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 Mosc. Math. J., 2003, Volume 3, Number 4, Pages 1209–1221 (Mi mmj128)

Frequent representations

V. I. Arnol'dab

a Steklov Mathematical Institute, Russian Academy of Sciences
b Université Paris-Dauphine

Abstract: Given a unitary representation $T$ of a finite group $G$ in $\mathbb C^n$, write $M$ for the variety of such representations which are unitary equivalent to $T$. The representation $T$ is said to be frequent if the dimension of the variety $M$ is maximal (among all representations of $G$ in the same complex space). We prove that the irreducible representations are distributed, in the frequent representation (of large dimension), asymptotically in the same way as in the fundamental representation in the space of functions on $G$: the frequencies of the irreducible components are proportional to their dimensions.

Key words and phrases: Representations of finite groups, unitary representations, frequent representations.

DOI: https://doi.org/10.17323/1609-4514-2003-3-4-1209-1221

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Citation: V. I. Arnol'd, “Frequent representations”, Mosc. Math. J., 3:4 (2003), 1209–1221

Citation in format AMSBIB
\Bibitem{Arn03} \by V.~I.~Arnol'd \paper Frequent representations \jour Mosc. Math.~J. \yr 2003 \vol 3 \issue 4 \pages 1209--1221 \mathnet{http://mi.mathnet.ru/mmj128} \crossref{https://doi.org/10.17323/1609-4514-2003-3-4-1209-1221} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2058796} \zmath{https://zbmath.org/?q=an:1075.20004} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000208594400001} 

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• http://mi.mathnet.ru/eng/mmj/v3/i4/p1209

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This publication is cited in the following articles:
1. Aicardi F., “Empirical estimates of the average orders of orbits period lengths in Euler groups”, C. R. Math. Acad. Sci. Paris, 339:1 (2004), 15–20
2. Arnold V., “Number-theoretical turbulence in Fermat-Euler arithmetics and large young diagrams geometry statistics”, J. Math. Fluid Mech., 7, suppl. 1 (2005), S4–S50
3. V. I. Arnol'd, “Statistics of Young diagrams of cycles of dynamical systems for finite tori automorphisms”, Mosc. Math. J., 6:1 (2006), 43–56
4. “Vladimir Igorevich Arnol'd (on his 70th birthday)”, Russian Math. Surveys, 62:5 (2007), 1021–1030
5. Ramacher P., “Reduced Weyl asymptotics for pseudodifferential operators on bounded domains I. The finite group case”, J. Funct. Anal., 255:4 (2008), 777–818
6. V. I. Arnold, “Topological properties of eigenoscillations in mathematical physics”, Proc. Steklov Inst. Math., 273 (2011), 25–34