# Determining explicitly the action on the exterior products of a vector space

Let $V$ be a 2-dimensional complex vector space with basis $e_1,e_2$. Consider the endomorphism $f:V\to V$ given by $f(e_1) = e_2$ and $f(e_2) = -e_1$ with matrix $$\left( \begin{matrix} 0 & -1 \\ 1 & 0 \end{matrix} \right).$$

Let $d$ be a positive integer and let $j=0,\ldots,2d$. How does the matrix of $\Lambda^j f^{\oplus d} : \Lambda^j V^{\oplus d} \to \Lambda^j V^{\oplus d}$ look like?

Here $f^{\oplus d}:V^{\oplus d} \to V^{\oplus d}$ denotes the direct sum of the morphism $f$ with itself $d$-times.

I can do it by hand for $d=2$. But it gets messy for bigger $d$. There should be a simple pattern.

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