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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access August 26, 2016

An update on a few permanent conjectures

Fuzhen Zhang
From the journal Special Matrices

Abstract

We review and update on a few conjectures concerning matrix permanent that are easily stated, understood, and accessible to general math audience. They are: Soules permanent-on-top conjecture†, Lieb permanent dominance conjecture, Bapat and Sunder conjecture† on Hadamard product and diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on permanent of permanents, and several other conjectures. Some of these conjectures are recently settled; some are still open.We also raise a few new questions for future study. (†conjectures have been recently settled negatively.)

References

[1] T. Ando, Inequalities for permanents, Hokkaido Math. J. 10 (1981), Special Issue, 18–36. Search in Google Scholar

[2] R.B. Bapat, Recent developments and open problems in the theory of permanents,Math. Student 76 (2007), no. 1-4, 55–69. Search in Google Scholar

[3] R.B. Bapat and V.S. Sunder, On majorization and Schur products, Linear Algebra Appl. 72 (1985) 107–117. 10.1016/0024-3795(85)90147-8Search in Google Scholar

[4] R.B. Bapat and V.S. Sunder, An extremal property of the permanent and the determinant, Linear Algebra Appl. 76 (1986) 153–163. 10.1016/0024-3795(86)90220-XSearch in Google Scholar

[5] L.B. Beasley, An inequality on permanents of Hadamard products, Bull. Korean Math. Soc. 37 (2000), no. 3, 633–639. Search in Google Scholar

[6] R. Brualdi, Permanent of the direct product of matrices, Paciffic J. Math. 16 (1966) 471–482. 10.2140/pjm.1966.16.471Search in Google Scholar

[7] D.K. Chang, A note on a conjecture of T.H. Foregger, Linear Multilinear Algebra 15 (1984), no. 3-4, 341–344. Search in Google Scholar

[8] D.K. Chang, On two permanental conjectures, Linear Multilinear Algebra 26 (1990), no. 3, 207–213. Search in Google Scholar

[9] G.-S. Cheon and I.-M. Wanless, An update on Minc’s survey of open problems involving permanents, Linear Algebra Appl. 403 (2005) 314–342. 10.1016/j.laa.2005.02.030Search in Google Scholar

[10] J. Chollet, Unsolved Problems: Is There a Permanental Analogue to Oppenheim’s Inequality? Amer.Math. Monthly 89 (1982), no. 1, 57–58. Search in Google Scholar

[11] D.Z. Djokovic, Simple proofs of a theorem on permanents, Glasgow Math. J. 10 (1969) 52–54. 10.1017/S0017089500000525Search in Google Scholar

[12] J. Drew and C. Johnson, The maximum permanent of a 3-by-3 positive semideffinite matrix, given the eigenvalues, Linear Multilinear Algebra 25 (1989), no. 3, 243–251. Search in Google Scholar

[13] J. Drew and C. Johnson, Counterexample to a conjecture of Mehta regarding permanental maximization, Linear Multilinear Algebra 25 (1989), no. 3, 253–254. Search in Google Scholar

[14] S. Drury, A counterexample to a question of Bapat and Sunder, Electronic Journal of Linear Algebra, Volume 31, pp. 69-70 (2016). 10.13001/1081-3810.3178Search in Google Scholar

[15] S. Drury, A real counterexample to two inequalties involving permanents, Mathematical Inequalities & Applications, in press. Search in Google Scholar

[16] S. Drury, Two open problems on permanents, a private communication, May 30, 2016. Search in Google Scholar

[17] R.J. Gregorac and I.R. Hentzel, A note on the analogue of Oppenheim’s inequality for permanents, Linear Algebra Appl. 94 (1987) 109–112. 10.1016/0024-3795(87)90082-6Search in Google Scholar

[18] R. Grone, An inequality for the second immanant, Linear Multilinear Algebra 18 (1985), no. 2, 147–152. Search in Google Scholar

[19] R. Grone, C. Johnson, E. de S? and H. Wolkowicz, A note onmaximizing the permanent of a positive deffinite Hermitianmatrix, given the eigenvalues, Linear Multilinear Algebra 19 (1986), no. 4, 389–393. Search in Google Scholar

[20] R. Grone and R. Merris, Conjectures on permanents, Linear Multilinear Algebra 21 (1987), no. 4, 419–427. Search in Google Scholar

[21] R. Grone, S. Pierce and W. Watkins, Extremal correlation matrices, Linear Algebra Appl. 134 (1990) 63–70. 10.1016/0024-3795(90)90006-XSearch in Google Scholar

[22] P. Heyfron, Immanant dominance orderings for hook partitions, Linear Multilinear Algebra 24 (1988), no. 1, 65–78. Search in Google Scholar

[23] R.A. Horn, The Hadamard product, Proceedings of Symposia in AppliedMathematics, Vol. 40, edited by C.R. Johnson, Amer. Math. Soc., Providence, RI, pp. 87–169, 1990. 10.1090/psapm/040/1059485Search in Google Scholar

[24] G. James, Immanants, Linear Multilinear Algebra 32 (1992), no. 3-4, 197–210. Search in Google Scholar

[25] G. James and M. Liebeck, Permanents and immanants of Hermitian matrices, Proc. London Math. Soc. (3) 55 (1987), no. 2, 243–265. Search in Google Scholar

[26] C.R. Johnson, The permanent-on-top conjecture: a status report, in Current Trends in Matrix Theory, edited by F. Uhlig and R. Grone, Elsevier Science Publishing Co., New York, pp. 167–174, 1987. Search in Google Scholar

[27] C.R. Johnson and F. Zhang, Erratum: The Robertson-Taussky Inequality Revisted, Linear Multilinear Algebra 38 (1995) 281– 282. 10.1080/03081089508818363Search in Google Scholar

[28] E.H. Lieb, Proofs of some conjectures on permanents, J. Math. and Mech. 16 (1966) 127–134. Search in Google Scholar

[29] M. Marcus, Permanents of direct products, Proc. Amer. Math. Soc. 17 (1966) 226–231. 10.1090/S0002-9939-1966-0191910-1Search in Google Scholar

[30] M. Marcus, Finite Dimensional Multilinear Algebra. Part I. (Pure and applied mathematics 23), Marcel Dekker, New York, 1973. Search in Google Scholar

[31] M. Marcus and H. Minc, Inequalities for general matrix functions, Bull. Amer. Math. Soc. 70 (1964) 308–313. 10.1090/S0002-9904-1964-11136-8Search in Google Scholar

[32] M. Marcus and H. Minc, Permanents, Amer. Math. Monthly 72 (1965) 577–591. 10.1080/00029890.1965.11970575Search in Google Scholar

[33] M. Marcus and M. Sandy, Bessel’s inequality in tensor space, Linear Multilinear Algebra 23 (1988), no. 3, 233–249. Search in Google Scholar

[34] R. Merris, Extensions of the Hadamard determinant theorem, Israel J. Math. 46 (1983), no. 4, 301–304. Search in Google Scholar

[35] R. Merris, The permanental dominance conjecture, Current trends in matrix theory : proceedings of the Third Auburn Matrix Theory Conference, March 19-22, 1986 at Auburn University, Auburn, Alabama, U.S.A., editors, Frank Uhlig, Robert Grone. Search in Google Scholar

[36] R. Merris, The permanental dominance conjecture, in Current Trends in Matrix Theory, edited by F. Uhlig and R. Grone, Elsevier Science Publishing Co., New York, pp. 213–223, 1987. Search in Google Scholar

[37] R. Merris, Multilinear Algebra, Gordon & Breach, Amsterdam, 1997. 10.1201/9781498714907Search in Google Scholar

[38] R. Merris and W. Watkins, Inequalities and identities for generalized matrix functions, Linear Algebra Appl. 64 (1985) 223– 242. 10.1016/0024-3795(85)90279-4Search in Google Scholar

[39] H. Minc, Permanents, Addison-Wesley, New York, 1978. Search in Google Scholar

[40] H. Minc, Theory of permanents 1978-1981, Linear Multilinear Algebra 12 (1983) 227–263. 10.1080/03081088308817488Search in Google Scholar

[41] H. Minc, Theory of permanents 1982-1985, Linear Multilinear Algebra 21 (1987) 109–148. 10.1080/03081088708817786Search in Google Scholar

[42] A. Oppenheim, Inequalities Connected with Deffinite Hermitian Forms, J. London Math. Soc. S1-5, no. 2, 114–119. 10.1112/jlms/s1-5.2.114Search in Google Scholar

[43] T.H. Pate, An extension of an inequality involving symmetric products with an application to permanents, Linear Multilinear Algebra 10 (1981), no. 2, 103–105. Search in Google Scholar

[44] T.H. Pate, Inequalities relating groups of diagonal products in a Gram matrix, Linear Multilinear Algebra 11 (1982), no. 1, 1–17. Search in Google Scholar

[45] T.H. Pate, An inequality involving permanents of certain direct products, Linear Algebra Appl. 57 (1984) 147–155. 10.1016/0024-3795(84)90183-6Search in Google Scholar

[46] T.H. Pate, Permanental dominance and the Soules conjecture for certain right ideals in the group algebra, LinearMultilinear Algebra 24 (1989), no. 2, 135–149. Search in Google Scholar

[47] T.H. Pate, Partitions, irreducible characters, and inequalities for generalized matrix functions, Trans. Amer. Math. Soc. 325 (1991), no. 2, 875–894. Search in Google Scholar

[48] T.H. Pate, Rowappendingmaps, ζ -functions, and immanant inequalities for Hermitian positive semi-deffinitematrices, Proc. London Math. Soc. (3) 76 (1998), no. 2, 307–358. Search in Google Scholar

[49] T.H. Pate, Tensor inequalities, ζ -functions and inequalities involving immanants, Linear Algebra Appl. 295 (1999), no. 1-3, 31–59. Search in Google Scholar

[50] S. Pierce, Permanents of correlation matrices, in Current Trends in Matrix Theory, edited by F. Uhlig and R. Grone, Elsevier Science Publishing Co., New York, pp. 247–249, 1987. Search in Google Scholar

[51] I. Schur, Über endliche Gruppen und Hermitesche Formen (German), Math. Z. 1 (1918), no. 2-3, 184–207. Search in Google Scholar

[52] V.S. Shchesnovich, The permanent-on-top conjecture is false, Linear Algebra Appl. 490 (2016) 196–201. 10.1016/j.laa.2015.10.034Search in Google Scholar

[53] G. Soules,Matrix functions and the Laplace expension theorem, Ph.D. Dissertation, University of California - Santa Barbara, July, 1966. Search in Google Scholar

[54] G. Soules, An approach to the permanental-dominance conjecture, Linear Algebra Appl. 201 (1994) 211–229. 10.1016/0024-3795(94)90117-1Search in Google Scholar

[55] R.C. Thompson, A determinantal inequality, Canad. Math. Bull., vol.4, no.1, Jan. 1961, pp. 57–62. 10.4153/CMB-1961-010-9Search in Google Scholar

[56] J.H. van Lint and R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 2001. 10.1017/CBO9780511987045Search in Google Scholar

[57] F. Zhang, Notes on Hadamard products of matrices, Linear Multilinear Algebra 25 (1989) 237–242. 10.1080/03081088908817946Search in Google Scholar

[58] F. Zhang, An analytic approach to a permanent conjecture, Linear Algebra Appl. 438 (2013) 1570–1579. 10.1016/j.laa.2011.09.034Search in Google Scholar

[59] F. Zhang, J. Liang, and W. So, Two conjectures on permanents in Report on Second Conference of the International Linear Algebra Society (Lisbon, 1992) by J.A.Dias da Silva, Linear Algebra Appl. 197/198 (1994) 791–844. Search in Google Scholar

Received: 2016-3-18
Accepted: 2016-7-27
Published Online: 2016-8-26

©2016 Fuzhen Zhang

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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