Symmetric Product Space of 2-Forms

• The symmetric product space of 2-forms is defined as the subspace of Λ²(V*) ⊗ Λ²(V*) consisting of tensors invariant under permutation, forming a foundation for analyzing curvature tensors.
• It underlies universal curvature identities and enables coordinate-free expressions of classical invariants such as determinants and Gauss–Bonnet curvatures, vital in finite element discretizations.
• This framework bridges multilinear algebra, differential geometry, and representation theory, offering practical tools for studying topological invariants and advancing spinorial methods.

The symmetric product space of 2-forms is the subspace of the tensor product $\Lambda^2(V) \otimes \Lambda^2(V)$ consisting of elements that are invariant under permutation of the two $\Lambda^2$ factors, that is, all tensors $\psi$ such that $\psi(X, Y; Z, W) = \psi(Z, W; X, Y)$. This space arises naturally in multilinear algebra, differential geometry, and finite element discretizations, playing a key role in the paper of curvature tensors, invariants such as the determinant, and topological invariants like Pontrjagin and Gauss-Bonnet curvatures.

1. Algebraic Structure and Definition

Let $V$ be an $n$-dimensional vector space. The space of double forms is given by $$D(V^*) = \bigoplus_{p, q} \Lambda^p(V^*) \otimes \Lambda^q(V^*)$$. A double form of type $(p, q)$ is a multilinear map that is alternating in the first $p$ and last $q$ arguments separately. The symmetric product space of 2-forms, denoted as $\operatorname{Sym}^2(\Lambda^2)$, consists of elements in $\Lambda^2(V^*) \otimes \Lambda^2(V^*)$ that are symmetric under exchange: $\psi(X, Y; Z, W) = \psi(Z, W; X, Y)$.

This space is characterized by its invariance under the permutation of argument pairs and its connection to algebraic curvature tensors. For example, the Riemann curvature tensor $R$ is a $(2,2)$-double form that lies in this space, subject also to the first Bianchi identity $R(X, Y; Z, W) + R(Y, Z; X, W) + R(Z, X; Y, W) = 0$ (Berchenko-Kogan et al., 22 May 2025).

2. Operations: Exterior and Composition Products

Double forms admit two fundamental products:

• Exterior Product $\wedge$: For double forms $w_1 = \omega_1 \otimes \omega_2$ and $w_2 = \theta_1 \otimes \theta_2$, $$w_1 \wedge w_2 = (\omega_1 \wedge \theta_1) \otimes (\omega_2 \wedge \theta_2)$$.
• Composition Product $\circ$: Derived via the identification of $D(V^*)$ with endomorphisms of $\Lambda V$, for $w_1, w_2$ the product is $$w_1 \circ w_2 = \langle \omega_2, \theta_1 \rangle (\omega_1 \otimes \theta_2)$$, provided the degrees match for contraction.

Key interaction identities in the symmetric product space of 2-forms include:

• The Greub–Vanstone identity: For symmetric $h$ and $k$ in $A^2(V^*)$,

$h^p \circ k^p = p! (h \circ k)^p$

which expresses how symmetric exterior powers and composition products intertwine (Belkhirat et al., 2014). This identity is crucial for calculations related to characteristic classes.

3. Classical Invariants and Multilinear Extensions

Classical matrix invariants—determinant, Laplace expansion, Cayley–Hamilton theorem—admit coordinate-free extensions in the field of double forms via the exterior product. For a bilinear form $h$ on $V$, viewed as a $(1,1)$-double form, its $k$-th exterior power is

$$h^k(x_1, \dots, x_k; y_1, \dots, y_k) = k! \det[h(x_i, y_j)]$$

and the determinant arises for $k = n$ (Labbi, 2011).

This formalism extends naturally to symmetric product spaces of 2-forms, with the Riemann curvature tensor $R$ serving as a canonical example for $(2,2)$-double forms. Critical invariants such as the Gauss–Bonnet curvature,

$$h_{2k}(R) = \frac{1}{(n-2k)!} \star (g^{n-2k} \wedge R^k)$$

are built using these exterior powers, where $\star$ is the double Hodge star, and $g$ is the metric treated as a $(1,1)$-double form.

4. Geometric and Topological Applications

The symmetric product space of 2-forms is central to the algebraic description of curvature identities and topological invariants in differential geometry:

• Universal Curvature Identities: Algebraic identities derived from the exterior and composition products generate universal constraints, such as the vanishing of Pontrjagin classes for manifolds with pure curvature when $n \geq 4k$ (Belkhirat et al., 2014).
• Gauss–Bonnet–Chern Theorem: The Cayley–Hamilton polynomial for the Riemann curvature as a double form vanishes, reflecting the topological invariance of Euler characteristic via vanishing cofactor transformations (Labbi, 2011).
• Finite Element Discretizations: The symmetric product space provides the correct setting for discrete curvature tensors in computational PDEs. Finite element spaces $\mathcal{P}_r \Lambda^{2,2}_0(\mathcal{T})$ are constructed to reflect the algebraic symmetries of curvature tensors, enabling robust geometric discretizations for elasticity and relativity (Berchenko-Kogan et al., 22 May 2025).

5. Explicit Cohomological and Algebraic Realizations

In algebraic geometry, symmetric product spaces of 2-forms appear in the paper of holomorphic symmetric differentials. On complex varieties, spaces of symmetric 2-forms (global sections of $S^2 \Omega_X$) can be identified via cohomological injections, and their dimension is sensitive to deformation of the underlying equations:

$$\operatorname{im} \varphi = \bigcap_{i=1}^c \left( \ker(\cdot F_i) \cap \bigcap_{j=1}^k \ker(\cdot dF_i^{\{j\}}) \right)$$

where $F_i$ are defining polynomials and $dF_i$ their differentials (Brotbek, 2014). A consequence is that

$h^0(Y_0, S^m_{Y_0}) > h^0(Y_t, S^m_{Y_t})$

is possible for families, demonstrating that symmetric product spaces are not deformation invariant for $m \geq 2$.

Furthermore, these explicit descriptions allow for verification of conjectures such as the ampleness of the cotangent bundle under specific cohomological and degree conditions.

6. Representation-Theoretic and Spinorial Perspectives

Analogous structures occur in the representation theory of the Lorentz group and spin geometry. Symmetric product spaces of 2-forms correspond, in the spinor formalism, to spaces of symmetric 2-spinors closed under a symmetric product $\odot$ with prescribed contractions and symmetrizations:

$(\phi \overset{i,j}{\odot} \psi) = \text{symmetrization after $i, j$contractions with the spin metric$\epsilon$}$

enabling construction of irreducible representations and simplifying computations in space–time calculus (Aksteiner et al., 2022). The symmetric product preserves the irreducibility and encodes graded anti-commutativity and non-associativity via combinatorial coefficients.

The development of computer algebra tools (e.g., SymSpin in the xAct suite) further enables automated manipulations adhering to the symmetries and contractions intrinsic to symmetric product spaces.

7. Summary Table: Algebraic Objects and Their Roles

Object	Symmetry Condition	Geometric Role
$\operatorname{Sym}^2(\Lambda^2)$	$\psi(X, Y; Z, W) = \psi(Z, W; X, Y)$	Algebraic curvature tensors, Riemann tensor
$(2,2)$-double forms	Alternating in each argument pair; Bianchi identity	Curvature tensors, Gauss–Bonnet formulas
Symmetric $(1,1)$ double forms	Matrix invariants under exchange	Determinants, Laplace expansion, cofactor

These structures unify multilinear algebra and geometry, systematizing both invariants and computational techniques in Riemannian geometry and mathematical physics.