{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:47:07Z","timestamp":1773251227104,"version":"3.50.1"},"reference-count":24,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2019,3,20]],"date-time":"2019-03-20T00:00:00Z","timestamp":1553040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["MTM2017-84098-P"],"award-info":[{"award-number":["MTM2017-84098-P"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>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\u2019s 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\u2019s 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\u2019s description of quantum mechanical systems.<\/jats:p>","DOI":"10.3390\/sym11030414","type":"journal-article","created":{"date-parts":[[2019,3,21]],"date-time":"2019-03-21T04:11:56Z","timestamp":1553141516000},"page":"414","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["On the Structure of Finite Groupoids and Their Representations"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0580-5858","authenticated-orcid":false,"given":"Alberto","family":"Ibort","sequence":"first","affiliation":[{"name":"ICMAT and Department of Mathematics, Universidad Carlos III, Avenida de la Universidad 30, 28911 Legan\u00e9s, Spain"}]},{"given":"Miguel","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Dpto. de F\u00edsica Te\u00f3rica, Univ. Complutense de Madrid, Plaza de las Ciencias 1, 28040 Madrid, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2019,3,20]]},"reference":[{"key":"ref_1","unstructured":"Connes, A. (1994). Noncommutative Geometry, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2917","DOI":"10.1016\/j.jfa.2018.09.004","article-title":"Spectral continuity for aperiodic quantum systems I. General Theory","volume":"275","author":"Beckus","year":"2018","journal-title":"J. Funct. Anal."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Landsman, N.P. (1998). Mathematical Topics between Classical and Quantum Mechanics, Springer.","DOI":"10.1007\/978-1-4612-1680-3"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1006\/jfan.1996.3001","article-title":"Graphs, groupoids and Cuntz-Krieger algebras","volume":"144","author":"Kumjian","year":"1997","journal-title":"J. Funct. Anal."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1142\/S0219887806001211","article-title":"A survey of Lagrangian mechanics and control on Lie algebroids and groupoids","volume":"3","author":"Marrero","year":"2006","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1023\/A:1011965919259","article-title":"Lagrangian mechanics on Lie algebroids","volume":"67","year":"2001","journal-title":"Acta Applicandae Mathematica"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1850122","DOI":"10.1142\/S0217732318501225","article-title":"A gentle introduction to Schwinger\u2019s formulation of quantum mechanics: The groupoid picture","volume":"33","author":"Ciaglia","year":"2018","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_8","first-page":"744","article-title":"Groupoids: Unifying Internal and External Symmetry. A Tour trough Some Examples","volume":"43","author":"Weinstein","year":"1996","journal-title":"Not. AMS"},{"key":"ref_9","unstructured":"Higgings, P.J. (1971). Groupoids and Categories, Van Nostrand Reinhold Studies."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Mackenzie, K. (2005). General Theory of Lie Groupoids and Lie Algebroids, Cambridge University Press.","DOI":"10.1017\/CBO9781107325883"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"621","DOI":"10.2140\/pjm.1968.27.621","article-title":"Harmonic Analysis on groupoids","volume":"27","author":"Westman","year":"1968","journal-title":"Pac. J. Math."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Renault, J. (1980). A Groupoid Approach to C*-Algebras, Springer.","DOI":"10.1007\/BFb0091072"},{"key":"ref_13","unstructured":"Bos, R. (arXiv, 2006). Continuous representations of groupoids, arXiv."},{"key":"ref_14","first-page":"661","article-title":"Groupoids, their representations and imprimitivity systems","volume":"37","author":"Pysiak","year":"2004","journal-title":"Demonstratio Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1515\/dema-2013-0294","article-title":"Imprimitive theorem for groupoid representations","volume":"44","author":"Pysiak","year":"2011","journal-title":"Demonstratio Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1515\/dema-2017-0004","article-title":"Multiciplicity formulas for representations of transformation groupoids","volume":"50","author":"Pysiak","year":"2017","journal-title":"Demonstratio Mathematica"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"741","DOI":"10.4310\/JSG.2017.v15.n3.a5","article-title":"VB-groupoids and representation theory of Lie groupoids","volume":"15","year":"2017","journal-title":"J. Symplectic Geom."},{"key":"ref_18","first-page":"305","article-title":"Groupoids in combinatorics\u2014Applications of a theory of local symmetries","volume":"Volume 423","author":"Athanasiadis","year":"2007","journal-title":"Algebraic and Geometric Combinatorics"},{"key":"ref_19","unstructured":"Schwinger, J. (1970). Quantum Kinematics and Dynamics, W.A. Benjamin, Inc.. Frontiers in Physics."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Ciaglia, F.M., Ibort, A., and Marmo, G. (2019). Schwinger\u2019s Picture of Quantum Mechanics: Groupoids, Preprint.","DOI":"10.1142\/S0219887819501196"},{"key":"ref_21","first-page":"411","article-title":"An introduction tBAo Tannaka duality and quantum groups","volume":"Volume 1488","author":"Carboni","year":"1991","journal-title":"Part II of Category Theory, Como 1990"},{"key":"ref_22","unstructured":"Kirillov, A.A. (2012). Elements of the Theory of Representations, Springer Science & Business Media."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Etingof, P.I., Golberg, O., Hensel, S., Liu, T., Schwendner, A., Vaintrob, D., and Yudovina, E. (2011). Introduction to Representation Theory, Student Mathematical Library, AMS.","DOI":"10.1090\/stml\/059"},{"key":"ref_24","unstructured":"Gardner, M. (1959). Mathematical Puzzles of Sam Loyd, Dover Publications Inc."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/414\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:39:23Z","timestamp":1760186363000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/3\/414"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,3,20]]},"references-count":24,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,3]]}},"alternative-id":["sym11030414"],"URL":"https:\/\/doi.org\/10.3390\/sym11030414","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,3,20]]}}}