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Algebra Logika, 2005, Volume 44, Number 1, Pages 24–43 (Mi al69)  

This article is cited in 8 scientific papers (total in 8 papers)

Zeros in tables of characters for the groups $S_n$ and $A_n$

V. A. Belonogov

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: In the representation theory of symmetric groups, for each partition $\alpha$ of a natural number $n$, the partition $h(\alpha)$ of $n$ is defined so as to obtain a certain set of zeros in the table of characters for $S_n$. Namely, $h(\alpha)$ is the greatest (under the lexicographic ordering $\leq$) partition among $\beta\in P(n)$ such that $\chi^\alpha(g_\beta)\ne0$. Here, $\chi^\alpha$ – is an irreducible character of $S_n$, indexed by a partition $\alpha$, and $g_\beta$ is a conjugacy class of elements in $S_n$, indexed by a partition $\beta$. We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition $\alpha\in P(n)$ the partition $f(\alpha)$ of $n$ is defined so that $f(\alpha)$ is greatest among the partitions $\beta$ of $n$ which are opposite in sign to $h(\alpha)$ and are such that $\chi^\alpha(g_\beta)\ne0$ (Thm. 1). Also, for any self-associated partition $\alpha$ of $n>1$, we construct a partition $\tilde f(\alpha)\in P(n)$ such that $\tilde f(\alpha)$ is greatest among the partitions $\beta$ of $n$ which are distinct from $h(\alpha)$ and are such that $\chi^\alpha(g_\beta)\ne0$ (Thm. 2).

Keywords: symmetric group, table of characters, partition

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English version:
Algebra and Logic, 2005, 44:1, 13–24

Bibliographic databases:

UDC: 512.54
Received: 05.04.2004

Citation: V. A. Belonogov, “Zeros in tables of characters for the groups $S_n$ and $A_n$”, Algebra Logika, 44:1 (2005), 24–43; Algebra and Logic, 44:1 (2005), 13–24

Citation in format AMSBIB
\Bibitem{Bel05}
\by V.~A.~Belonogov
\paper Zeros in tables of characters for the groups $S_n$ and~$A_n$
\jour Algebra Logika
\yr 2005
\vol 44
\issue 1
\pages 24--43
\mathnet{http://mi.mathnet.ru/al69}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2165871}
\zmath{https://zbmath.org/?q=an:1096.20015}
\transl
\jour Algebra and Logic
\yr 2005
\vol 44
\issue 1
\pages 13--24
\crossref{https://doi.org/10.1007/s10469-005-0002-3}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-17444379009}


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    This publication is cited in the following articles:
    1. V. A. Belonogov, “Zeros in Tables of Characters for the Groups $S_n$ and $A_n$. II”, Algebra and Logic, 44:6 (2005), 357–369  mathnet  crossref  mathscinet  zmath
    2. V. A. Belonogov, “Irreducible characters with equal roots in the groups $S_n$ and $A_n$”, Algebra and Logic, 46:1 (2007), 1–15  mathnet  crossref  mathscinet  zmath  isi
    3. V. A. Belonogov, “Certain pairs of irreducible characters of the groups $S_n$ and $A_n$”, Proc. Steklov Inst. Math. (Suppl.), 257, suppl. 1 (2007), S10–S46  mathnet  crossref  mathscinet  elib
    4. V. A. Belonogov, “The young diagrams of a pair of irreducible characters of $S_n$ with the same zero set on $S^\varepsilon_n$”, Siberian Math. J., 49:5 (2008), 784–795  mathnet  crossref  mathscinet  isi
    5. V. A. Belonogov, “On irreducible characters of the group $S_n$ that are semiproportional on $A_n$ or $S_n\setminus A_n$. I”, Proc. Steklov Inst. Math. (Suppl.), 263, suppl. 2 (2008), S150–S171  mathnet  crossref  zmath  isi  elib
    6. V. A. Belonogov, “On irreducible characters of the group $S_n$ that are semiproportional on $A_n$ or $S_n\setminus A_n$. VI”, Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S14–S35  mathnet  crossref  isi  elib
    7. Belonogov V.A., “On character tables and abstract structure of finite groups”, Character Theory of Finite Groups, Contemporary Mathematics, 524, 2010, 1–10  crossref  mathscinet  zmath  isi
    8. V. A. Belonogov, “O neprivodimykh kharakterakh gruppy $S_n$, poluproportsionalnykh na $A_n$ ili na $S_n\setminus A_n$. VII”, Tr. IMM UrO RAN, 17, no. 1, 2011, 3–16  mathnet  elib
  • Алгебра и логика Algebra and Logic
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