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

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

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
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
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
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
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
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
7. Belonogov V.A., “On character tables and abstract structure of finite groups”, Character Theory of Finite Groups, Contemporary Mathematics, 524, 2010, 1–10
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
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