Set-theoretic defining equations of the variety of principal minors of symmetric matrices

Algebra and Number Theory

Volumn 5, Issue 1, 2011, Pages 75-109

  Oeding, Luke ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Determinant; G module; G variety; Hyperdeterminant; Hyperdeterminantal module; Principal minors; Relations among minors; Representation theory; Symmetric matrices; Variety of principal minors

Indexed keywords

EID: 80052069932     PISSN: 19370652     EISSN: None     Source Type: Journal    
DOI: 10.2140/ant.2011.5.75     Document Type: Article

References (32)

1. E. S. Allman and J. A. Rhodes, "Phylogenetic ideals and varieties for the general Markov model", Adv. in Appl. Math. 40:2 (2008), 127-148. MR 2008m:60145 Zbl 1131.92046
2. Borcea, P. Brändén, and T. M. Liggett, "Negative dependence and the geometry of polynomials", J. Amer. Math. Soc. 22:2 (2009), 521-567. MR 2010b:62215
3. A. Borodin and E. M. Rains, "Eynard-Mehta theorem, Schur process, and their Pfaffian analogs", J. Stat. Phys. 121:3-4 (2005), 291-317. MR 2006k:82039 Zbl 1127.82017
4. P. Bürgisser, M. Clausen, and M. A. Shokrollahi, Algebraic complexity theory, Grundlehren der MathematischenWissenschaften 315, Springer, Berlin, 1997. MR 99c:68002 Zbl 1087.68568
5. P. Comon and M. Rajih, "Blind identification of under-determined mixtures based on the characteristic function", Signal processing 86:9 (2006), 2271-2281. Zbl 1172. 65304
6. P. Comon, G. Golub, L.-H. Lim, and B. Mourrain, "Symmetric tensors and symmetric tensor rank", SIAM J. Matrix Anal. Appl. 30:3 (2008), 1254-1279. MR 2009i:15039 Zbl 1181.15014
7. D. Cox, J. Little, and D. O'Shea, Ideals, varieties, and algorithms, 3rd ed., Springer, New York, 2007. MR 2007h:13036 Zbl 1118.13001
8. W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics 129, Springer, New York, 1991. MR 93a:20069 Zbl 0744.22001
9. I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser, Boston, 1994. MR 95e:14045 Zbl 0827. 14036
10. R. Goodman and N. R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Applications 68, Cambridge University Press, 1998. MR 99b:20073 Zbl 0901.22001
11. K. Griffin and M. J. Tsatsomeros, "Principal minors, II: The principal minor assignment problem", Linear Algebra Appl. 419:1 (2006), 125-171. MR 2008h:15015 Zbl 1110.65035
12. Harris, Algebraic geometry, Graduate Texts in Mathematics 133, Springer, New York, 1992. MR 93j:14001 Zbl 0779.14001
13. O. Holtz, "Not all GKK _matrices are stable", Linear Algebra Appl. 291:1-3 (1999), 235-244. MR 2000a:15011 Zbl 0968.15014
14. O. Holtz and H. Schneider, "Open problems on GKK _-matrices", Linear Algebra Appl. 345 (2002), 263-267. MR 1883278 Zbl 1160.15305
15. O. Holtz and B. Sturmfels, "Hyperdeterminantal relations among symmetric principal minors", J. Algebra 316:2 (2007), 634-648. MR 2009c:15032 Zbl 1130.15005
16. J. M. Landsberg, "Geometry and the complexity of matrix multiplication", Bull. Amer. Math. Soc.N.S./ 45:2 (2008), 247-284. MR 2009b:68055 Zbl 1145.68054
17. J. M. Landsberg, "The geometry of tensors: applications to complexity, statistics and other areas", in preparation.
18. J. M. Landsberg and L. Manivel, "On the projective geometry of rational homogeneous varieties", Comment. Math. Helv. 78:1 (2003), 65-100. MR 2004a:14050 Zbl 1048.14032
19. J. M. Landsberg and L. Manivel, "On the ideals of secant varieties of Segre varieties", Found. Comput. Math. 4:4 (2004), 397-422. MR 2005m:14101 Zbl 1068.14068
20. J. M. Landsberg and J. Weyman, "On tangential varieties of rational homogeneous varieties", J. Lond. Math. Soc.2/76:2 (2007), 513-530. MR 2008m:14101 Zbl 1127.14045
21. L. de Lathauwer and A. de Baynast, "Blind deconvolution of DS-CDMA signals by means of decomposition in rank-.1; L; L/ terms", IEEE Trans. Signal Process. 56:4 (2008), 1562-1571. MR 2516571
22. S. Lin and B. Sturmfels, "Polynomial relations among principal minors of a 4_4-matrix", J. Algebra 322:11 (2009), 4121-4131. MR 2010m:15014 Zbl 1198.14049
23. T. Mikkonen, "The ring of graph invariants-graphic values", preprint, 2007. arXiv 0712.0146
24. E. J. Nanson, "On the relations between the coaxial minors of a determinant", Philosophical Magazine 44:5 (1897), 362-367. JFM 28.0140.01
25. L. A. Oeding, G-varieties and the principal minors of symmetric matrices, ProQuest LLC, Ann Arbor, MI, 2009. MR 2713672
26. L. Oeding, "Set-theoretic defining equations of the tangential variety of the Segre variety", J. Pure Appl. Algebra 215:6 (2011), 1516-1527. MR 2769247
27. L. Pachter and B. Sturmfels, Algebraic statistics for computational biology, Cambridge University Press, New York, 2005. MR 2006i:92002 Zbl 1108.62118
28. J. Sidman and S. Sullivant, "Prolongations and computational algebra", Canad. J. Math. 61:4 (2009), 930-949. MR 2010h:14084 Zbl 1174.13042
29. E. B. Stouffer, "On the independence of principal minors of determinants", Trans. Amer. Math. Soc. 26:3 (1924), 356-368. MR 1501282 JFM 50.0065.01
30. B. Stouffer, "Expressions for the general determinant in terms of its principal minors", Amer. Math. Monthly 35:1 (1928), 18-21. MR 1521341 JFM 54.0108.06
31. D. G. Wagner, "Negatively correlated random variables and Mason's conjecture for independent sets in matroids", Ann. Comb. 12:2 (2008), 211-239. MR2009f:05053 Zbl 1145.05003
32. H. Weyl, The classical groups, Princeton University Press, 1997. MR 98k:01049 Zbl 1024.20501