Does someone of you know an easy representation of $S_3$ that is reducible and indecomposable in characteristic 2?
2026-05-03 15:14:26.1777821266
Representation of $S_3$ that is reducible and indecomposable
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The standard example of a reducible indecomposable representation in characteristic $p$ is the two dimensional representation of the cyclic group $C_p$ that sends the generator to the matrix $A=\begin{pmatrix}1 &1\\0&1\end{pmatrix}$. Since $S_3$ has $C_2$ as a quotient, we can compose the quotient map with the representation of $C_2$ to get the desired representation of $S_3$.
It wil send every transposition to $A$ and every other element to the identity.