representation of the group theory

Group Representation

Two representations are considered equivalent if they are similar . For example, performing similarity transformations of the above matrices by

This entry contributed by Todd Rowland

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Rowland, Todd . "Group Representation." From MathWorld --A Wolfram Web Resource, created by Eric W. Weisstein . https://mathworld.wolfram.com/GroupRepresentation.html

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Representation of a group

2020 Mathematics Subject Classification: Primary: 20B Secondary: 22F05 [ MSN ][ ZBL ]

A homomorphism of the group into the group of all invertible transformations of a set $V$.

A permutation representation is a homomorphism to the symmetric group $S_V$: a group action of $G$ on $V$: cf. Permutation group .

A representation $\rho$ of a group $G$ is called linear if $V$ is a vector space over a field $k$ and if the transformations $\rho(g)$, $g\in G$, are linear. Often, linear representations are, for shortness, simply termed representations (cf. Representation theory ). In the theory of representations of abstract groups the theory of finite-dimensional representations of finite groups is best developed (cf. Finite group, representation of a ; Representation of the symmetric groups ).

If $G$ is a topological group, then one considers continuous linear representations of $G$ on a topological vector space $V$ (cf. Continuous representation ; Representation of a topological group ). If $G$ is a Lie group and $V$ is a finite-dimensional space over $\mathbf R$ or $\mathbf C$, then a continuous linear representation is automatically real analytic. Analytic and differentiable representations of a Lie group are defined also in the infinite-dimensional case (cf. Analytic representation ; Infinite-dimensional representation ). To each differentiable representation $\rho$ of a Lie group $G$ corresponds some linear representation of its Lie algebra — the differential representation of $\rho$ (cf. Representation of a Lie algebra ). If $G$ is moreover connected, then its finite-dimensional representations are completely determined by their differentials. The most developed branch of the representation theory of topological groups is the theory of finite-dimensional linear representations of semi-simple Lie groups, which is often formulated in the language of Lie algebras (cf. Finite-dimensional representation ; Representation of the classical groups ; Cartan theorem on the highest weight vector), the representation theory of compact groups, and the theory of unitary representations (cf. Representation of a compact group ; Unitary representation ).

For algebraic groups one has the theory of rational representations (cf. Rational representation ), which is in many aspects analogous to the theory of finite-dimensional representations of Lie groups.

• Group theory and generalizations
• Topological groups, Lie groups
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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71 . What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

• All volumes

On input and Langlands parameters for epipelagic representations HTML articles powered by AMS MathViewer

• Anne-Marie Aubert , Paul Baum , Roger Plymen , and Maarten Solleveld , Depth and the local Langlands correspondence , Arbeitstagung Bonn 2013, Progr. Math., vol. 319, Birkhäuser/Springer, Cham, 2016, pp. 17–41. MR 3618046 , DOI 10.1007/978-3-319-43648-7_{2}
• Anne-Marie Aubert , Paul Baum , Roger Plymen , and Maarten Solleveld , The local Langlands correspondence for inner forms of $\mathrm {SL}_n$ , Res. Math. Sci. 3 (2016), Paper No. 32, 34. MR 3579297 , DOI 10.1186/s40687-016-0079-4
• Moshe Adrian , The Langlands parameter of a simple supercuspidal representation: odd orthogonal groups , J. Ramanujan Math. Soc. 31 (2016), no. 2, 195–214. MR 3518182
• Moshe Adrian , Guy Henniart , Eyal Kaplan , and Masao Oi , Simple supercuspidal L-packets of split special orthogonal groups over dyadic fields , arXiv: 2305.09076 (2023).
• Moshe Adrian and Eyal Kaplan , The Langlands parameter of a simple supercuspidal representation: symplectic groups , Ramanujan J. 50 (2019), no. 3, 589–619. MR 4031300 , DOI 10.1007/s11139-018-0060-5
• Moshe Adrian and Eyal Kaplan , On the Langlands parameter of a simple supercuspidal representation: even orthogonal groups , Israel J. Math. 246 (2021), no. 1, 459–485. MR 4358290 , DOI 10.1007/s11856-021-2259-1
• Anne-Marie Aubert , Sergio Mendes , Roger Plymen , and Maarten Solleveld , On $L$-packets and depth for $\textrm {SL}_2(K)$ and its inner form , Int. J. Number Theory 13 (2017), no. 10, 2545–2568. MR 3713091 , DOI 10.1142/S1793042117501421
• James Arthur , The endoscopic classification of representations , American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. MR 3135650 , DOI 10.1090/coll/061
• Anne-Marie Aubert and Yujie Xu , The explicit local Langlands correspondence for $G_2$ , arXiv: 2208.12391v2 (2022).
• Colin J. Bushnell and Guy Henniart , The local Langlands conjecture for $\rm GL(2)$ , Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120 , DOI 10.1007/3-540-31511-X
• Roger W. Carter , Simple groups of Lie type , Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Reprint of the 1972 original; A Wiley-Interscience Publication. MR 1013112
• Jessica Fintzen , On the Moy-Prasad filtration , J. Eur. Math. Soc. (JEMS) 23 (2021), no. 12, 4009–4063. MR 4321207 , DOI 10.4171/jems/1098
• Jessica Fintzen , Types for tame $p$-adic groups , Ann. of Math. (2) 193 (2021), no. 1, 303–346. MR 4199732 , DOI 10.4007/annals.2021.193.1.4
• Benedict H. Gross and Mark Reeder , Arithmetic invariants of discrete Langlands parameters , Duke Math. J. 154 (2010), no. 3, 431–508. MR 2730575 , DOI 10.1215/00127094-2010-043
• Wee Teck Gan and Gordan Savin , The local Langlands conjecture for $G_2$ , Forum Math. Pi 11 (2023), Paper No. e28, 42. MR 4658199 , DOI 10.1017/fmp.2023.27
• Guy Henniart , Une preuve simple des conjectures de Langlands pour $\textrm {GL}(n)$ sur un corps $p$-adique , Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446 , DOI 10.1007/s002220050012
• Kaoru Hiraga , Atsushi Ichino , and Tamotsu Ikeda , Formal degrees and adjoint $\gamma$-factors , J. Amer. Math. Soc. 21 (2008), no. 1 , 283–304. MR 2350057 , DOI 10.1090/S0894-0347-07-00567-X
• Guy Henniart and Masao Oi , Simple supercuspidal $L$-packets of symplectic groups over dyadic fields , arXiv: 2207.12985v1 (2022).
• Kaoru Hiraga and Hiroshi Saito , On $L$-packets for inner forms of $SL_n$ , Mem. Amer. Math. Soc. 215 (2012), no. 1013, vi+97. MR 2918491 , DOI 10.1090/S0065-9266-2011-00642-8
• Michael Harris and Richard Taylor , The geometry and cohomology of some simple Shimura varieties , Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
• N. Iwahori and H. Matsumoto , On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups , Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016 , DOI 10.1007/BF02684396
• Tasho Kaletha , Simple wild $L$-packets , J. Inst. Math. Jussieu 12 (2013), no. 1, 43–75. MR 3001735 , DOI 10.1017/S1474748012000631
• Bertram Kostant , The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group , Amer. J. Math. 81 (1959), 973–1032. MR 114875 , DOI 10.2307/2372999
• P. C. Kutzko , Mackey’s theorem for nonunitary representations , Proc. Amer. Math. Soc. 64 (1977), no. 1 , 173–175. MR 442145 , DOI 10.1090/S0002-9939-1977-0442145-3
• Allen Moy and Gopal Prasad , Unrefined minimal $K$-types for $p$-adic groups , Invent. Math. 116 (1994), no. 1–3, 393–408., DOI 10.1007/BF01231566
• Allen Moy and Gopal Prasad , Jacquet functors and unrefined minimal $K$-types , Comment. Math. Helv. 71 (1996), no. 1, 98–121., DOI 10.1007/BF02566411
• David Mumford , Stability of projective varieties , Enseign. Math. (2) 23 (1977), no. 1-2, 39–110. MR 450272
• Masao Oi , Endoscopic lifting of simple supercuspidal representations of $\textrm {SO}_{2n+1}$ to $\textrm {GL}_{2n}$ , Amer. J. Math. 141 (2019), no. 1, 169–217. MR 3904769 , DOI 10.1353/ajm.2019.0004
• Masao Oi , Simple supercuspidal L-packets of quasi-split classical groups , arXiv: 1805.01400v2 (2021).
• Dmitri I. Panyushev , On invariant theory of $\theta$-groups , J. Algebra 283 (2005), no. 2, 655–670. MR 2111215 , DOI 10.1016/j.jalgebra.2004.03.032
• Mark Reeder , Paul Levy , Jiu-Kang Yu , and Benedict H. Gross , Gradings of positive rank on simple Lie algebras , Transform. Groups 17 (2012), no. 4, 1123–1190. MR 3000483 , DOI 10.1007/s00031-012-9196-3
• Beth Romano , On the local Langlands correspondence: New examples from the epipelagic zone , ProQuest LLC, Ann Arbor, MI, 2016. Thesis (Ph.D.)–Boston College. MR 3517860
• Sean Rostami , On the canonical representatives of a finite Weyl group , arXiv: 1505.07442v3 (2016).
• Mark Reeder and Jiu-Kang Yu , Epipelagic representations and invariant theory , J. Amer. Math. Soc. 27 (2014), no. 2, 437–477. MR 3164986 , DOI 10.1090/S0894-0347-2013-00780-8
• Jean-Pierre Serre , Cohomologie des groupes discrets , Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970) Ann. of Math. Stud., No. 70, Princeton Univ. Press, Princeton, NJ, 1971, pp. 77–169 (French). MR 385006
• Jean-Pierre Serre , Linear representations of finite groups , Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 450380 , DOI 10.1007/978-1-4684-9458-7
• Shaun Stevens , The supercuspidal representations of $p$-adic classical groups , Invent. Math. 172 (2008), no. 2, 289–352. MR 2390287 , DOI 10.1007/s00222-007-0099-1
• J. Tits , Reductive groups over local fields , Proc. Sympos. Pure Math. 33 (1979), no. 1, 29–69., DOI 10.1090/pspum/033.1/546588
• È. B. Vinberg , The Weyl group of a graded Lie algebra , Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, 709 (Russian). MR 430168
• Jiu-Kang Yu , Construction of tame supercuspidal representations , J. Amer. Math. Soc. 14 (2001), no. 3 , 579–622. MR 1824988 , DOI 10.1090/S0894-0347-01-00363-0
• Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2020): 22E50 , 11S37
• Beth Romano
• Affiliation: Department of Mathematics, King’s College London, WC2R 2LS, United Kingdom
• MR Author ID: 1229203
• Email: [email protected]
• Received by editor(s): March 18, 2023
• Received by editor(s) in revised form: September 29, 2023, and December 5, 2023
• Published electronically: February 12, 2024
• © Copyright 2024 Copyright by the Authors
• Journal: Represent. Theory 28 (2024), 90-111
• DOI: https://doi.org/10.1090/ert/668
• MathSciNet review: 4704423

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Title: hilbert series of representations of categories of $g$-sets.

Abstract: Let $G$ be a finite group. A contravariant functor from the category of finite free $G$-sets to vector spaces has an associated Hilbert series, which records the underlying sequence of $G^n$ representations, $n \in \mathbb N$. We prove that this Hilbert series is rational with denominator given by linear polynomials with coefficients in the field generated by the character table of $G$.

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