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I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.

1)How is it connected to the plethysms in Schur functions.

2)Do the multiplicities of the irreducible decomposition and the corresponding schur expansion match? (reference?)

Is there a simple way to understand the plethysm in regular representation of $S_n$. I tried to read around and found that involves 'wreath products' I don't fully understand this, as my background is in physics.

3)Can any one explain in 'simple terms' using perhaps a simple example $s_2 \circ s_2$ (as representation of $S_4$)?

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  • $\begingroup$ This seems close to your previous question: mathoverflow.net/questions/209536/…. But I don't quite understand the question because I don't know what you mean by "plethysm [...] as a module in the regular representation of the symmetric group". $\endgroup$ – Sam Hopkins Nov 23 '15 at 19:47
  • $\begingroup$ Here is an explanation of $s_2\circ s_2$ in ‘simple terms‘. Here we have permutation representations: $s_2$ is the characteristic of $S_2$ acting on the single set $\{1,2\}$ by permutating 1 and 2. This is the trivial representation, since $S_2$ is acting on a single element. The plethysm $s_2\circ s_2$ is the characteristic of $S_4$ acting on partitions of $\{1,2,3,4\}$ into two blocks of size 2; i.e. acting on the three partitions $\{\{1,2\},\{3,4\}\},$ $\{\{1,3\},\{2,4\}\},$ and $\{\{1,4\},\{2,3\}\},$ by permuting 1, 2, 3, and 4. $\endgroup$ – Ira Gessel Nov 23 '15 at 20:14
  • $\begingroup$ Thanks for your answer that makes things clearer.. @SamHopkins perhaps my use of the word 'module' is misleading I wanted say, what would be the 'vector space' associated to the regular representation corresponding to the plethysm $s_\lambda \circ s_\mu$. For example, for irreducible representation $\lambda$ it would be $CS_n c_\lambda$ where $c_\lambda$ is the young symmetriser. $\endgroup$ – phys Nov 24 '15 at 17:25

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