A row removal theorem for the Ext$^1$ 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 $\operatorname {Ext}^1$ 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 $p$. This proof lets us demonstrate that first-row removal induces an injection on Ext$^1$ 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 $\textrm {Ext}^1_{\Sigma _r}(D_\lambda ,D_\lambda )=0$ for any simple module $D_\lambda$ in characteristic $p \neq 2$. The proof makes use of an interesting fixed-point functor from $\Sigma _r$-modules to $\Sigma _{r-m}$-modules about which little seems to be known.