We construct and decompose the Eisenbud-Fløystad-Weyman complex of type (0,2,3,6) over a polynomial ring in 3 variables. The ring can be identified with $Sym(E)$, where $E$ is a complex vector space of dimension 3. The ring and the complex carry an action of $SL(E)$.
The complex is constructed using the package PieriMaps. For more information on these complexes, we invite the reader to consult the documentation of that package and the accompanying article.
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The matrix above is a presentation of the module whose resolution is the complex in the title. The rows of the matrix are indexed by standard tableaux of shape $(2,2)$ and entries from $\{0,1,2\}$. The weight of one such tableau is $m_0*L_0+m_1*L_1+m_2*L_2$, where $m_i$ is the multiplicity of $i$ in the tableau. The command below generates all the weights.
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Next we generate the resolution and obtain its decomposition.
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We conclude that with the action of $SL(E)$ the complex has the following structure: $$S_{2,2} E \otimes R \leftarrow S_{4,2} E \otimes R(-2) \leftarrow S_{4,3} E \otimes R(-3) \leftarrow E \otimes R(-6) \leftarrow 0$$ where $S_\lambda$ denotes the Schur functor associated with the partition $\lambda$.