I am given the pair $(n, \lambda)$ where $\lambda$ is a partition of $n$ such that 6 is not a part in $\lambda$. I am told to let $\lambda^*$ represent the partition of $n$ conjugate to $\lambda$. Now we are supposing $(n, \lambda)$ has the following property: there exists a $\theta \in S_n$, the set of permutations of $\{1,2, \dots, n\}$, and $\theta^* \in S_n$ such that both $\theta, \theta^*$ have order 6, $\theta$ has cycle-structure $\lambda$, and $\theta^*$ has cycle-structure $\lambda^*$. I am asked to determine the possible values of n. I am not sure where to start here. Any help would be great.
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$\begingroup$ Yes, thank you, I have edited the post. $\endgroup$– Ethan DeakinsCommented Dec 6, 2019 at 1:30
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Hint: the partition must consist of some parts of size $2$ and $3$ (in order for the associated permutation to have order $6$), it could also have some parts of size $1$.
The dual partition is obtained by reflecting in the diagonal, & this tableau will need to only have parts of size $1,2,3$ or $6$ (If it does not have $6$, then it must have at least one $2$ and $3$ (in order for the associated permutation to have order $6$)).
What shapes are possible ?
answered Dec 6, 2019 at 1:49
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$\begingroup$ Are you talking about the overall shapes? $\endgroup$ Commented Dec 6, 2019 at 1:56
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$\begingroup$ I'm not sure I understand your question. ... I guess so. $\endgroup$ Commented Dec 6, 2019 at 2:01
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$\begingroup$ Haha I didn't understand your reference to "what shapes are possible." I guess we are both a tad bit confused. $\endgroup$ Commented Dec 6, 2019 at 2:12
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$\begingroup$ I bet the confusion is about identifying partitions with shapes of tableaux. Then there's also whether you draw partitions/tableaux the "French" way (as here) or the "English" way (parts correspond to rows, largest at the top). $\endgroup$ Commented Dec 6, 2019 at 2:18
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$\begingroup$ I have $4$ possible solutions written in my note book ... but there could be more ... let me know what you think. $\endgroup$ Commented Dec 6, 2019 at 2:19