Spectrahedrality Under the Eigenvalue-Set Transformation

Definitions and mathematical statement

Let $n\geq 1$ be an integer. Write $\mathcal S^n=\{A\in\mathbb R^{n\times n}:A^{\mathsf T}=A\}$. For $A\in\mathcal S^n$, let $\lambda(A)=(\lambda_1(A),\ldots,\lambda_n(A))$, where $\lambda_1(A)\leq\cdots\leq\lambda_n(A)$ are its real eigenvalues, counted with multiplicity.

For a finite-dimensional real vector space $V$, a set $S\subseteq V$ is called a spectrahedron if there exist an integer $r\geq1$, a matrix $B_0\in\mathcal S^r$, and a linear map $L:V\to\mathcal S^r$ such that \[ S=\{x\in V:B_0+L(x)\succeq0\}. \] Here $B\succeq0$ means $u^{\mathsf T}Bu\geq0$ for every $u\in\mathbb R^r$. The matrix size and affine map may differ for different spectrahedral representations. This draft uses the standard affine linear matrix inequality definition; the supplied excerpt does not explicitly specify its convention.

Let $M\subseteq\mathbb R^n$ satisfy \[ (x_1,\ldots,x_n)\in M \quad\Longrightarrow\quad (x_{\sigma(1)},\ldots,x_{\sigma(n)})\in M \qquad\text{for every permutation }\sigma\text{ of }\{1,\ldots,n\}. \] Define \[ \Lambda(M)=\{A\in\mathcal S^n:\lambda(A)\in M\}. \] Permutation invariance makes membership independent of the chosen ordering of the eigenvalues.

Conjecture attributed in the supplied record to Mario Kummer. For every integer $n\geq1$ and every coordinate-permutation-invariant set $M\subseteq\mathbb R^n$, is it true that \[ M\text{ is a spectrahedron in }\mathbb R^n \quad\Longleftrightarrow\quad \Lambda(M)\text{ is a spectrahedron in }\mathcal S^n? \] No additional convexity, closedness, dimension, or boundary hypotheses on $M$ are specified in the supplied statement. The registry labels the problem open; this is a supplied status assessment, not an independently verified mathematical conclusion.

Short English statement

For a set $M$ invariant under permutations of its coordinates, form the set of real symmetric matrices whose eigenvalue vectors belong to $M$. The conjecture asks whether $M$ admits an affine linear matrix inequality representation exactly when this matrix set does. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004822$, title Spectrahedrality Under the Eigenvalue-Set Transformation, collection Oberwolfach Reports Open Problems, exact source problem code catalogue entry.
• Supplied report citation: MFO-RIMS Tandem Workshop: Symmetries on Polynomial Ideals and Varieties (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/43, pp. 2365--2396. https://doi.org/10.4171/owr/2021/43. The original excerpt associates the conjecture with Mario Kummer and the contribution title Symmetric spectrahedra with respect to the permutation and orthogonal group. The registry separately gives proposed year 2021 and the tag owr-year:2021; the supplied year discrepancy remains unresolved.
• Supplied literature source labeled "Kummer": https://doi.org/10.1016/j.aim.2021.107749. No fuller bibliographic description is supplied for this link.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that known sufficient conditions and derivative-relaxation cases do not establish the equivalence for all symmetric sets, and describes a connection with the unresolved generalized Lax conjecture. Its listed sources are the OWR report and Kummer links above. This is supplied dated curation, not independently verified; time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly and definitions restored. These are supplied provenance claims; no source was independently visited or verified in preparing this draft.