Timeline for A maximal element, where Schur gives a minimal element
Current License: CC BY-SA 3.0
14 events
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| S May 10, 2016 at 10:21 | comment | added | Keith McNulty | that the Permanent was the maximal Immanant for all Hermitian Positive Semidefinite matrices of rank at most 2. The second part of the thesis focused on Totally Positive Matrices and set out a computational structure (obtained via the Bruhat partial order on the Symmetric Group) to test matrix inequalities on this class of matrices. As a corollary I was able to show that the J-matrices were test matrices for all matrix function inequalities on $n x n$ totally positive matrices for $n \leq 5$. | |
| S May 10, 2016 at 10:21 | comment | added | Keith McNulty | @Suvrit. Sadly I do not have the TeX file any more. I have a hard copy and it is also available at the library of Imperial College London. The proof generalized a result by Tom Pate (Article: Immanant Inequalities, Induced Characters, and Rank Two Partitions Thomas H. Pate Full-text · Article · Feb 1994 · Journal of the London Mathematical Society). Through the generalization and establishing a link between the rank of a partition in character theory of the Symmetric Group and the rank of a Hermitian Positive Semidefinite matrix, it was possible to show (continued) | |
| May 9, 2016 at 15:30 | history | edited | Suvrit | CC BY-SA 3.0 |
replaced dropbox link by LAA link.
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| May 9, 2016 at 14:37 | comment | added | Suvrit | @Keith: very interesting! do you have a version of your proof available electronically? | |
| May 9, 2016 at 10:21 | comment | added | Keith McNulty | This is fascinating. I found it interesting that matrices with rank as low as 2 were found as counterexamples. I know that this would not be possible for Lieb's conjecture as I proved Lieb for all $n \times n$ matrices of rank 2 during my PhD research in 1998 (although I never published the work as I left academia after my PhD). | |
| Oct 13, 2015 at 14:00 | comment | added | Suvrit | Yes, precisely. The POT (which I refer to as "Soules' conjecture" above) is falsified, but the permanental dominance asked in your question is still open (and I would be surprised if it were false!). The key insight (embarrassingly simple in hindsight) of the linked counterexample was to use complex matrices. I wrote a Matlab script to generate counterexamples, and it yields several more rank-2 $5\times5$ matrices as counterexamples. Though the relative errors I observed are typically around 1 to 1.5 percent.. | |
| Oct 13, 2015 at 13:36 | comment | added | Denis Serre | Just to make sure : the counterexample invalidates the Permanent on top (POT) conjecture by Soules, but it leaves open the permanent dominance conjecture. By the way, what is quantitatively important in the counterexample is the relative excess $(\lambda_\max-{\rm per})/{\rm per}$, which is about $2$ percent here. | |
| Oct 13, 2015 at 12:59 | comment | added | Suvrit | I added the links; the countex is very simple: use $v=(4-2i,2+3i,-4+4i,-3-4i,1)$ and $w=(2+4i,3i,2+4i,3i,-5+7i)$. Then set $A=vv^*+ww^*$. With this choice $\lambda_{\max}(\Pi(A))-\text{per}(A) \approx 1.37\times 10^7$, here $\Pi(A)$ denotes the Schur-product matrix. | |
| Oct 13, 2015 at 12:57 | history | edited | Suvrit | CC BY-SA 3.0 |
changed link; also put it on my dropbox
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| Oct 13, 2015 at 6:30 | comment | added | Denis Serre | Many thanks for the additional references ! But the second one (the counterexample) is on a restricted site. Do you have a copy of it ? | |
| Oct 12, 2015 at 21:31 | history | edited | Suvrit | CC BY-SA 3.0 |
removed misclaim on my part!
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| Oct 12, 2015 at 21:05 | history | edited | Suvrit | CC BY-SA 3.0 |
added related new information about soules!
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| Oct 1, 2012 at 10:54 | vote | accept | Denis Serre | ||
| Oct 1, 2012 at 9:27 | history | answered | Suvrit | CC BY-SA 3.0 |