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For dual finite-dimensional vector spaces $V,V^*$ the "mixed exterior algebra" $$\textstyle\bigwedge(V^*,V)=\bigwedge V^*\otimes\bigwedge V$$ is a powerful tool for studying linear transformations of $V$ (see here for a brief intro and example). It's also possible to use the algebra $$\textstyle\bigwedge(V^*\oplus V)=\bigwedge V^*\mathbin{\widehat{\otimes}}\bigwedge V$$ which only differs by sign factors. The power stems in part from the fact that the diagonal subalgebra -- which contains the linear transformations of $V$ as well as their exterior powers -- is commutative under the mixed exterior product, and there are a number of identities connecting this product to the usual (non-commutative) composition product.

I'm curious whether the "mixed symmetric algebra" $$\textstyle\bigvee(V^*\oplus V)=\bigvee V^*\otimes\bigvee V$$ has any applications to linear algebra, or anything else.

I've searched for this but haven't found anything yet, possibly because I'm using the wrong terminology. Any pointers or references are appreciated.

blargoner
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    It can be thought of as the algebra of functions on the cotangent bundle $T^{\ast}(V) \cong V \oplus V^{\ast}$, and accordingly it is one of the simplest examples of a Poisson algebra. It is also the associated graded of the Weyl algebra on $V$. I'm not aware of applications to linear algebra per se, though. – Qiaochu Yuan Oct 25 '23 at 04:10

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