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Jun 24, 2020 at 7:36 comment added Maithreya Sitaraman David Ridout kindly informed me that the answer is in: arxiv.org/pdf/1605.05159.pdf. This is Theorem 3.10. The notation of that theorem is as follows: $S_k$ is the standard representation, $P_k$ is the projective cover of the irreducible obtained by quotienting by the radical (the kernel of the bilinear form). $B^l_k$ and $T^{l'}_k$ were new to me, and are defined in Section 3.1 (page 20).
May 8, 2020 at 22:03 history asked Maithreya Sitaraman CC BY-SA 4.0