Specht Modules of Trivial Source and the Endomorphism Ring of the Lie Module
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The source of a Specht module is not generally known, unless it is a projective or irreducible module and in this case the source is trivial. Are these the only Specht modules with trivial source? To address this question, we classify the Specht modules of FΣ2p and FΣ2p+1 with trivial source for p≥3. It turns out that a Specht module of FΣ2p and FΣ2p+1 corresponding to a weight two partition has trivial source if and only if it is irreducible. On the other hand, there are Specht modules corresponding to weight one partitions that are not irreducible but have trivial source, these can be identified by ordering the partitions (using the lexicographic ordering) in the block to which they belong. The result concerning Specht modules corresponding to weight one partitions with trivial source is not unique to FΣ2p and FΣ2p+1 , but it generalizes to Specht modules of FΣn for any 5≤p≤n.