Eigenvector Varieties

Abstract: Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.

• The paper introduces eigenvector varieties as projective subvarieties derived from linear matrix spaces, extending classical spectral theory.
• The paper formulates a determinantal framework that computes explicit dimensions, degrees, and irreducibility criteria for these varieties.
• The paper applies incidence theory, computational algebra techniques, and Lie algebra representations to analyze quantum Hamiltonians and related models.

Summary of "Eigenvector Varieties" (2606.20432)

Introduction and Conceptual Framework

The paper provides a systematic algebraic-geometric study of eigenvector varieties associated to linear spaces $H$ of $n \times n$ matrices over $C$. An eigenvector variety $E(H)$ consists of projective points $x$ such that $Hx = \lambda x$ for some $\lambda$ and $H \in H$. This framework yields a subvariety of $P^{n-1}$ for each linear matrix space, extending classical spectral theory to multidimensional parameter spaces. The study emphasizes explicit descriptions for generic matrix spaces and explores key cases including Lie algebras and quantum Hamiltonians.

Multiple incarnations of eigenvector varieties are introduced:

• $E(H)$: eigenvectors for generic $n \times n$0
• $n \times n$1: Zariski closure of all eigenvectors (nonzero eigenvalue) for $n \times n$2

Subtle distinctions are established between $n \times n$3 and $n \times n$4; generically they coincide, but may diverge for special matrix spaces (e.g., diagonal matrices).

Algebraic Structure and Generic Case

For $n \times n$5 spanned by $n \times n$6 generic $n \times n$7 matrices, $n \times n$8 is identified as the zero locus of $n \times n$9 minors of an augmented matrix $C$0 formed from $C$1. The variety's dimension is $C$2, and its degree is $C$3 for $C$4, with irreducibility for $C$5. Notably:

• For $C$6, $C$7 consists of $C$8 points (the standard spectral case).
• For $C$9, higher-dimensional determinantal varieties emerge (e.g., curves and surfaces in projective space), with explicit genus and degree calculations.

A determinantal hypersurface interpretation is given for $E(H)$0, connecting to Calabi–Yau geometry.

Incidence Varieties and Characteristic Polynomial Factorization

The incidence variety $E(H)$1 in $E(H)$2 encodes all $E(H)$3 such that $E(H)$4, with saturation removing non-generic eigenvectors (e.g., for singular $E(H)$5). Its irreducible horizontal components correspond bijectively to irreducible factors of the characteristic polynomial $E(H)$6, with the algebraic/geometric multiplicities governing component dimensions.

The elimination-theoretic perspective allows algorithmic computation of $E(H)$7 (via Gröbner bases or homotopy continuation), which is practically significant for large-scale systems.

Rank Stratification and Low-Dimensional Eigenvector Varieties

The rank stratification of matrix $E(H)$8 (formed from $E(H)$9) governs the dimension of $x$0 components: for squarefree $x$1, each irreducible component dimension is $x$2 generically. Syzygy methods provide explicit equations for $x$3 in low-rank matrix spaces. Classical results on compression spaces and low-rank matrix spaces (Eisenbud-Harris) yield explicit families of eigenvector varieties with controlled dimension.

Lie Algebra Representations

Lie algebras are treated as a source of matrix spaces, with eigenvector varieties described via weight decompositions:

• For a reductive Lie algebra $x$4 acting on $x$5, $x$6 is the union of orbit closures of projectivized weight spaces under the Lie group $x$7.
• For minuscule representations, $x$8 is the closed $x$9-orbit of a highest weight vector, leading to classical varieties (e.g., Grassmannians and Lagrangian varieties) as eigenvector varieties for compound matrices.

Symmetric and exterior power representations are analyzed, with eigenvector varieties for additive compound matrices shown to coincide with Grassmannians in Plücker coordinates. Extensions to classical Lie algebras yield orthogonal and symplectic Grassmannians.

For symmetric power representations, the eigenvector variety decomposes as a union of refined Chow varieties indexed by integer partitions of $Hx = \lambda x$0, with maximal partitions corresponding to irreducible components.

Multidegree and Intersection-Theoretic Analysis

The multidegree of the horizontal incidence variety is computed in the Chow ring; it encodes dimension and degree data for all linear sections/subspaces. For minuscule representations, Chern class computations yield the degrees for linear sections of the eigenvector variety, enabling explicit degree formulas for Grassmannians and related varieties.

Application to Quantum Hamiltonians

The algebraic theory is applied to Hamiltonians in quantum chemistry and physics:

• Fermionic systems: The one-body operator's eigenvector variety equals the Grassmannian $Hx = \lambda x$1 in its Plücker embedding (contradicting a potentially expected smaller variety), while the two-body operator yields an irreducible eigenvector variety whose dimension is sharply upper bounded and conjectured to be tight (as numerically verified for large $Hx = \lambda x$2).
• Bosonic systems: The eigenvector variety of the bosonic one-body operator decomposes as a union of orbit closures (refined Chow varieties) indexed by integer partitions, with the Veronese component distinguished as corresponding to ground states.

The Bose-Hubbard model and other explicit quantum Hamiltonians are analyzed, demonstrating the structure and complexity of eigenvector varieties arising in physical contexts.

Conclusion

The study initiates a rigorous algebraic-geometric theory of eigenvector varieties for linear matrix spaces, integrating determinantal geometry, representation theory, and computational algebraic geometry. Explicit dimension and degree results, orbit-theoretic descriptions, and intersection-theoretic tools are presented, with strong numerical verification for conjectured tightness in physical models. The implications are multifaceted:

• Theoretically, the results open avenues to classify possible eigenvector varieties for matrix spaces, including their relation to classical projective varieties and orbit closures.
• Practically, the theory provides tools for analyzing quantum Hamiltonians and characterizing ground state varieties.
• Future directions include extending the framework to bosonic two-body operators, refining dimension bounds (Conjecture~\ref{conj:dim}), and further studying the connection between algebraic structures of matrix spaces and their eigenvector varieties.

The paper establishes foundational techniques and results for future exploration into the algebraic geometry of spectral theory for parameterized matrix spaces and quantum systems.