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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