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daveh
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I see no reason in general if $\operatorname{Ext}_P^1(4,1) \neq 0$ that, on restriction to R the four dimensional simple can't drop down and have a 2-dimensional submodule. I.e. On restriction to R it looks like:

\begin{equation}\begin{array}{c} 2\\ 2 \end{array} \oplus 1 \end{equation}

If you knew on restriction to KR the {\em only}only irreducible submodule was two dimensional then this would be ruled out.

I see no reason in general if $\operatorname{Ext}_P^1(4,1) \neq 0$ that, on restriction to R the four dimensional simple can't drop down and have a 2-dimensional submodule. I.e. On restriction to R it looks like:

\begin{equation}\begin{array}{c} 2\\ 2 \end{array} \oplus 1 \end{equation}

If you knew on restriction to KR the {\em only} irreducible submodule was two dimensional then this would be ruled out.

I see no reason in general if $\operatorname{Ext}_P^1(4,1) \neq 0$ that, on restriction to R the four dimensional simple can't drop down and have a 2-dimensional submodule. I.e. On restriction to R it looks like:

\begin{equation}\begin{array}{c} 2\\ 2 \end{array} \oplus 1 \end{equation}

If you knew on restriction to KR the only irreducible submodule was two dimensional then this would be ruled out.

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daveh
  • 136
  • 1
  • 7

I see no reason in general if $\operatorname{Ext}_P^1(4,1) \neq 0$ that, on restriction to R the four dimensional simple can't drop down and have a 2-dimensional submodule. I.e. On restriction to R it looks like:

\begin{equation}\begin{array}{c} 2\\ 2 \end{array} \oplus 1 \end{equation}

If you knew on restriction to KR the {\em only} irreducible submodule was two dimensional then this would be ruled out.