The tetrahedral Horn problem and asymptotics of U(n) 6j symbols

TL;DR

This work addresses the additive tetrahedral Horn problem by connecting eigenvalue configurations of six Hermitian matrices to the semiclassical behavior of $U(n)$ $6j$-symbols. It develops a suite of representation-theoretic tools, including Schur–Weyl duality and a family of $\varphi$-functions, to derive necessary and sufficient inequalities for tetrahedral eigenvalues, plus a quantitative distance bound that becomes exact as the degree grows. The paper then proves two key asymptotic results for $U(n)$ $6j$-symbols: an inverse-polynomial lower bound along tetrahedral sequences and an exponential upper bound when tetrahedral feasibility fails, revealing a sharp geometry–quantum correspondence. Overall, the work advances understanding of how tensor-product multiplicities and quantum marginal-like constraints govern classical eigenvalue problems, with potential implications for geometric invariant theory and quantum information.

Abstract

Horn's problem is concerned with characterizing the eigenvalues $(a,b,c)$ of Hermitian matrices $(A,B,C)$ satisfying the constraint $A+B=C$ and forming the edges of a triangle in the space of Hermitian matrices. It has deep connections to tensor product invariants, Littlewood-Richardson coefficients, geometric invariant theory and the intersection theory of Schubert varieties. This paper concerns the tetrahedral Horn problem which aims to characterize the tuples of eigenvalues $(a,b,c,d,e,f)$ of Hermitian matrices $(A,B,C,D,E,F)$ forming the edges of a tetrahedron, and thus satisfying the constraints $A+B=C$, $B+D=F$, $D+C=E$ and $A+F=E$. Here we derive new inequalities satisfied by the Schur-polynomials of such eigenvalues and, using eigenvalue estimation techniques from quantum information theory, prove their satisfaction up to degree $k$ implies the existence of approximate solutions with error $O(\ln k / k)$. Moreover, the existence of these tetrahedra is related to the semiclassical asymptotics of the $6j$-symbols for the unitary group $U(n)$, which are maps between multiplicity spaces that encode the associativity relation for tensor products of irreducible representations. Using our techniques, we prove the asymptotics of norms of these $6j$-symbols are either inverse-polynomial or exponential depending on whether there exists such tetrahedra of Hermitian matrices.

Figures (2)

• Figure 1: (Left) The Horn problem concerns the eigenvalues of three Hermitian matrices, $A,B,$ and $C$, forming a triangle in $\mathcal{H}_{n}$. (Right) The tetrahedral Horn problem concerns the eigenvalues of six Hermitian matrices, $A,B,C,D,E,$ and $F$, forming a tetrahedron in $\mathcal{H}_{n}$.
• Figure 2: The interior green region depicts a slice of the space of side-lengths $(\ell_a,\ell_b,\ell_c,\ell_d,\ell_e,\ell_f)$ of tetrahedra where $(\ell_a,\ell_b,\ell_d) = (5,7,6)$ and thus depicts a slice of $\mathop{\mathrm{Tetra}}\nolimits(2)$. The exterior blue polytope depicts the values of $(\ell_c,\ell_e,\ell_f)$ which merely satisfy the triangle inequalities, while the interior green region additionally satisfies the Cayley-Menger determinant inequality.

Theorems & Definitions (63)

• Theorem 1
• Theorem 2
• Proposition 3
• proof
• Proposition 4
• proof
• Theorem 5: Binomial theorem
• proof
• Proposition 6
• proof