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It is well known that the irreducible representations of the Lie algebra $\mathfrak{sl}(n)$ are symmetric powers of a vector space of dimension $n$. This can be viewed for instance using Young tableaux. I wonder if a similar result exists for the special linear supergroup $\mathfrak{sl}(n|m)$. I tried to look into the literature about it and did not find, and it seems that classical proof doesn't work (That is, the irreducible representations are symmetric powers of a super vector space of dimension $n|m$). It would be quite interesting if there could be other examples of representations besides the classical ones.

Does anyone have any insight?

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    $\begingroup$ Some of the irreducibles for $\mathfrak{sl}(n)$ are symmetric powers, but most are not. $\endgroup$
    – Dave Benson
    Commented Sep 27 at 12:16
  • $\begingroup$ Do you have any reference for this? $\endgroup$
    – User43029
    Commented Sep 27 at 12:37
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    $\begingroup$ Try Jim Humphreys' "Introduction to Lie algebras and representation theory". At least in characteristic zero, irreducibles for $\mathfrak{gl}(n)$ are given by Schur functors, labelled by partitions with at most $n$ rows, and the partitions with one row correspond to the symmetric powers of the natural module. $\endgroup$
    – Dave Benson
    Commented Sep 27 at 12:44
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    $\begingroup$ I strongly suggest you learn the $\mathfrak{gl}(n)$ theory properly before trying to understand $\mathfrak{sl}(n|m)$. $\endgroup$
    – Dave Benson
    Commented Sep 27 at 13:00

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