Geodesic
On extensions and branching rules for modules close to completely splittable
Sbornik. Mathematics, Tome 196 (2005) no. 8, pp. 1209-1249

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The modules $D^\lambda{\downarrow}_{\Sigma_{n-1}}$ and $D^\lambda{\uparrow}^{\Sigma_{n+1}}$ are described for certain simple $K\Sigma_n$-modules $D^\lambda$ (the completely splittable ones or close to them), where $K$ is a field of characteristic $p>0$ and $\Sigma_n$ is the symmetric group of degree $n$. This result is based on an upper bound for the dimensions of the $\operatorname{Ext}^1$-spaces between certain simple modules. 
@article{SM_2005_196_8_a4,
     author = {V. V. Shchigolev},
     title = {On extensions and branching rules for modules close to completely splittable},
     journal = {Sbornik. Mathematics},
     pages = {1209--1249},
     publisher = {mathdoc},
     volume = {196},
     number = {8},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_8_a4/}
}
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V. V. Shchigolev. On extensions and branching rules for modules close to completely splittable. Sbornik. Mathematics, Tome 196 (2005) no. 8, pp. 1209-1249. http://geodesic.mathdoc.fr/item/SM_2005_196_8_a4/