Ok, today we start our march towards Schubert Calculus.  Before we start, we’ll review the Grassmannian variety itself, because it’s central to the story.  A lot of this will consist of setting up notation, and there will be two different notations for the “same” Grassmannians, so the notation will describe how we’re going to think of the object.  For this series, we’ll assume that everything is done over $\mathbb{C}$, because we’re going to use a bit of algebraic topology along the way.

Let $V$ be a vector space of dimension $m$.  We define for $0\leq d\leq m$ the set $Gr^d(V)$ to be the set of subspaces of $V$ with codimension $d$ and by $Gr_d(V)$ the set of subspaces of dimension $d$.  We’re mostly going to use the first, though the second is more intuitive.  A second way to think of $Gr^dV$ is as the set of $d$-dimensional quotients of $V$.  Some examples:

$Gr^1(V)=\mathbb{P}^*(V)$, the space of all linear functionals on $V$.

$Gr^{m-1}(V)=\mathbb{P}(V)$, the space of all lines in $V$.

This is important: projective space is a special case of a Grassmannian.

Now, the Grassmannian is a projective variety.  This is due to the Plucker Embedding, described in this old post.  Now, it’s worth noting that the details on the Plucker embedding tell us that a Grassmannian is always nonsingular, and is always the intersection of quadrics.  In particular, for $Gr^2(\mathbb{C}^4)$, the first Grassmannian to not be a projective space, it will be a quadric hypersurface in $\mathbb{P}^5$.

The example above will be dimension four.  Let’s see what the dimension of a Grassmannian will be in general: let $\Gamma\subset V$ be a subspace of codimension $k$, that is, a point in $Gr^k(V)$ and let $V$ have dimension $n$.  Then it corresponds to $\omega\in\bigwedge^kV^*\cong \bigwedge^{n-k}V$, and we get a dense open subset of the Grassmannian where $\omega$ is nonvanishing.  However, this will just be the set of $k$-dimensional subspaces that are complementary to $\Gamma$, and we can view this as the graph of a map $\Gamma\to V/\Gamma$, and so the open set is isomorphic to $\hom(\Gamma,V/\Gamma)$, which is an affine space of dimension $k(n-k)$, which then must be the dimension of the Grassmannian.

Now, a bit of fun.  Way back, I talked about the Veronese map.  It broke down to looking at the $d$th graded piece of the coordinate ring for $\mathbb{P}^n$.  Well, we can generalize this to Grassmannians!

Let $S=\mathbb{C}[x_0,x_1,\ldots,x_n]$, the homogeneous coordinate ring of $\mathbb{P}^n$, and let the $d$th graded piece be denoted by $S_d$.  Now, let $\Lambda\subset\mathbb{P}^n$.  It has ideal $I(\Lambda)$ and is a point of $Gr^{n-k}(\mathbb{C}^{n+1})$.  (From here on out, when the vector space is $\mathbb{C}^m$, we’ll just write $m$ for convenience.)

Now, look at the degree $d$ part.  That is, we have $I_d(\Lambda)\subset S_d$.  Now, $I_d(\Lambda)$ is a codimension $\binom{k+d}{d}$ subspace of $\binom{n+d}{d}$ dimensional space, so it gives us a point of $Gr^{\binom{k+d}{d}}\left(\binom{n+d}{d}\right)$.

Now, I’m not sure how USEFUL this $d$-uple embedding of Grassmannians is in cases other than projective space, but it’s nice how it generalized.  Next time, we’ll put together some nice subvarieties of the Grassmannian, called Schubert Varieties, which will be essential to our study of Schubert Calculus.