Marcus’s block-permanent inequality

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Minc’s list · Conjecture 9

Marcus’s block-permanent inequality

Open

Posed by
Marcus
First posed
1965

Statement

Posed by M. Marcus in the 1965 survey of Marcus and Minc; it is Conjecture 9 in Minc’s catalogue.

Conjecture 9. Let \(A\) be an \(mk\times mk\) positive semidefinite Hermitian matrix partitioned into \(k\times k\) blocks \(A_{ij}\), \(1\le i,j\le m\), and let \(G\) be the \(m\times m\) matrix whose \((i,j)\) entry is \(\operatorname{per}A_{ij}\). Then

\[ \operatorname{per}A\ \ge\ \operatorname{per}G. \]

If the diagonal blocks \(A_{ii}\) are positive definite, equality holds if and only if \(A=A_{11}\oplus A_{22}\oplus\cdots\oplus A_{mm}\).

The equality clause needs \(k\ge2\): for \(k=1\) the matrix \(G\) is \(A\) itself. The inequality is the case of Lieb’s Conjecture 42 for the wreath-product subgroup \(S_k\wr S_m\) of \(S_{mk}\) with its trivial character.

Prior progress

  • Lieb (1966) proved the case of two blocks, \(m=2\), for every \(k\), with the equality clause; Djoković (1969) gave another proof.
  • Pate (1981) proved the weaker bound \(\operatorname{per}A\ge\operatorname{per}G/k!\) for all \(m\) and \(k\).
  • Pate (1982) proved the inequality for real matrices with a fixed number of blocks, for all but finitely many block sizes \(k\).
  • Rodtes (2024) proved Lieb’s dominance on positive semidefinite matrices whose diagonal products \(\prod_i a_{i\sigma(i)}\) are all nonnegative, which covers, for example, \(vv^*+D\) with \(D\ge0\) diagonal.

Our progress so far

  • Three \(2\times2\) blocks at rank at most two. For every \(6\times6\) positive semidefinite Hermitian \(A\) of rank at most \(2\), partitioned into \(2\times2\) blocks, \[ \operatorname{per}A-\operatorname{per}G\ \ge\ \tfrac{14}{9}\,\operatorname{per}A_{11}\operatorname{per}A_{22}\operatorname{per}A_{33}, \] and the constant \(14/9\) is best possible.
  • Rank reduction. If the conjecture holds for \(m-1\) blocks of size \(k\), then any counterexample with \(m\) blocks can be compressed to one of rank at most \((m-1)k\). Together with the previous result, a counterexample with three \(2\times2\) blocks would have to have rank \(3\) or \(4\).
  • Near diagonal matrices. For all \(m,k\ge2\), the inequality and its equality clause hold in a whole neighbourhood of every positive definite diagonal matrix.
  • Rank one plus diagonal. The conjecture holds for \(A=vv^*+D\) with \(D\ge0\) diagonal; when \(D\) is positive definite and \(k\ge2\), equality holds only for block-diagonal \(A\).

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.
  3. D. Ž. Djoković, Simple proofs of a theorem on permanents, Glasgow Math. J. 10 (1969), 52–54.
  4. T. H. Pate, An extension of an inequality involving symmetric products with an application to permanents, Linear and Multilinear Algebra 10 (1981), 103–105.
  5. T. H. Pate, Inequalities relating groups of diagonal products in a Gram matrix, Linear and Multilinear Algebra 11 (1982), 1–17. doi:10.1080/03081088208817427
  6. K. Rodtes, Some remarks on permanental dominance conjecture, Adv. in Appl. Math. 160 (2024), 102758. doi:10.1016/j.aam.2024.102758
  7. I. M. Wanless, Lieb’s permanental dominance conjecture, preprint (2022). arXiv:2202.01867