Lieb’s permanental dominance conjecture

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Minc’s list · Conjecture 42

Lieb’s permanental dominance conjecture

Open

Posed by
Lieb
First posed
1966

Statement

Posed by E. H. Lieb in 1966, extending Problem 2 of Marcus and Minc to characters of any degree; it is Conjecture 42 in Minc’s catalogue.

Conjecture 42. Let \(G\) be a subgroup of \(S_n\) and let \(\chi\) be a character of \(G\). Then

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

for every positive semidefinite Hermitian \(n\times n\) matrix \(A=(a_{ij})\), where \(e\) is the identity permutation.

The right-hand side is the generalized matrix function of \(A\) attached to \((G,\chi)\); the conjecture is the upper counterpart of Schur’s 1918 theorem that the determinant lies below every such normalized function. It suffices to treat irreducible characters. The immanant case is \(G=S_n\) with \(\chi\) irreducible; the case of characters of degree one is Problem 2, and the block inequality of Conjecture 9 is another special case.

Prior progress

  • Lieb (1966) proved the case of the subgroups \(S_p\times S_q\) with the trivial character, the permanental Fischer inequality; iterating it gives every Young subgroup with the trivial character.
  • For immanants, James and Liebeck (1987), Heyfron (1988) and a long series of papers by Pate (1989–1999) proved the inequality for large families of partitions, which together cover every immanant for \(n\le13\).
  • For arbitrary subgroups the conjecture was known for \(n\le3\), and Rodtes (2024) proved it on the matrices all of whose diagonal products are nonnegative.
  • After we started, Zeng (preprint, August 2026) proved the full conjecture for \(n=4\), and Y. Li (preprint, 11 September 2026) proved the immanant case for all \(n\le15\) together with infinite families of partitions.

Our progress so far

  • Every immanant through order \(17\). For every \(n\le17\), every partition \(\lambda\vdash n\) and every positive semidefinite Hermitian \(A\), \(\operatorname{Imm}_\lambda(A)\le\chi_\lambda(e)\operatorname{per}A\). Fourteen partitions of \(14\)–\(17\) lay outside the published families, and we settled them. We obtained this in early September 2026; it is unpublished, and Li’s later preprint independently covers \(n\le15\).
  • Characters of degree one. For characters of degree one of arbitrary subgroups, we proved the conjecture for all \(n\le4\), for \(38\) of the \(45\) subgroup–character classes at \(n=5\), and for infinite families of wreath products; see Problem 2.
  • Low rank beyond order \(17\). For five near-rectangular partitions of orders \(18\) to \(34\), such as \((4,4,4,3,3)\), we proved the immanant inequality on all matrices of rank at most \(5\) or \(6\).
  • A quadratic strengthening. Our paper A Conjecture on the Permanent of Positive Semidefinite Matrices states a quadratic inequality that is implied by Soules’ permanent-on-top conjecture and implies the immanant case at each order, and proves it through order five. It also settles the previously open order-four case of the permanent-on-top conjecture.

References

  1. E. H. Lieb, Proofs of some conjectures on permanents, J. Math. Mech. 16 (1966), 127–134. jstor.org/stable/24901474
  2. G. D. James and M. W. Liebeck, Permanents and immanants of Hermitian matrices, Proc. London Math. Soc. 55 (1987), 243–265. doi:10.1093/plms/s3-55_2.243
  3. P. Heyfron, Immanant dominance orderings for hook partitions, Linear and Multilinear Algebra 24 (1988), 65–78.
  4. T. H. Pate, Tensor inequalities, ξ-functions and inequalities involving immanants, Linear Algebra Appl. 295 (1999), 31–59. doi:10.1016/S0024-3795(99)00035-X
  5. I. M. Wanless, Lieb’s permanental dominance conjecture, preprint (2022). arXiv:2202.01867
  6. K. Rodtes, Some remarks on permanental dominance conjecture, Adv. in Appl. Math. 160 (2024), 102758. doi:10.1016/j.aam.2024.102758
  7. S. Zeng, The general subgroup permanental-dominance conjecture in order four, preprint (2026). arXiv:2608.21749
  8. Y. Li, Lieb’s permanental dominance conjecture for ordinary immanants through order fifteen, preprint (2026). arXiv:2609.13412