Tetrahedral Horn Problem

• Tetrahedral Horn Problem is a generalization of Horn’s classical eigenvalue challenge to six Hermitian matrices with specific additive and geometric constraints.
• It employs Schur polynomial inequalities and representation theory to derive necessary conditions for non-convex spectral configurations.
• The problem bridges geometric combinatorics, quantum information, and space-tiling criteria, offering new insights for spectral analysis and optimization.

The Tetrahedral Horn Problem refers to a collection of interrelated questions at the intersection of linear algebra, geometric combinatorics, spectral theory, and polyhedral geometry, seeking necessary and sufficient conditions for tuples—typically eigenvalues or geometric invariants—associated to the edges of a tetrahedron, subject to a prescribed set of additive or geometric constraints. This topic generalizes Horn’s problem for matrices from the triangular case to the tetrahedral setting, with deep connections to geometric invariant theory, representation-theoretic combinatorics (notably Schur polynomials), and the geometry of convex polytopes. The literature encompasses combinatorial, analytic, and geometric frameworks, including space-tiling tetrahedra, polyhedral tiling criteria, volume optimization, connection to 6j-symbols in representation theory, and the probabilistic structure underlying random spectral sums within the tetrahedral context.

1. Mathematical Formulation and Generalization of Horn's Problem

Horn’s classical problem asks: Given Hermitian matrices $A$, $B$, and $C$ with $A+B=C$, what are the necessary and sufficient conditions on their spectra $(a, b, c)$ for such matrices to exist? The set $Horn(n)$ of admissible triples in dimension $n$ forms a convex polyhedral cone defined by linear inequalities (Horn inequalities).

The tetrahedral Horn problem investigates the existence of Hermitian matrix sextuples $(A,B,C,D,E,F) \in \mathcal{H}_n^6$ corresponding to the six edges of a tetrahedron, subject to four additive relations:

$A+B = C,\quad B+D = F,\quad D+C = E,\quad A+F = E$

and seeks to characterize the set

$$Tetra(n) = \{ (eig\,A,\,eig\,B,\,eig\,C,\,eig\,D,\,eig\,E,\,eig\,F) \mid \text{above constraints hold} \}$$

Unlike $Horn(n)$, $Tetra(n)$ is neither convex nor polyhedral. The additive relations correspond to the combinatorial structure of the four faces of a tetrahedron, with each triangle enforcing a spectrum constraint akin to the classical triangle problem but in a higher-arity setting (Alekseev et al., 6 Oct 2025).

2. Schur-Polynomial Inequalities and Representation-Theoretic Certificates

Central to the solution strategy is the use of Schur polynomials $s_\lambda$ and their rescaled versions $\varphi_\lambda(X) = s_\lambda(eig\,X)/H_\lambda$ where $H_\lambda$ is the product of hook lengths of the Young diagram $\lambda$. The inequalities derived in (Alekseev et al., 6 Oct 2025) generalize the Horn inequalities to the tetrahedral case: for all test triples of Young diagrams, one has

$$[\varphi(a)\,\varphi(b)\,\varphi(d)]^{2/3} \leq \sum_{\text{test indices}} [\varphi(f)\,\varphi(c)\,\varphi(e)]^{1/3}$$

These certificates provide necessary and, in the limit $k\to\infty$, sufficient conditions for the existence of a tetrahedral configuration of Hermitian matrices with the prescribed spectra. The error in approximating actual tetrahedral eigenvalues given satisfaction up to degree $k$ is bounded as $O(\sqrt{\ln k/k})$, indicating rapid convergence of approximate solutions (Alekseev et al., 6 Oct 2025).

This approach reflects the algebraic structure of the intersection theory of Schubert varieties and encodes the associativity constraints from tensor product invariants in representation theory.

3. Quantum Information Theory and Spectral Estimation Techniques

The spectrum estimation techniques employed derive from quantum information theory, specifically the concentration properties of the Schur-Weyl distribution on Young diagrams, which is sharply focused around the true spectrum for large tensor powers. The Schur-Weyl duality decomposes tensor powers of Hermitian matrices, enabling sharp concentration bounds and approximate recovery of admissible tuples from spectral statistics (Alekseev et al., 6 Oct 2025).

Explicitly, for candidate eigenvalue sextuples, the distance to a true tetrahedral configuration

$$D\{a,b,c,d,e,f\} \equiv \min \Big( \sum_{matrices} \|\text{eig}(matrix) - \text{candidate}\|^2 \Big)^{1/2}$$

obeys

$$(1/\text{Tr}(c))\,D\{a,b,c,d,e,f\} \leq 6\sqrt{3}n\sqrt{(\ln k)/k}$$

provided all certificate inequalities up to degree $k$ hold (Alekseev et al., 6 Oct 2025).

4. Connection to 6j-Symbols and Semiclassical Asymptotics

The tetrahedral Horn problem is intrinsically linked to the asymptotics of $U(n)$ 6j-symbols—quantities controlling the associativity of tensor product representations. The paper (Alekseev et al., 6 Oct 2025) demonstrates that the existence of a tetrahedral configuration is equivalent to the non-exponential, i.e. inverse-polynomial, decay of norms of the corresponding 6j-symbols in the semiclassical (large-$k$) limit. In contrast, the lack of an admissible tetrahedral configuration forces exponential decay.

This correspondence extends the known links between Horn’s problem, Littlewood–Richardson coefficients, and quantum invariants: whether a given sextuple is realized geometrically manifests in the polynomial versus exponential asymptotics of representation-theoretic quantities.

5. Geometric and Polyhedral Aspects: Tiling and Volume Formulas

The classification of space-tiling tetrahedra refines the geometric understanding of the problem. Rational dihedral angle tetrahedra are classified into two one-parameter families (Hill and a new family) and 59 sporadic cases (Chentouf et al., 2023). The tiling problem delineates precisely which tetrahedra tile $\mathbb{R}^3$, with every member of the Hill family tiling, while only one in the new family, plus at most 40 sporadic examples, satisfy tiling criteria. The Dehn invariant is necessary but not sufficient for tiling—contradicting Debrunner’s conjecture. Detailed combinatorial and edge-alignment criteria are used to test for tileability.

Volume optimization for horn-shaped subregions carved from tetrahedra is informed by explicit formulas, e.g.

$$V_{KLMN} = \frac{|1 - x y z t|}{(1+x)(1+y)(1+z)(1+t)}$$

$$V_{PQRS} = \frac{|1 - x y z t|^3}{(1 + x + x y + x y z) (1 + y + y z + y z t) (1 + z + z t + z t x) (1 + t + t x + t x y)}$$

from (Litvinov et al., 2014), thereby enabling geometric optimization and concurrency analysis for sub-tetrahedral regions possibly encountered in horn-type configurations.

6. Combinatorial, Probabilistic, and Algorithmic Interpretations

Recent work (Gangopadhyay et al., 16 Oct 2024) interprets the randomized Horn problem (spectra of sums of random Hermitian matrices) in terms of “hives” and lozenge tilings. The tetrahedral Horn problem is recast as a three-dimensional extension, assembling hive boundary data on tetrahedral faces with propagation governed by the octahedron recurrence. The large-$n$ large deviation rate is given by a surface-tension minimization over continuum hives:

$$- \frac{2}{n^2} \ln P[\text{spec}(Z_n) \in B^n_I(\nu_n, \varepsilon)] \simeq I(\gamma')$$

$$I(\gamma') = \ln \frac{V(\lambda) V(\mu)}{V(\nu')} + \inf_{h} \int_T \sigma(- (h)_{ac}) dx$$

Bounds and exact expressions for the entropy (integrated surface tension) are derived, e.g.

$$\inf_{h}\int_T \sigma(- (h)_{ac})\, dx = -\frac{5}{4} - \ln \frac{4\,\Delta(\sigma_\lambda, \sigma_\mu,\sigma_\nu)^2}{\sigma_\lambda \sigma_\mu \sigma_\nu}$$

where $\Delta$ is the triangle area with side lengths derived from GUE-type boundary spectra. Exact-sampling algorithms based on the octahedron recurrence provide numerical access to the admissible polytope hulls for the tetrahedral Horn context.

7. Geometric Realization and Angular Constraints

Problems in spherical and solid geometry (Rieck, 2022) establish equivalence between constraints for constructing specific tetrahedral faces and configurations of great circles on the sphere, with practical applications in pose estimation and computer vision. Given three edge directions at a vertex, the opposite face can be realized as any acute or right triangle, provided certain angular inequalities hold. These constructions are directly leveraged in the Tetrahedral Horn context to analyze feasibility and uniqueness of spatial configurations given fixed horn directions.

Summary Table: Principal Constraints and Properties

Aspect	Horn Problem (Triangle)	Tetrahedral Horn Problem (Tetrahedron)
Objects	3 Hermitian matrices ($A, B, C$)	6 Hermitian matrices ($A,B,C,D,E,F$)
Additive Relations	$A+B=C$	$A+B=C$, $B+D=F$, $D+C=E$, $A+F=E$
Solution Set	Convex polyhedral cone	Non-convex, non-polyhedral set
Characterization	Horn inequalities (Schur, LR)	Schur-polynomial inequalities, combinatorial
Asymptotic Invariant	Littlewood–Richardson coeff.	U(n) 6j-symbol norm (semiclassical regime)
Space-tiling Classification	Not relevant	Combinatorial and angular criteria (tiling)
Probabilistic Interpretation	Horn polytope volume	Surface tension minimization in hive context

Concluding Remarks

The Tetrahedral Horn Problem synthesizes a spectrum of geometric, combinatorial, algebraic, and probabilistic techniques to extend the classical Horn problem to higher-combinatorial settings relevant for spectral theory, polyhedral tiling, and quantum invariants. The solution frameworks converge on Schur polynomial inequalities, representation-theoretic asymptotics, and exact combinatorial algorithms, with empirical and theoretical evidence showing precision and depth in relating geometric configuration existence, spectral sum inequalities, and the structures underlying space-filling and optimization problems. The non-convexity and higher-arity constraints herald new directions for research in spectral theory, geometric combinatorics, and their computational realization.