Permanent dominance for degree-one characters

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Minc’s list · Problem 2

Permanent dominance for degree-one characters

Open

Posed by
Marcus and Minc
First posed
1965

Statement

Posed by M. Marcus and H. Minc in their 1965 survey of permanents; it is Problem 2 in Minc’s catalogue.

Problem 2. Let \(H\) be a subgroup of \(S_n\) and let \(\chi\) be a character of degree \(1\) of \(H\). Under what conditions on \(\chi\) does the inequality

\[ \sum_{\sigma\in H}\chi(\sigma)\prod_{i=1}^n a_{i\sigma(i)}\ \le\ \operatorname{per}A \]

hold for all positive semidefinite Hermitian \(n\times n\) matrices \(A\)?

For \(H=S_n\) the trivial character gives the permanent itself and the sign character gives \(\det A\le\operatorname{per}A\). Marcus and Minc’s original wording asks for conditions on both \(\chi\) and \(H\). Lieb’s Conjecture 42 predicts that the inequality always holds; this problem is its case of characters of degree one.

Prior progress

  • Lieb (1966) proved the inequality for the subgroups \(S_p\times S_q\) and for the wreath product \(S_k\wr S_2\), both with the trivial character.
  • The inequality holds for every subgroup and character when \(n\le3\). James (1992) found a \(4\times4\) matrix at which a degree-one character of the alternating group \(A_4\) attains equality.
  • Rodtes (2024) proved it for all subgroups on the matrices whose diagonal products are all nonnegative.
  • After we started, Zeng (preprint, August 2026) proved the inequality for every character of every subgroup of \(S_4\).

Our progress so far

  • Complete answer for \(n=4\). For every subgroup of \(S_4\) and every character of degree one the inequality holds. For the hardest case, the two non-real characters of \(A_4\), we also found the equality cases for correlation matrices. We obtained this in early August 2026, before Zeng’s more general preprint.
  • Order five, \(38\) of \(45\) classes. Up to conjugacy there are \(45\) pairs \((H,\chi)\) with \(H\le S_5\). We proved the inequality for \(38\) of them. The seven that remain come from the cyclic group \(C_5\), the dihedral group \(D_{10}\) and the affine group \(\mathrm{AGL}(1,5)\) of order \(20\); each of them is known to hold on the matrices \(vv^*+D\) with \(D\ge0\) diagonal.
  • Wreath products. For \(S_k\wr S_m\le S_{km}\) with \(k,m\ge2\), both degree-one characters that restrict to the sign on the top factor \(S_m\) satisfy the inequality, and for \(m=2\) all four degree-one characters do. The remaining trivial-top cases include Conjecture 9.

References

  1. M. Marcus and H. Minc, Permanents, Amer. Math. Monthly 72 (1965), 577–591. doi:10.1080/00029890.1965.11970575
  2. E. H. Lieb, Proofs of some conjectures on permanents, J. Math. Mech. 16 (1966), 127–134. jstor.org/stable/24901474
  3. G. James, Immanants, Linear and Multilinear Algebra 32 (1992), 197–210. doi:10.1080/03081089208818163
  4. I. M. Wanless, Lieb’s permanental dominance conjecture, preprint (2022). arXiv:2202.01867
  5. K. Rodtes, Some remarks on permanental dominance conjecture, Adv. in Appl. Math. 160 (2024), 102758. doi:10.1016/j.aam.2024.102758
  6. S. Zeng, The general subgroup permanental-dominance conjecture in order four, preprint (2026). arXiv:2608.21749