A row removal theorem for the Ext$^1$ quiver of symmetric groups and Schur algebras

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 ${\rm Ext}^1_{\Sigma_r}(D_\lambda,D_\lambda)=0$ 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.