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 tableus. 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 has any insight?

Thank you.

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?

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 tableus. I wonder if a similar result exists for the special linear supergroup $\mathfrak{sl}(n|m)$. I tried 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 has any insight?

Thank you.

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 tableus. 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 has any insight?

Thank you.