A row removal theorem for the Ext¹ quiver of symmetric groups and Schur algebras

Abstract

In 1981, G. D. James proved two theorems about the decomposition matrices of Schur algebras involving the removal of the first row or column from a Young diagram. He established corresponding results for the symmetric group using the Schur functor. We apply James’ techniques to prove that row removal induces an injection on the corresponding Ext 1 \operatorname {Ext}^1 between simple modules for the Schur algebra. We then give a new proof of James’ symmetric group result for partitions with the first part less than p p . This proof lets us demonstrate that first-row removal induces an injection on Ext 1 ^1 spaces between these simple modules for the symmetric group. We conjecture that our theorem holds for arbitrary partitions. This conjecture implies the Kleshchev-Martin conjecture that Ext Σ r 1 ( D λ , D λ ) = 0 \textrm {Ext}^1_{\Sigma _r}(D_\lambda ,D_\lambda )=0 for any simple module D λ D_\lambda in characteristic p ≠ 2 p \neq 2 . The proof makes use of an interesting fixed-point functor from Σ r \Sigma _r -modules to Σ r − m \Sigma _{r-m} -modules about which little seems to be known.