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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 show that it carries a variety structure, but the key result is that it not only has a variety structure but also preserves the relative position of each \(k\)-plane in \(\mathbb{A}^n\), so it behaves as we want without much effort.
Example 2 For example, \(\Gr(1, n+1)\) is the space of lines in the \((n+1)\)-dimensional vector space \(\mathbb{K}^{n+1}\), so by definition it equals \(\mathbb{P}^n\). Once we define the variety structure on the Grassmannian, we will see that these two structures are exactly the same.
The simplest example that did not appear before is \(\Gr(2,4)\). This is the collection of \(2\)-dimensional subspaces of a \(4\)-dimensional space. When dealing with Grassmannians, this example will serve as a toy example.
As always, to give a variety structure it suffices to think of an affine cover and work affine-locally. For this purpose we make the following definition.
Definition 3 For each set of \(k\) indices \(I = \{i_1 < \cdots < i_k\}\), we define an open set \(U_I\) by
\[U_I = \{W \in \Gr(k, V) \mid \text{projection } W \to \operatorname{span}(e_{i_1}, \ldots, e_{i_k}) \text{ is an isomorphism}\}\]Fixing a basis \(e_1,\ldots, e_n\) of \(V\) and using vectors \(w_1,\ldots, w_k\) spanning \(W\), we see that \(W\) is the row space of the following \(k\times n\) matrix
\[\begin{pmatrix}w_1\\\vdots\\w_k\end{pmatrix}=\begin{pmatrix}w_{1,1}&w_{1,2}&\cdots &w_{1,n}\\ \vdots&\vdots&\ddots&\vdots\\ w_{k,1}&w_{k,2}&\cdots&w_{k,n}\end{pmatrix}\]Then the condition defining \(U_I\) is equivalent to the \(k\times k\) matrix formed using the columns \(i_1,\ldots, i_k\) corresponding to the index set \(I\) being invertible. Then the following holds.
Proposition 4 Each \(U_I \cong \mathbb{A}^{k(n-k)}\).
Proof
Without loss of generality, we show the case \(I = \{1, 2, \ldots, k\}\). That is, for the \(k \times n\) matrix \(A\) representing \(W \in U_I\), the left \(k \times k\) minor is nonzero. Using row operations, we bring this minor to the form
\[A = \begin{pmatrix} I_k & B \end{pmatrix}\]Here \(B\) is a \(k \times (n-k)\) matrix. Then the \(k(n-k)\) entries of \(B\) completely determine \(W\), and there is no constraint among them. Therefore \(U_I \cong \mathbb{A}^{k(n-k)}\).
As seen in this proof, the coordinate system on \(U_I\) consists of \(k(n-k)\) free parameters. They correspond to the “non-trivial part” of the matrix representing \(W\). That is, once the \(k \times k\) block defined by \(I\) is fixed to be the identity, the remaining \((n-k) \times k\) block can vary freely.
Then it is obvious that for any \(W\in \Gr(k,V)\) there exists an affine open cover containing \(W\). Moreover, since it is also obvious that the transition maps from \(U_I\) to \(U_J\) are regular maps, this endows \(\Gr(k,V)\) with a variety structure. Of course, to show that this is quasi-projective one needs an explicit projective embedding, but for now we have the following.
Proposition 5 \(\dim \Gr(k, V) = k(n - k)\).
Plücker Embedding
We now show that the Grassmannian is a quasi-projective variety. That is, we define an embedding from the Grassmannian 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
\[\iota(W) = [v_1 \wedge v_2 \wedge \cdots \wedge v_k]\]Then the following holds.
Proposition 7 The Plücker embedding is well-defined and injective.
Proof
That the Plücker embedding is well-defined means the above value does not change when we choose a different basis of \(W\). But if we choose a different basis of \(W\), then \(v_1\wedge\cdots\wedge v_k\) is only scaled by the determinant of the change-of-basis matrix, so when sent to \(\mathbb{P}(\bigwedge^k V)\) it specifies the same point anyway. Running a similar argument in reverse, injectivity is also easily shown.
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 decomposable vectors, i.e., vectors of the form
\[v_1\wedge\cdots\wedge v_k\]Therefore, to show that the image of \(\iota\) is a closed subvariety, it suffices to define polynomials having these as their zero set, and this follows from the properties of the wedge product via the following Plücker relations
\[\sum_{r=1}^{k+1} (-1)^r p_{i_1 \cdots i_{k-1} j_r} p_{j_1 \cdots \widehat{j_r} \cdots j_{k+1}} = 0\tag{$\ast$}\]Here \(i_1 < \cdots < i_{k-1}\) and \(j_1 < \cdots < j_{k+1}\) are arbitrary subsets of \(\{1, \ldots, n\}\), and \(\widehat{j_r}\) means omitting \(j_r\). These equations hold for all possible choices of \(i\)’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 \(p_{12}, p_{13}, p_{14}, p_{23}, p_{24}, p_{34}\), and these are the homogeneous coordinates of \(\mathbb{P}^5\). Then the Plücker relation is given by the unique \(3\)-term relation
\[p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0\]This is a quadratic equation, so \(\Gr(2, 4)\) is a quadric hypersurface in \(\mathbb{P}^5\). If the dimension of \(V\) increases, more such equations will appear, and if \(k\) increases, each equation will have more terms.
Schubert Varieties
The Grassmannian is equipped with a kind of cell structure, so it can be understood from a combinatorial viewpoint. For this purpose 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
\[F_\bullet:\qquad 0 = F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_n = V\]where \(\dim F_i = i\).
Example 11 When the standard basis \(e_1, \ldots, e_n\) is given 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 \(W\), an element of \(\Gr(k, V)\), we can track step by step how this \(W\) meets the flag \(F_\bullet\). Consider the sequence
\[0 = \dim(W \cap F_0) \leq \dim(W \cap F_1) \leq \cdots \leq \dim(W \cap F_n) = k\]If we consider this sequence, the dimension increases by at most \(1\) at each step. To represent this information concisely, we use a partition.
Definition 12 A sequence \(\lambda = (\lambda_1, \ldots, \lambda_k)\) of \(k\) integers satisfying the conditions
\[\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0,\qquad \lambda_1 \leq n - k\]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 a Young diagram. This consists of a first row with \(\lambda_1\) boxes, a second row with \(\lambda_2\) boxes, …, and a \(k\)th row with \(\lambda_k\) boxes. This facilitates computations called Schubert calculus, but since these are needed only when performing intersections, or multiplication in cohomology, we do not introduce them yet. Instead, we define the following.
Definition 13 For a flag \(F_\bullet\) and a partition \(\lambda = (\lambda_1, \ldots, \lambda_k)\), the Schubert variety \(\Omega_\lambda(F_\bullet)\) is defined as the set of \(W \in \Gr(k, V)\) satisfying the conditions
\[\dim(W \cap F_{n - k + i - \lambda_i}) \geq i \quad\text{for all } 1 \leq i \leq k\]This condition means that the dimensions of the intersections of \(W\) with the flag follow a specific pattern. Specifically, \(W\) 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 the defining conditions are given by the zero set of regular functions.
To compute the dimension, we consider the (open) Schubert cell \(\Omega_\lambda^\circ(F_\bullet)\) of \(\Omega_\lambda(F_\bullet)\). This is obtained by changing the inequalities in the defining conditions to equalities:
\[\dim(W \cap F_{n - k + i - \lambda_i}) = i \quad\text{for all } 1 \leq i \leq k\]and it is an open dense subset of \(\Omega_\lambda(F_\bullet)\). Computing the dimension of this cell yields \(\lambda_1 + \cdots + \lambda_k = \lvert \lambda \rvert\), and therefore the dimension of \(\Omega_\lambda(F_\bullet)\) is also \(\lvert \lambda \rvert\).
Schubert varieties provide a cell decomposition of the Grassmannian. That is, the Schubert cells \(\Omega_\lambda^\circ(F_\bullet)\) corresponding to different 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, 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
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