Question

Let $V$ be a finite dimensional vector space over some field (say, $\mathbb C$). Consider the set $GLI(V)$ of all linear isomorphisms between subspaces of $V$. This is a monoid under natural multiplication (in fact an inverse monoid). Its elements can be represented by triples: two elements of the Grassmannian of $V$ of degree $k\le n$ representing the domain and the range, and a non-singular $k\times k$-matrix representing the map. I am interested in developing a theory of representations of finite inverse monoids (pseudogroups) in $GLI(V)$. What is the structure of $GLI(V)$ from the algebraic geometry or geometric topology point of view?

Edit: It looks like the question is not completely clear. For comparison, if somebody gives me a group and asks what can I say about it, I would try to decide whether the group is finite or infinite, solvable or not, hyperbolic or not, what is the derived subgroup and the lower central series, is it residually finite and what is the profinite competion, etc. I want a similar analysis of $GLI$ (but from the algebraic geometry point of view). One of the goals is to study representation varieties of groupoids (=pseudogroups, inverse semigroups). These varieties are complicated even for easy finite groupoids. The starting point would be to understand $GLI$ itself.

Answer 1

Some small comments.

Let $n=dim(V)$, so I'll think of $V$ as $\mathbb R^n$, then as a space, $GLI(V)$ you could think of as

$$ V_{n,k} \times_{O_k} V_{n,k} $$

where $V_{n,k}$ is the Stiefel manifold of orthonormal $k$-frames in the vector space $V$. i.e. this is the space $V_{n,k}^2$ mod the diagonal action of $O_k$.

So you could view it as a bundle over $G_{n,k}^2$ with fiber $O_k$, or as a bundle over $G_{n,k}$ with fiber $V_{n,j}$. $G_{n,k}$ is the Grassmannian of $k$-dimensional subspaces of $V$.

The map $V_{n,k} \times_{O_k} V_{n,k}$ to $GLI(V)$ is given by sending a pair $(A,B) \in V_{n,k} \times V_{n,k}$ to:

The span of $A$, the span of $B$ and the corresponding linear isometry represented by $B\circ A^{-1}$ where we think of $A$ and $B$ as representing isometric embeddings $\mathbb R^k \to \mathbb R^n$.

So the homotopy-type of this space is at least fairly reasonable as $V_{n,k}$ is highly connected. I think this bundle likely has a lot of other nice properties lurking near the surface. Is this the kind of thing you're asking about? In particular as a bundle over $G_{n,k}^2$ you'd have some nice Schubert-cell type constructions. i.e. you could view $V_{n,k} \times_{O_k} V_{n,k}$ as the "diagonal" $V_{n,k}$ subspace union "Schubert cells".

Answer 2

I'm no expert, but paging through "Linear Algebraic Monoids" by Lex Renner suggests to me that it has a lot of information you could use.