Covering Numbers of Some Irreducible Characters of the Symmetric Group

The covering number of a non-linear character of a finite group G is the least positive integer k such that every irreducible character of G occurs in ᵏ. 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, 1^n/2-1) when n is even, and the self-conjugate hooks ( (n+1) /2, 1^ (n-1) /2) when n is odd.