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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2005, Volume 44, Number 6, Pages 643–663 (Mi al135)

Zeros in Tables of Characters for the Groups $S_n$ and $A_n$. II

V. A. Belonogov

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

Abstract: Let $P(n)$ be the set of all partitions of a natural number $n$. In the representation theory of symmetric groups, for every partition $\alpha\in P(n)$, the partition $h(\alpha)\in P(n)$ is defined so as to produce a certain set of zeros in the character table for $S_n$. Previously, the analog $f(\alpha)$ of $h(\alpha)$ was obtained pointing out an extra set of zeros in the table mentioned. Namely, $h(\alpha)$ is greatest (under the lexicographic ordering $\le$) of the partitions $\beta$ of $n$ such that $\chi^\alpha(g_\beta)\ne0$, and $f(\alpha)$ is greatest of the partitions $\gamma$ of $n$ that are opposite in sign to $h(\alpha)$ and are such that $\chi^\alpha(g_\gamma)\ne0$, where $\chi^\alpha$ is an irreducible character of $S_n$, indexed by $\alpha$, and $g_\beta$ is an element in the conjugacy class of $S_n$, indexed by $\beta$. For $\alpha\in P(n)$, under some natural restrictions, here, we construct new partitions $h'(\alpha)$ and $f'(\alpha)$ of $n$ possessing the following properties.
(A) Let $\alpha\in P(n)$ and $n\geqslant 3$. Then $h'(\alpha)$ is identical is sign to $h(\alpha)$, $\chi^\alpha(g_{h'(\alpha)})\ne0$, but $\chi^\alpha(g_\gamma)=0$ for all $\gamma\in P(n)$ such that the sign of $\gamma$ coincides with one of $h(\alpha)$, and $h'(\alpha)<\gamma<h(\alpha)$.
(B) Let $\alpha\in P(n)$, $\alpha\ne\alpha'$, and $n\geqslant4$. Then $f'(\alpha)$ is identical in sign to $f(\alpha)$, $\chi^\alpha(g_{f'(\alpha)})\ne0$, but $\chi^\alpha(g_\gamma)=0$ for all $\gamma\in P(n)$ such that the sign of $\gamma$ coincides with one of $f(\alpha)$, and $f'(\alpha)<\gamma<f(\alpha)$. The results obtained are then applied to study pairs of semiproportional irreducible characters in $A_n$.

Keywords: symmetric group, alternating group, character table of a group

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English version:
Algebra and Logic, 2005, 44:6, 357–369

Bibliographic databases:

UDC: 512.54

Citation: V. A. Belonogov, “Zeros in Tables of Characters for the Groups $S_n$ and $A_n$. II”, Algebra Logika, 44:6 (2005), 643–663; Algebra and Logic, 44:6 (2005), 357–369

Citation in format AMSBIB
\Bibitem{Bel05} \by V.~A.~Belonogov \paper Zeros in Tables of Characters for the Groups~$S_n$ and~$A_n$.~II \jour Algebra Logika \yr 2005 \vol 44 \issue 6 \pages 643--663 \mathnet{http://mi.mathnet.ru/al135} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2213300} \zmath{https://zbmath.org/?q=an:1104.20012} \transl \jour Algebra and Logic \yr 2005 \vol 44 \issue 6 \pages 357--369 \crossref{https://doi.org/10.1007/s10469-005-0035-7} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-28644438768} 

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This publication is cited in the following articles:
1. V. A. Belonogov, “Irreducible characters with equal roots in the groups $S_n$ and $A_n$”, Algebra and Logic, 46:1 (2007), 1–15
2. 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
3. V. A. Belonogov, “Irreducible characters of the group $S_n$ that are semiproportional on $A_n$”, Algebra and Logic, 47:2 (2008), 77–90
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. Belonogov V.A., “On character tables and abstract structure of finite groups”, Character Theory of Finite Groups, Contemporary Mathematics, 524, 2010, 1–10
7. 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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