Vtyjkyuk

Consider symmetric group $S_n$.

Let $lambda=(lambda_1,lambda_2,cdots,lambda_k)$ be a partition of $n$ with $lambda_1gelambda_2cdots$ (so $lambda_1+lambda_2+cdots + lambda_k=n$).

Let $mu=(mu_1,mu_2,cdots,mu_r)$ be another partition of $n$.

In the representation theory of symmetric group, one important relation on partitions comes is the dominance relation:
$$lambda trianglerighteq mu mbox if (lambda_1 + lambda_2 + lambda_i) ge (mu_1 + mu_2 + cdots + mu_i) mbox for all ige 1.$$

Let $lambda,mu$ be two partitions of $n$ as above. Then the partition of $lambda$ determines a conjugacy class of subgroups of $S_n$, isomorphic to
$$S_lambda_1times S_lambda_2times cdots times S_lambda_k $$
where

• $S_lambda_1$ is symmetric group on first $lambda_1$ symbols $1,2,cdots,lambda_1$,




• $S_lambda_2$ is the permutation group on next $lambda_2$ symbols, and so on.

Question. If $lambdatrianglerighteq mu$, then is it true that some conjugate of $S_mu_1times S_mu_2times cdots times S_mu_r$ is contained in some conjugate of $S_lambda_1times S_lambda_2times cdots times S_lambda_k$? (conjugates taken by may be different permutations in $S_n$ for these two subgroups.)

No: $(4,2)$ dominates $(3,3)$ but the order of $S_3times S_3$ is divisible by $3^2$ while that of $S_4×S_2$ is not. – Stephen yesterday