Introduction
The Grassmannian stands as one of the most important geometric objects in modern mathematics, serving as a meeting point for algebraic geometry, differential geometry, representation theory, and mathematical physics. At its core, the Grassmannian \(G(k,n)\) parametrizes all \(k\)-dimensional linear subspaces of an \(n\)-dimensional vector space. This seemingly simple definition gives rise to remarkably rich structure: algebraically, it is a smooth projective variety defined by quadratic equations; differentially, it is a compact homogeneous space with a natural Riemannian structure. This article explores these dual perspectives in detail, revealing how they complement and enrich each other.
1. Algebraic Geometry: The Grassmannian as a Projective Variety
1.1 Definition and Basic Structure
For a field \(K\) (typically \(\mathbb{R}\) or \(\mathbb{C}\)) and integers \(0 \leq k \leq n\), the Grassmannian \(G(k,n)\) is defined as:
\[G(k,n) = \{V \subseteq K^n \mid V \text{ is a } k\text{-dimensional linear subspace}\}.\]
When \(k = 1\), we recover projective space: \(G(1,n) \cong \mathbb{P}^{n-1}\). For \(k = n-1\), we obtain the dual projective space. Thus Grassmannians naturally generalize projective spaces.
There is an equivalent projective interpretation: via projectivization, \(k\)-dimensional subspaces of \(K^n\) correspond to \((k-1)\)-dimensional linear subspaces of \(\mathbb{P}^{n-1}\). This correspondence is bijective and inclusion-preserving.
1.2 The Plücker Embedding
The fundamental construction that reveals the algebraic nature of Grassmannians is the Plücker embedding. For \(V \in G(k,n)\), choose a basis \(v_1, \dots, v_k\) and form the \(k \times n\) matrix \(M\) with these vectors as rows. For each \(k\)-element subset \(I = \{i_1 < \cdots < i_k\} \subseteq \{1,\dots,n\}\), define the Plücker coordinate:
\[x_I(V) = \det(M_I),\]
where \(M_I\) is the \(k \times k\) submatrix consisting of columns indexed by \(I\).
The Plücker map is then defined as:
\[P_{k,n}: G(k,n) \longrightarrow \mathbb{P}^{\binom{n}{k}-1}, \quad V \mapsto [x_I(V)]_{|I|=k}.\]
Proposition 1.2.1 The Plücker map is well-defined and injective.
Proof: To show well-definedness, note that changing the basis of \(V\) replaces \(M\) by \(AM\) for some \(A \in GL(k)\). Then each minor transforms as \(\det((AM)_I) = \det(A)\det(M_I)\). Since \(\det(A) \neq 0\), this scales all Plücker coordinates by the same nonzero factor, leaving the projective point unchanged.
For injectivity, suppose \(P_{k,n}(V) = P_{k,n}(W)\). Choose bases so that their matrix representations \(M_V\) and \(M_W\) have the same Plücker coordinates. After possibly rescaling, we may assume there exists \(I_0\) with \(x_{I_0}(V) = x_{I_0}(W) = 1\). Perform column operations (which correspond to changes of basis in \(K^n\)) to make both matrices have the identity in columns \(I_0\). The remaining columns are then uniquely determined by the other Plücker coordinates through the relations:
\[x_{(I_0\setminus\{i\})\cup\{j\}} = \pm(\text{entry in row }i,\text{ column }j),\]
which follows from expanding the determinant along column \(j\). Thus \(M_V = M_W\), so \(V = W\). ∎
1.3 Plücker Relations and the Defining Ideal
The image of \(P_{k,n}\) is not all of \(\mathbb{P}^{\binom{n}{k}-1}\); it satisfies quadratic relations known as Plücker relations.
Theorem 1.3.1: For any \(I \subseteq \{1,\dots,n\}\) with \(|I| = k-1\) and \(J \subseteq \{1,\dots,n\}\) with \(|J| = k+1\), define:
\[f_{I,J} = \sum_{j \in J} (-1)^{\sigma(I,j)} x_{I \cup \{j\}} x_{J \setminus \{j\}},\]
where \(\sigma(I,j)\) is the sign of the permutation that orders \(I \cup \{j\}\). Then:
\[P_{k,n}(G(k,n)) = V\left(\{f_{I,J}\}_{|I|=k-1, |J|=k+1}\right) \subseteq \mathbb{P}^{\binom{n}{k}-1}.\]
Proof: We prove both inclusions.
(\(\subseteq\)) Let \(V \in G(k,n)\) with matrix representation \(M\). Fix \(I,J\) as above. Consider the \((k+1) \times (k+1)\) matrix formed by taking rows of \(M\) corresponding to some basis and columns \(I \cup J\) (with appropriate repetitions). This matrix has rank at most \(k\), so all \((k+1) \times (k+1)\) minors vanish. The Laplace expansion of such a minor along a row yields exactly the Plücker relation \(f_{I,J}(P_{k,n}(V)) = 0\).
More concretely: Let \(M'\) be the \((k+1) \times n\) matrix obtained by adding an extra row \(w\) to \(M\). For any \(j \in J\), let \(M'_{I \cup \{j\}}\) be the \((k+1) \times (k+1)\) submatrix with columns \(I \cup \{j\}\). The determinant \(\det(M'_{I \cup \{j\}})\) can be computed by expanding along the last row, giving a linear combination of \(k \times k\) minors of \(M\). Summing over \(j\) with appropriate signs gives zero, which is exactly \(f_{I,J}\).
(\(\supseteq\)) Suppose \(a = [a_I] \in \mathbb{P}^{\binom{n}{k}-1}\) satisfies all Plücker relations. Choose \(I_0\) with \(a_{I_0} \neq 0\) and rescale so \(a_{I_0} = 1\). Construct a \(k \times n\) matrix \(M\) as follows: set columns \(I_0\) to be the identity matrix. For \(j \notin I_0\) and \(i \in I_0\), define the \((i,j)\)-entry as:
\[m_{ij} = (-1)^{\sigma(i,j)} a_{(I_0 \setminus \{i\}) \cup \{j\}},\]
where \(\sigma(i,j)\) is determined by ordering considerations. Let \(V\) be the row space of \(M\). One verifies that \(P_{k,n}(V) = a\) by showing that for any \(I\), \(\det(M_I) = a_I\). This follows by induction on \(|I \setminus I_0|\), using the Plücker relations to express \(a_I\) in terms of coordinates with smaller symmetric difference. ∎
Example 1.3.2: For \(G(2,4)\), with coordinates \((x_{12}, x_{13}, x_{14}, x_{23}, x_{24}, x_{34})\), the only independent Plücker relation (up to sign) is:
\[x_{12}x_{34} - x_{13}x_{24} + x_{14}x_{23} = 0.\]
Thus \(G(2,4)\) is a smooth quadric hypersurface in \(\mathbb{P}^5\), known as the Klein quadric.
1.4 Irreducibility, Dimension, and Smoothness
Proposition 1.4.1: \(G(k,n)\) is an irreducible algebraic variety of dimension \(k(n-k)\).
Proof: For each \(k\)-element subset \(I\), define the affine open:
\[U_I = \{ [x_J] \in G(k,n) \mid x_I \neq 0 \}.\]
On \(U_I\), we can normalize \(x_I = 1\). Any \(V \in U_I\) has a unique matrix representation of the form:
\[M = \begin{pmatrix} I_k &|& A \end{pmatrix},\]
where \(I_k\) is the \(k \times k\) identity matrix (in columns \(I\)), and \(A\) is a \(k \times (n-k)\) matrix. The entries of \(A\) provide coordinates on \(U_I\), giving an isomorphism \(U_I \cong \mathbb{A}^{k(n-k)}\).
Since the \(U_I\) cover \(G(k,n)\) and each is irreducible, \(G(k,n)\) is irreducible. The dimension is \(k(n-k)\) because each \(U_I\) is an affine space of that dimension. ∎
The affine coordinates also show that \(G(k,n)\) is smooth: transition functions between charts are given by rational functions (specifically, by solving linear equations), hence regular.
2. Differential Geometry: The Grassmannian as a Homogeneous Space
2.1 Smooth Manifold Structure
When \(K = \mathbb{R}\) or \(\mathbb{C}\), the Grassmannian naturally carries the structure of a smooth (resp. complex) manifold. The affine charts \(U_I \cong \mathbb{A}^{k(n-k)}\) provide local coordinate charts. Transition functions between charts are given by matrix operations: if \(V \in U_I \cap U_J\), then the matrix \(A\) in the \(I\)-chart and the matrix \(B\) in the \(J\)-chart are related by:
\[B = (M_I^{-1} M_J)_{\text{complement}},\]
where \(M\) is a matrix representing \(V\), and the subscript indicates taking the appropriate submatrix. These are rational functions with nonzero denominators, hence smooth (holomorphic in the complex case).
Thus \(G(k,n;\mathbb{R})\) is a smooth manifold of dimension \(k(n-k)\), and \(G(k,n;\mathbb{C})\) is a complex manifold of complex dimension \(k(n-k)\) (hence real dimension \(2k(n-k)\)).
2.2 Homogeneous Space Structure
A fundamental insight is that Grassmannians are homogeneous spaces of classical Lie groups.
Theorem 2.2.1: There are natural diffeomorphisms:
\[\begin{aligned} G(k,n;\mathbb{R}) &\cong O(n)/(O(k) \times O(n-k)), \\ G(k,n;\mathbb{C}) &\cong U(n)/(U(k) \times U(n-k)). \end{aligned}\]
Proof: Consider the real case. The orthogonal group \(O(n)\) acts transitively on \(G(k,n;\mathbb{R})\): given any \(k\)-dimensional subspace \(V\), choose an orthonormal basis for \(V\) and extend to an orthonormal basis for \(\mathbb{R}^n\). The orthogonal matrix sending the standard basis to this basis maps the standard \(k\)-plane \(\mathbb{R}^k \times \{0\}\) to \(V\).
The stabilizer of the standard \(k\)-plane consists of orthogonal matrices that preserve this subspace, which are precisely block-diagonal matrices with blocks in \(O(k)\) and \(O(n-k)\). Thus by the orbit-stabilizer theorem, \(G(k,n;\mathbb{R}) \cong O(n)/(O(k) \times O(n-k))\).
The complex case is analogous with \(U(n)\) acting on \(\mathbb{C}^n\). ∎
This description has several important consequences: - \(G(k,n)\) is compact (as a quotient of compact groups). - \(G(k,n)\) has a natural Riemannian metric induced from the bi-invariant metric on \(O(n)\) or \(U(n)\). - The tangent space at a point \(V\) can be identified with \(\operatorname{Hom}(V, V^\perp)\).
2.3 Tangent Space and Dimension
Let \(V \in G(k,n)\). A curve \(V(t)\) in \(G(k,n)\) with \(V(0) = V\) can be described by a family of linear maps. Specifically, choose an orthonormal basis \(e_1, \dots, e_k\) for \(V\) and extend to an orthonormal basis for \(\mathbb{R}^n\). For small \(t\), we can write:
\[e_i(t) = e_i + t \sum_{j=k+1}^n a_{ij} e_j + O(t^2),\]
where \(A = (a_{ij})\) is a \(k \times (n-k)\) matrix. Differentiating at \(t=0\) gives a linear map \(V \to V^\perp\) represented by \(A\). Thus:
Proposition 2.3.1: There is a natural isomorphism:
\[T_V G(k,n) \cong \operatorname{Hom}(V, V^\perp) \cong \mathbb{R}^{k(n-k)}.\]
This provides an intrinsic, coordinate-free description of the tangent space and confirms the dimension \(k(n-k)\).
2.4 Volume Formulas
The homogeneous space structure allows computation of volumes with respect to natural invariant metrics. We derive these formulas through a sequence of fibrations.
Lemma 2.4.1 (Sphere volumes):
\[\operatorname{vol}(S^{m-1}) = \frac{2\pi^{m/2}}{\Gamma(m/2)}.\]
Proof: This follows from integrating the volume form on the sphere or from the known formula for surface area. ∎
Theorem 2.4.2 (Classical group volumes):
\[\begin{aligned} \operatorname{vol}(O(n)) &= \frac{2^n}{\prod_{j=1}^n \Gamma(j/2)} \pi^{\frac{n(n+1)}{4}}, \\ \operatorname{vol}(U(n)) &= \frac{2^n}{\prod_{j=1}^n \Gamma(j)} \pi^{\frac{n(n+1)}{2}}, \\ \operatorname{vol}(Sp(n)) &= \frac{2^n}{\prod_{j=1}^n \Gamma(2j)} \pi^{n(n+1)}. \end{aligned}\]
Proof for \(O(n)\): Consider the fibration:
\[O(n-1) \hookrightarrow O(n) \xrightarrow{\pi} S^{n-1},\]
where \(\pi\) maps an orthogonal matrix to its first column. This is a Riemannian submersion with respect to the bi-invariant metric on \(O(n)\) (induced from the Killing form \(\langle X, Y \rangle = \operatorname{tr}(X^TY)\)). The fibers are isometric to \(O(n-1)\).
By the coarea formula (or the fact that in a Riemannian submersion, volumes multiply):
\[\operatorname{vol}(O(n)) = \operatorname{vol}(O(n-1)) \cdot \operatorname{vol}(S^{n-1}).\]
Applying this recursively:
\[\operatorname{vol}(O(n)) = \prod_{m=1}^n \operatorname{vol}(S^{m-1}) = \prod_{m=1}^n \frac{2\pi^{m/2}}{\Gamma(m/2)}.\]
Simplifying:
\[\prod_{m=1}^n 2\pi^{m/2} = 2^n \pi^{\sum_{m=1}^n m/2} = 2^n \pi^{\frac{n(n+1)}{4}},\]
and the denominator is \(\prod_{j=1}^n \Gamma(j/2)\). ∎
The proofs for \(U(n)\) and \(Sp(n)\) are similar, using fibrations:
\[U(n-1) \hookrightarrow U(n) \twoheadrightarrow S^{2n-1}, \quad Sp(n-1) \hookrightarrow Sp(n) \twoheadrightarrow S^{4n-1}.\]
Corollary 2.4.3 (Grassmannian volumes):
\[\begin{aligned} \operatorname{vol}(G(k,n;\mathbb{R})) &= \frac{\operatorname{vol}(O(n))}{\operatorname{vol}(O(k)) \operatorname{vol}(O(n-k))} \\ &= \frac{\prod_{j=1}^n \Gamma(j/2)}{\prod_{j=1}^k \Gamma(j/2) \prod_{j=1}^{n-k} \Gamma(j/2)} \pi^{\frac{k(n-k)}{2}}, \\ \operatorname{vol}(G(k,n;\mathbb{C})) &= \frac{\operatorname{vol}(U(n))}{\operatorname{vol}(U(k)) \operatorname{vol}(U(n-k))} \\ &= \frac{\prod_{j=1}^n \Gamma(j)}{\prod_{j=1}^k \Gamma(j) \prod_{j=1}^{n-k} \Gamma(j)} \pi^{k(n-k)}. \end{aligned}\]
Proof: This follows from the quotient representation \(G(k,n) \cong O(n)/(O(k) \times O(n-k))\) and the fact that the natural projection \(O(n) \to G(k,n)\) is a Riemannian submersion with fiber isometric to \(O(k) \times O(n-k)\). Thus:
\[\operatorname{vol}(G(k,n)) = \frac{\operatorname{vol}(O(n))}{\operatorname{vol}(O(k) \times O(n-k))} = \frac{\operatorname{vol}(O(n))}{\operatorname{vol}(O(k)) \operatorname{vol}(O(n-k))}.\]
The simplification of the Gamma factors uses properties of the Gamma function and careful index manipulation. The exponent of \(\pi\) comes from:
\[\frac{n(n+1)}{4} - \frac{k(k+1)}{4} - \frac{(n-k)(n-k+1)}{4} = \frac{k(n-k)}{2}.\]
The complex case is analogous. ∎
3. Interplay: Schubert Calculus and Enumerative Geometry
The unity of algebraic and differential geometry is beautifully illustrated by enumerative problems on Grassmannians. A classic example is:
Problem: Given four lines in \(\mathbb{P}^3\) in general position, how many lines intersect all four?
Solution using Grassmannians: Lines in \(\mathbb{P}^3\) correspond to points in \(G(2,4)\). Given a fixed line \(L \subset \mathbb{P}^3\), the set of lines meeting \(L\) is:
\[X_L = \{ \ell \in G(2,4) \mid \ell \cap L \neq \emptyset \}.\]
Under the Plücker embedding \(G(2,4) \hookrightarrow \mathbb{P}^5\), one can show that \(X_L\) is cut out by a linear equation. Specifically, if \(L\) has Plücker coordinates \((p_{ij})\) and \(\ell\) has coordinates \((x_{ij})\), then:
\[\ell \cap L \neq \emptyset \iff \sum_{i<j} \varepsilon_{ij} p_{ij} x_{kl} = 0,\]
where \(\varepsilon_{ij}\) are appropriate signs and \(\{k,l\} = \{1,2,3,4\} \setminus \{i,j\}\). Thus \(X_L = G(2,4) \cap H_L\) where \(H_L\) is a hyperplane in \(\mathbb{P}^5\).
For four general lines \(L_1, L_2, L_3, L_4\), the set of lines meeting all four is:
\[X = G(2,4) \cap H_1 \cap H_2 \cap H_3 \cap H_4,\]
where \(H_i = H_{L_i}\). Generically, the intersection of four hyperplanes in \(\mathbb{P}^5\) is a line \(\mathbb{P}^1\). Since \(G(2,4)\) is a quadric hypersurface, a line typically intersects it in 2 points (by Bézout's theorem). Thus there are exactly 2 lines meeting all four given lines.
This reasoning combines: - Algebraic geometry: The Plücker embedding realizes \(G(2,4)\) as a concrete variety. - Differential geometry: Transversality ensures the intersection is well-behaved for generic choices. - Enumerative geometry: Bézout's theorem provides the count.
This is a simple example of Schubert calculus, the intersection theory on Grassmannians, which has deep connections to representation theory, combinatorics, and topology.
Reference
[1] Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1994.
[2] Robin Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, Vol. 52, Springer-Verlag, New York-Heidelberg, 1977.