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

V. A. Belonogov

Algebra i logika, Tome 44 (2005) no. 6, pp. 643-663

Résumé

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 $h'(\alpha)\gamma$. (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 $f'(\alpha)\gamma$. The results obtained are then applied to study pairs of semiproportional irreducible characters in $A_n$.

Détail

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