The conjugacy classes are indexed by partitions like $[6]$ and $[2,2,2] = [2^3]$. What happens when you take products of two such elements? I saw in a paper, $$[6][2^3] = 6[3,1^3] + 8[2^2,1^2]+5[5,1]+4[4,2]+3[3^2]$$ Which I take to mean if you multiply a 6-cycle (abcdef) and a product of disjoint 3-cycles (pq)(rs)(tv), you can get

• a three-cycle (abc),
• two two-cycles (ab)(cd),
• a five-cycles (abcde),
• a four-cycles and a two-cycle (abcd)(ef),
• two three cycles (abc)(def)

With certain multiplicities. Is it predictable what kinds of conjugacy classes you get? And there an interpretation of this as the intersection cohomology of some moduli space?