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I need to construct a representation of a symmetric group $S_n$, in which a character of the conjugacy class $(n)$ (a class of permutations, which are cycles of a maximal possible length $n$) would be different from all other characters. I am looking for a representation of a minimal possible dimensionality. It can be reducible. Characters of other classes can coincide.

Would be grateful for ideas.

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  • $\begingroup$ Do you need a proof that the dimension is minimal, or just a construction that works for all $n$ and with small dimension for practical reasons? Obv the former is ideal, just thinking that the latter may be a lot more feasible $\endgroup$
    – Kevin Casto
    Commented Dec 9, 2022 at 16:30
  • $\begingroup$ What I need is an explicit construction together with a proof that it indeed provides the answer. I actually need it for some numeric computation $\endgroup$
    – V. Asnin
    Commented Dec 9, 2022 at 16:40
  • $\begingroup$ No answers, but some data from what I think was an exhaustive search: S_3: (2,1) distinguishes (3), dimension is 2; S_4: (2,1,1,) distinguishes (4), dimension is 3; S_5: (2,1,1,1) distinguishes (5), dimension is 4; S_6: (2,2,2) $\oplus$ sign distinguishes (6), dimension is 6. $(2,1^n)$ was briefly looking promising, but it went bad quickly. S_7 I'm not finding anything with my method of staring at character tables while my students take a final. $\endgroup$
    – coolpapa
    Commented Dec 9, 2022 at 18:21
  • $\begingroup$ Thank you for your answer. I tried for some time to write examples explicitly, but was unable to see any clear pattern. That is why I am asking this question. $\endgroup$
    – V. Asnin
    Commented Dec 9, 2022 at 18:49
  • $\begingroup$ I checked the case of $S_7$, there is no representation of dimension 7 with the required properties, so the assumption of coolpapa doesn't work $\endgroup$
    – V. Asnin
    Commented Dec 10, 2022 at 7:04

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