On extensions of simple modules over symmetric and algebraic groups
Let \(G\) be a connected reductive algebraic group \(G\) over an algebraically closed field \(F\) of characteristic \(p>0\). Further, let \(L(\lambda)\) and \(L(\mu)\) be simple \(G\)-modules corresponding to dominant weights \(\lambda\) and \(\mu\). Under the standard partial ordering on weights, if \(\mu\) does not strictly dominate \(\lambda\), the group of extensions of \(L(\lambda)\) by \(L(\mu)\) is isomorphic to the group of homomorphisms from the radical of \(\Delta(\lambda)\) to \(L(\mu)\). (Here \(\Delta(\lambda)\) is the Weyl module with highest weight \(\lambda\).) That is, \(\text{Ext}^1(L(\lambda),L(\mu))\cong\text{Hom}_G(\text{rad }\Delta(\lambda),L(\mu))\). The main result of this paper is an analogous isomorphism for extensions of \(F\Sigma_n\)-modules. For the symmetric group, \(\lambda\) and \(\mu\) are partitions of \(n\) and the Weyl module is replaced by the corresponding Specht module. The analogous isomorphism is shown to hold when \(p>2\), \(\lambda\) and \(\mu\) are partitions of \(n\) with at most \(p-1\) parts, and \(\lambda\) does not strictly dominate \(\mu\) (relative to the standard partial ordering of partitions). Note here the reversal of the dominance condition. The authors give examples to show that the conditions on \(p\) and the partitions are in fact necessary. The isomorphism is then used to make computations of extensions between simple modules whose corresponding partitions are two row partitions of \(n\). In the process, some computations of extensions are also made for algebraic groups.
scientific article; zbMATH DE number 1400117