Grassmannians

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

We introduce a special variety and conclude our introduction to the basic objects of study in algebraic geometry.

By definition, projective space \(\mathbb{P}^n\) is the space of lines in \(\mathbb{A}^{n+1}\). The Grassmannian introduced in this post generalizes this: it is the space of \(k\)-dimensional linear subspaces of \(\mathbb{A}^n\).

Definition of Grassmannians

Definition 1 The set of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space \(V\) is called the Grassmannian \(\Gr(k, V)\) or \(\Gr(k, n)\).

Throughout this post we always assume that \(V\) is an \(n\)-dimensional space.

Of course, one must separately verify that this set carries a variety structure, but the essential point is that it not only admits one but also encodes the relative position of each \(k\)-plane in \(\mathbb{A}^n\) faithfully, so it behaves exactly as expected with little effort.

Example 2 For instance, \(\Gr(1, n+1)\) is the space of lines in the \((n+1)\)-dimensional vector space \(\mathbb{K}^{n+1}\), so by definition it is \(\mathbb{P}^n\). Once we define the variety structure on the Grassmannian, we will see that these two structures coincide exactly.

The simplest new example is \(\Gr(2,4)\), the collection of \(2\)-dimensional subspaces of a \(4\)-dimensional space. This example will serve as our toy model when working with Grassmannians.

As always, to endow a set with a variety structure it suffices to construct an affine cover and work affine-locally. To this end we make the following definition.

Definition 3 For each set of \(k\) indices \(I = \{i_1 < \cdots < i_k\}\), define the open set \(U_I\) by

\[U_I = \{W \in \Gr(k, V) \mid \text{projection } W \to \operatorname{span}(e_{i_1}, \ldots, e_{i_k}) \text{ is an isomorphism}\}\]

Fix a basis \(e_1,\ldots, e_n\) of \(V\), and let \(w_1,\ldots, w_k\) be vectors spanning \(W\). Then \(W\) is the row space of the \(k\times n\) matrix

\[\begin{pmatrix}w_1\\\vdots\\w_k\end{pmatrix}=\begin{pmatrix}w_{1,1}&w_{1,2}&\cdots &w_{1,n}\\ \vdots&\vdots&\ddots&\vdots\\ w_{k,1}&w_{k,2}&\cdots&w_{k,n}\end{pmatrix}\]

The condition defining \(U_I\) is then equivalent to the \(k\times k\) submatrix formed by the columns \(i_1,\ldots, i_k\) indexed by \(I\) being invertible. From this we obtain the following.

Proposition 4 Each \(U_I \cong \mathbb{A}^{k(n-k)}\).

Proof

Without loss of generality, we treat the case \(I = \{1, 2, \ldots, k\}\). For the \(k \times n\) matrix \(A\) representing \(W \in U_I\), the left \(k \times k\) minor is nonzero. By row operations we bring this minor to the form

\[A = \begin{pmatrix} I_k & B \end{pmatrix}\]

where \(B\) is a \(k \times (n-k)\) matrix. The \(k(n-k)\) entries of \(B\) then completely determine \(W\), and there are no constraints among them. Therefore \(U_I \cong \mathbb{A}^{k(n-k)}\).

As this proof shows, the coordinate system on \(U_I\) consists of \(k(n-k)\) free parameters. They correspond to the “non-trivial part” of the matrix representing \(W\): once the \(k \times k\) block determined by \(I\) is fixed to be the identity, the remaining \(k \times (n-k)\) block varies freely.

It is then immediate that every \(W\in \Gr(k,V)\) lies in some affine open cover. Moreover, the transition maps between \(U_I\) and \(U_J\) are evidently regular, so these charts endow \(\Gr(k,V)\) with a variety structure. Of course, proving quasi-projectivity requires an explicit projective embedding; nevertheless, the following is already clear.

Proposition 5 \(\dim \Gr(k, V) = k(n - k)\).

Plücker Embedding

We now show that the Grassmannian is a quasi-projective variety by defining an embedding into a suitable projective space.

Definition 6 The Plücker embedding \(\iota: \Gr(k, V) \to \mathbb{P}(\bigwedge^k V)\) is the map sending a \(k\)-dimensional subspace \(W = \operatorname{span}(v_1, \ldots, v_k)\) to the element

\[\iota(W) = [v_1 \wedge v_2 \wedge \cdots \wedge v_k]\]

(§Tensor Algebras, ⁋Definition 10)

We then have the following.

Proposition 7 The Plücker embedding is well-defined and injective.

Proof

That the Plücker embedding is well-defined means the value above does not change when a different basis of \(W\) is chosen. If we select another basis, the vector \(v_1\wedge\cdots\wedge v_k\) is merely scaled by the determinant of the change-of-basis matrix, so its image in \(\mathbb{P}(\bigwedge^k V)\) is the same point. Reversing this argument establishes injectivity just as easily.

Moreover, \(\iota\) realizes \(\Gr(k,V)\) as a closed subvariety of \(\mathbb{P}(\bigwedge^kV)\). To see this, observe that the image of \(\iota\) consists precisely of the decomposable vectors, i.e., those of the form

\[v_1\wedge\cdots\wedge v_k\]

Hence, to prove that the image is closed, it suffices to exhibit polynomials whose zero set is exactly these vectors. This follows from the properties of the wedge product via the Plücker relations

\[\sum_{r=1}^{k+1} (-1)^r p_{i_1 \cdots i_{k-1} j_r} p_{j_1 \cdots \widehat{j_r} \cdots j_{k+1}} = 0\tag{$\ast$}\]

where \(i_1 < \cdots < i_{k-1}\) and \(j_1 < \cdots < j_{k+1}\) are arbitrary subsets of \(\{1, \ldots, n\}\), and \(\widehat{j_r}\) indicates omission of \(j_r\). These equations hold for all possible choices of the \(i\)’s and \(j\)’s. From this we obtain the following.

Proposition 8 The image of the Plücker embedding is a closed subvariety of \(\mathbb{P}^{\binom{n}{k}-1}\), and therefore \(\Gr(k,V)\) is a projective variety.

Example 9 Let us examine the Plücker relation (\(\ast\)) for \(\Gr(2,4)\). The Plücker coordinates are \(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34}\), which serve as homogeneous coordinates on \(\mathbb{P}^5\). In this case the Plücker relation reduces to the single three-term equation

\[p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0\]

This is a quadratic equation, so \(\Gr(2, 4)\) is a quadric hypersurface in \(\mathbb{P}^5\). As the dimension of \(V\) grows, more such equations appear, and as \(k\) increases, each equation acquires more terms.

Schubert Varieties

The Grassmannian carries a natural cell structure and can be understood from a combinatorial viewpoint. To develop this, we first define the notions of flag and partition.

Definition 10 A flag in an \(n\)-dimensional vector space \(V\) is a chain of subspaces

\[F_\bullet:\qquad 0 = F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_n = V\]

with \(\dim F_i = i\) for each \(i\).

Example 11 Given the standard basis \(e_1, \ldots, e_n\) on \(V = \mathbb{K}^n\), the standard flag is defined by

\[F_i = \operatorname{span}(e_1, \ldots, e_i)\]

Now, given a \(k\)-dimensional subspace \(W \in \Gr(k, V)\), we can track step by step how \(W\) meets the flag \(F_\bullet\). Consider the sequence

\[0 = \dim(W \cap F_0) \leq \dim(W \cap F_1) \leq \cdots \leq \dim(W \cap F_n) = k\]

At each step the dimension increases by at most \(1\). To encode this information concisely we use a partition.

Definition 12 A sequence \(\lambda = (\lambda_1, \ldots, \lambda_k)\) of \(k\) integers satisfying

\[\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0,\qquad \lambda_1 \leq n - k\]

is called a partition. The size of a partition \(\lambda\) is defined as \(\lvert \lambda \rvert = \sum_{i=1}^{k} \lambda_i\).

Partitions can be visualized geometrically as Young diagrams: a first row with \(\lambda_1\) boxes, a second row with \(\lambda_2\) boxes, …, and a \(k\)th row with \(\lambda_k\) boxes. Such diagrams facilitate computations in Schubert calculus, but since these are needed only for computing intersections, or products in cohomology, we postpone their introduction. Instead, we define the following.

Definition 13 Given a flag \(F_\bullet\) and a partition \(\lambda = (\lambda_1, \ldots, \lambda_k)\), the Schubert variety \(\Omega_\lambda(F_\bullet)\) is the set of \(W \in \Gr(k, V)\) satisfying

\[\dim(W \cap F_{n - k + i - \lambda_i}) \geq i \quad\text{for all } 1 \leq i \leq k\]

This condition means that the dimensions of the intersections of \(W\) with the flag follow a specific pattern: \(W\) must meet \(F_{n-k+i-\lambda_i}\) in dimension at least \(i\). The partition condition \(\lambda_1 \leq n - k\) guarantees that \(n - k + 1 - \lambda_1 \geq 1\) in the first inequality \(\dim(W \cap F_{n - k + 1 - \lambda_1}) \geq 1\).

Proposition 14 The Schubert variety \(\Omega_\lambda(F_\bullet)\) is a closed subvariety of \(\Gr(k, V)\), and its dimension is \(\lvert \lambda \rvert\).

Proof

That \(\Omega_\lambda(F_\bullet)\) is closed follows because its defining conditions are given by the vanishing of regular functions.

To compute the dimension, consider the (open) Schubert cell \(\Omega_\lambda^\circ(F_\bullet)\) inside \(\Omega_\lambda(F_\bullet)\). It is obtained by replacing the inequalities in the defining conditions with equalities:

\[\dim(W \cap F_{n - k + i - \lambda_i}) = i \quad\text{for all } 1 \leq i \leq k\]

and forms an open dense subset of \(\Omega_\lambda(F_\bullet)\). The dimension of this cell is \(\lambda_1 + \cdots + \lambda_k = \lvert \lambda \rvert\), and therefore \(\dim \Omega_\lambda(F_\bullet) = \lvert \lambda \rvert\) as well.

Schubert varieties provide a cell decomposition of the Grassmannian. That is, the Schubert cells \(\Omega_\lambda^\circ(F_\bullet)\) corresponding to distinct partitions \(\lambda\) give \(\Gr(k, V)\) the structure of a cell complex, and each cell is isomorphic to the affine space \(\mathbb{A}^{\lvert \lambda \rvert}\). Through this decomposition one can study the topological and combinatorial properties of the Grassmannian.

---

References

[Harris] J. Harris, Algebraic Geometry: A First Course, Springer, 1992.
[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, 1978.
[Ful] W. Fulton, Young Tableaux, Cambridge University Press