Suppose that $V$ is a finite dimensional $\mathbb Q$-vector space. To each subspace $S$ of dimension $k$, we can associate the line from the origin of $\Lambda^k(V)$ through the point $s_1\wedge \ldots \wedge s_k$ where $\{s_1,\ldots,s_k\}$ is an arbitrary basis for $S$. This is essentially representing subspaces via the Plucker embedding.
Now, suppose that I have a pair of elements $\alpha \in \Lambda^k(V)$ and $\beta\in \Lambda^{k'}(V)$ corresponding to some subspaces $S$ and $S'$. These elements are given in the usual coordinates on the exterior algebra relative to some basis for $V$.
Is there a simple way to compute the element of the exterior algebra corresponding to the subspace $S+S'$?
If $S\cap S'=0$, then $S+S'$ is represented simply by $\alpha\wedge\beta$, but it's not clear how to do a similar calculation if $S$ and $S'$ intersect non-trivially
While there is a reasonably straightforwards way to do this - in particular, given $\alpha$ and $\beta$, we can find a basis for each of the $S$ and $S'$ and then, using the exterior algebra, we can relatively easily find a basis for $S+S'$ and then pass back into the exterior algebra. However, this seems rather inelegant, especially given the really nice formula that exists if the spaces intersect trivially.
It is also trivial to do this, using Hodge duality, if one were able to compute an element representing $S\cap S'$ from $\alpha$ and $\beta$. This problem seems analogous to finding a $\gcd$ of $\alpha$ and $\beta$ in $\Lambda(V)$ considered as a ring.
Mainly, I'm wondering about this since I'm trying to mechanistically do various computations on some arrangements of codimension $2$ subspaces and have found the exterior algebra to be an good tool for similar computations on arrangements of hyperplanes, but moving from codimension $1$ to codimension $2$ requires knowing how to carry out this computation in greater generality.
