hilbert Space Associated with a Unitary Representation of a Groupoid

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