G=schurComplex(lambda,F)This function computes the Schur complex associated to a partition $\lambda$ and a bounded complex $F_{\bullet}$ of finitely-generated free modules over a commutative ring.
The user inputs the partition $\lambda$ as a list and the chain complex $F_{\bullet}$.
In the following example, the complex F is the free resolution of the ideal $(x,y,z)\subset \mathbb{Z}[x,y,z]$, and lambda is the partition $(1,1)$ in the form of a List. In this case, the Schur complex G is the second exterior power of F.
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As a second example, we consider the ring of polynomial functions $R=\mathbb{Q}[x_{i,j}]$ on the space of 2 x 4 generic matrices. We set the complex F to be the map $R^4\to R^2$ given by the generic matrix $(x_{i,j})$. We compute the third symmetric power G of F, in which case lambda is the partition $(3)$. By Weyman "Cohomology of Vector Bundles and Syzygies", Exercise 6.34(d), the Schur complex G is exact except in degree zero. We verify this by computing the Hilbert series of each homology module of G.
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We compute a third example.
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The object schurComplex is a function closure.