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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 1(355), Pages 125–144 (Mi umn704)  

This article is cited in 17 scientific papers (total in 18 papers)

Rings of continuous functions, symmetric products, and Frobenius algebras

V. M. Buchstabera, E. G. Reesb

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Edinburgh

Abstract: A constructive proof is given for the classical theorem of Gel'fand and Kolmogorov (1939) characterising the image of the evaluation map from a compact Hausdorff space $X$ into the linear space $C(X)^*$ dual to the ring $C(X)$ of continuous functions on $X$. Our approach to the proof enabled us to obtain a more general result characterising the image of the evaluation map from the symmetric products $\operatorname{Sym}^n(X)$ into $C(X)^*$. A similar result holds if $X=\mathbb C^m$ and leads to explicit equations for symmetric products of affine algebraic varieties as algebraic subvarieties in the linear space dual to the polynomial ring. This leads to a better understanding of the algebra of multisymmetric polynomials.
The proof of all these results is based on a formula used by Frobenius in 1896 in defining higher characters of finite groups. This formula had no further applications for a long time; however, it has appeared in several independent contexts during the last fifteen years. It was used by A. Wiles and R. L. Taylor in studying representations and by H.-J. Hoehnke and K. W. Johnson and later by J. McKay in studying finite groups. It plays an important role in our work concerning multivalued groups. Several properties of this remarkable formula are described. It is also used to prove a theorem on the structure constants of Frobenius algebras, which have recently attracted attention due to constructions taken from topological field theory and singularity theory. This theorem develops a result of Hoehnke published in 1958. As a corollary, a direct self-contained proof is obtained for the fact that the 1-, 2-, and 3-characters of the regular representation determine a finite group up to isomorphism. This result was first published by Hoehnke and Johnson in 1992.

DOI: https://doi.org/10.4213/rm704

Full text: PDF file (350 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:1, 125–145

Bibliographic databases:

UDC: 515.42+517.982
MSC: Primary 46E25, 05E05; Secondary 05A18, 54C40, 20C15, 20C05
Received: 15.01.2004

Citation: V. M. Buchstaber, E. G. Rees, “Rings of continuous functions, symmetric products, and Frobenius algebras”, Uspekhi Mat. Nauk, 59:1(355) (2004), 125–144; Russian Math. Surveys, 59:1 (2004), 125–145

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. D. V. Gugnin, “On continuous and irreducible Frobenius $n$-homomorphisms”, Russian Math. Surveys, 60:5 (2005), 967–969  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. V. M. Buchstaber, “$n$-valued groups: theory and applications”, Mosc. Math. J., 6:1 (2006), 57–84  mathnet  crossref  mathscinet  zmath
    3. D. V. Gugnin, “Polynomially dependent homomorphisms. Uniqueness theorem for Frobenius $n$-homomorphisms”, Russian Math. Surveys, 62:5 (2007), 993–995  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. T. Voronov, H. M. Khudaverdian, “Generalized symmetric powers and a generalization of the Kolmogorov–Gel'fand–Buchstaber–Rees theory”, Russian Math. Surveys, 62:3 (2007), 623–625  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. Khudaverdian H.M., Voronov T.T., “Operators on superspaces and generalizations of the Gelfand-Kolmogorov theorem”, XXVI Workshop on Geometrical Methods in Physics, AIP Conference Proceedings, 956, 2007, 149–155  crossref  mathscinet  zmath  adsnasa  isi
    6. Vale R., Waldron Sh., “Tight frames generated by finite nonabelian groups”, Numer. Algorithms, 48:1-3 (2008), 11–27  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. Buchstaber V.M., Rees E.G., “Frobenius $n$-homomorphisms, transfers and branched coverings”, Math. Proc. Cambridge Philos. Soc., 144:1 (2008), 1–12  crossref  mathscinet  zmath  isi  elib
    8. D. V. Gugnin, “Polynomially Dependent Homomorphisms and Frobenius $n$-Homomorphisms”, Proc. Steklov Inst. Math., 266 (2009), 59–90  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    9. Domokos M., “Vector invariants of a class of pseudoreflection groups and multisymmetric syzygies”, J. Lie Theory, 19:3 (2009), 507–525  mathscinet  zmath  isi  elib
    10. H. M. Khudaverdian, T. T. Voronov, “A short proof of the Buchstaber-Rees theorem”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 369:1939 (2011), 1334  crossref  mathscinet  zmath  adsnasa  isi
    11. D. V. Gugnin, “Topological applications of graded Frobenius $n$-homomorphisms”, Trans. Moscow Math. Soc., 72 (2011), 97–142  mathnet  crossref  mathscinet  zmath  elib
    12. Domokos M., Puskas A., “Multisymmetric Polynomials in Dimension Three”, J. Algebra, 356:1 (2012), 283–303  crossref  mathscinet  zmath  isi  elib
    13. A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. K.W.. JOHNSON, EIRINI POIMENIDOU, “A formal power series attached to a class function on a group and its application to the characterisation of characters”, Math. Proc. Camb. Phil. Soc, 2013, 1  crossref  mathscinet  isi
    15. D. V. Gugnin, “Lower Bounds for the Degree of a Branched Covering of a Manifold”, Math. Notes, 103:2 (2018), 187–195  mathnet  crossref  crossref  mathscinet  isi  elib
    16. V. M. Buchstaber, A. P. Veselov, “Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups”, Russian Math. Surveys, 74:3 (2019), 387–430  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    17. Johnson K.W., “Group Matrices, Group Determinants and Representation Theory the Mathematical Legacy of Frobenius Preface”: Johnson, KW, Group Matrices, Group Determinants and Representation Theory: the Mathematical Legacy of Frobenius, Lect. Notes Math., Lecture Notes in Mathematics, 2233, Springer International Publishing Ag, 2019, IX+  mathscinet  isi
    18. A. V. Ovchinnikov, “O dvoistvennosti v teorii gladkikh mnogoobrazii”, MaterialyVserossiiskoinauchnoikonferentsii Differentsialnye uravneniyai ikh prilozheniya,posvyaschennoi 85-letiyu professoraM.T.Terekhina.Ryazanskii gosudarstvennyi universitet im. S.A. Esenina,Ryazan, 1718 maya2019 g. Chast 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 185, VINITI RAN, M., 2020, 132–136  mathnet  crossref
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