Covering Numbers of Some Irreducible Characters of the Symmetric Group

The covering number of a non-linear character χ of a finite group G is the least positive integer k such that every irreducible character of G occurs in χᵏ. We determine the covering numbers of irreducible characters of the symmetric group Sₙ indexed by certain two-row partitions (and their conjugates), namely $(n-2,2)$ and $((n+1)/2, (n-1)/2)$ when n is odd. We also determine the covering numbers of irreducible characters indexed by certain hook-partitions (and their conjugates), namely (n-2,1²), the almost self-conjugate hooks (n/2+1, 1n/2-1) when n is even, and the self-conjugate hooks ((n+1)/2, 1(n-1)/2) when n is odd.