Abstract
The representation category mathcal {A} = Rep(G,epsilon ) of a supergroup scheme G has a largest proper tensor ideal, the ideal mathcal {N} of negligible morphisms. If we divide mathcal {A} by mathcal {N} we get the semisimple representation category of a pro-reductive supergroup scheme Gred. We list some of its properties and determine Gred in the case GL(m|1).
Highlights
A fundamental fact about finite-dimensional algebraic representations of a reductive group over an algebraically closed field k of characteristic 0 is complete reducibility: Every representation decomposes into a direct sum of irreducible representations
Many standard techniques from Lie theory do not work for representations of supergroups
A lot of progress has been made on representations of special supergroups such as GL(m|n) and OSp(m|2n), many classical questions are still open, most notably the tensor product decompositon of two irreducible representations
Summary
A fundamental fact about finite-dimensional algebraic representations of a reductive group over an algebraically closed field k of characteristic 0 is complete reducibility: Every representation decomposes into a direct sum of irreducible representations. The question should be modified as follows: We should study the subcategory in Rep(Gred , ) generated by the images ω(L(λ)) of the irreducible representations of G. To determine this subcategory would amount to determine the tensor product decomposition of irreducible representations up to superdimension 0 and would give a parametrization of the indecomposable summands of non-vanishing superdimension. From the classification it is easy to determine the irreducible objects of Rep(Gred , ) in Lemma 5.2 We compute their tensor product decomposition in Theorem 5.11. To determine the subgroup of Gred corresponding to the irreducible representations is already very difficult for GL(m|n) and even more so for the supergroups OSp(m|2n) and P (n)
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