Optimizing over Nonzero Littlewood–Richardson Coefficients: The Quantum Max d-Cut of Complete Bipartite Graphs

Konvalinka asked (Oberwolfach Report 2/2026, p. 150) for the maximum of \(\Delta(\lambda,\mu,\nu)=\eta(\lambda)-\eta(\mu)-\eta(\nu)\), where \(\eta(\lambda)=|\lambda|^2+\frac{d(d-1)(2d-1)}{6}-\sum_{i=1}^{d}(\lambda_i-i+1)^2\), over the triples of partitions with \(|\lambda|=n\), \(|\mu|=k\), \(|\nu|=n-k\), at most \(d\) parts and nonzero Littlewood–Richardson coefficient \(c^{\lambda}_{\mu\nu}\). By the work of Klep, Štrekelj and Volčič this maximum is the quantum max \(d\)-cut of the complete bipartite graph \(K_{n-k,k}\), that is, the largest eigenvalue of the sum of \(2(I-\operatorname{SWAP}_{ij})\) over its edges acting on \((\mathbb{C}^d)^{\otimes n}\). They determined it for \(d\le3\) and for \(k\le4\) and conjectured that it is always attained at a triple in which at most one part of \(\lambda\) is the sum of a part of \(\mu\) and a part of \(\nu\) (their Conjecture 7.7). We determine the maximum for all \(n\), \(k\), \(d\). An explicit Littlewood–Richardson tableau and a level-set argument reduce the problem to the maximization of an elementary function \(\Psi(\mu,x)\) over partitions \(\mu\) of \(k\) with at most \(d\) parts and vectors \(x\in\mathbb{Z}_{\ge0}^d\) with sum \(n-k\), an optimal \(\lambda\) being the decreasing rearrangement of \(\mu+x\). Exchange and convexity arguments then give an optimum with at most one row in which both \(\mu\) and \(x\) are nonzero and with balanced remaining rows. Consequently, for \(d\ge2\) the maximum equals \(2k(n-k)+2\max\{\Psi_A,\Psi_B\}\) over an explicit list of at most \((d-1)(1+k(n-k))\) integers built from balanced partitions; it is \(0\) for \(d=1\) and \(4k(n-k)\) for \(d\ge n\). As a corollary, Conjecture 7.7 of Klep, Štrekelj and Volčič holds, with explicit optimal triples \((\lambda,\mu,\nu)=(P\uplus Z,P,Z)\) or \((P\uplus Z\uplus(\pi+y),\,P\uplus(\pi),\,Z\uplus(y))\), where \(P\) and \(Z\) are balanced partitions and \(\uplus\) denotes the union of the parts. The answer is a maximum over a candidate list of polynomial size, not a single closed formula. The known cases (\(d=2\), which is the Lieb–Mattis theorem; \(d=3\); \(k\le4\); the star graph) are recovered, and the results agree with brute-force computations over all Littlewood–Richardson coefficients for \(n\le22\) and with the reduction for \(n\le80\). The problem is Konvalinka's; the reduction of the quantum max \(d\)-cut of \(K_{n-k,k}\) to this maximum, the cases \(d\le3\) and \(k\le4\) and the conjecture are due to Klep, Štrekelj and Volčič. New, to our knowledge, are the general solution and the proof of the conjecture. This is an unrefereed note.