Let $\sigma$ and $\tau$ be two permutations in $S_n$ with partitions $\lambda$ and $\mu$ as their cycle type.

What is the cycle type of the product $\sigma \tau$ in terms of $\lambda$ and $\mu$?

Thank you.

  • $\begingroup$ I don't think you can tell. For example, $(12)(12) = 1$, but $(12)(23) = (123)$. Both examples are products of two transpositions, but the results have different cycle types. $\endgroup$ – Nick Aug 3 at 14:43
  • $\begingroup$ hmm. ok. Thanks. Some partial results with some assumptions on $\sigma$ and $\tau$? $\endgroup$ – GA316 Aug 3 at 15:24

The only intersting case is the one where the partitions are disjoint , i.e. $c$ (not a singleton) $\in \lambda \Rightarrow c \notin \mu$ and vice versa. The partition of the product is then the union of the partitions, example: $\sigma = (1,2)(5,6,7)$ and $\tau = (3,4)(8,9,10)$ then $\sigma\tau=\tau\sigma = (1,2)(3,4)(5,6,7)(8,9,10)$ in other cases the partitions all get mixed up.


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