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

• URL: http://arxiv.org/abs/2510.04877v1
• Date: Mon, 06 Oct 2025 15:00:25 GMT
• Title: The tetrahedral Horn problem and asymptotics of U(n) 6j symbols
• Abstract summary: Horn's problem is concerned with characterizing the eigenvalues $(a,b,c,d,e,f)$ of Hermitian matrices $(A,B,C,D,E,F)$.<n>This paper aims to characterize the tetrashedra eigenvalues $(a,b,c,d,e,f)$ of Hermitian matrices $(A,B,C,D,E,F)$ forming the edges of a tetrahedron.
• Score: 2.2633707287259868
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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