Universality of Rank 6 Plücker relations and Grassmann cone preserving maps
HTML articles powered by AMS MathViewer
- by Alex Kasman, Kathryn Pedings, Amy Reiszl and Takahiro Shiota
- Proc. Amer. Math. Soc. 136 (2008), 77-87
- DOI: https://doi.org/10.1090/S0002-9939-07-09122-8
- Published electronically: October 11, 2007
- PDF | Request permission
Abstract:
The Plücker relations define a projective embedding of the Grassmann variety $Gr(p,n)$. We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a certain finite set of linear maps $\bigwedge ^pk^n\to \bigwedge ^2k^4$, and pulling back the unique Plücker relation on $\bigwedge ^2k^4$. We also give a quadratic equation depending on $(p+2)$ parameters having the same properties.References
- Silvana Abeasis, On the Plücker relations for the Grassmann varieties, Adv. in Math. 36 (1980), no. 3, 277–282. MR 577305, DOI 10.1016/0001-8708(80)90017-1
- M. J. Bergvelt and A. P. E. ten Kroode, Partitions, vertex operator constructions and multi-component KP equations, Pacific J. Math. 171 (1995), no. 1, 23–88. MR 1362978, DOI 10.2140/pjm.1995.171.23
- N. Bourbaki, Algèbre. Chapitre III: Algèbre multilinéaire, Actualités Sci. Ind., no. 1044. Hermann et Cie., Paris, 1948
- Claude C. Chevalley, The algebraic theory of spinors, Columbia University Press, New York, 1954. MR 0060497, DOI 10.7312/chev93056
- S. Duzhin, Decomposable skew-symmetric functions, Mosc. Math. J. 3 (2003), no. 3, 881–888, 1198 (English, with English and Russian summaries). {Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday}. MR 2078565, DOI 10.17323/1609-4514-2003-3-3-881-888
- Michael Gekhtman and Alex Kasman, On KP generators and the geometry of the HBDE, J. Geom. Phys. 56 (2006), no. 2, 282–309. MR 2173898, DOI 10.1016/j.geomphys.2005.02.002
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523, DOI 10.1002/9781118032527
- W. V. D. Hodge and D. Pedoe, Methods of algebraic geometry. Vol. I, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Book I: Algebraic preliminaries; Book II: Projective space; Reprint of the 1947 original. MR 1288305, DOI 10.1017/CBO9780511623899
- S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, DOI 10.2307/2317421
- I. Krichever, “Characterizing Jacobians via trisecants of the Kummer Variety”, math.AG/0605625
- I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
- Tetsuji Miwa, On Hirota’s difference equations, Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), no. 1, 9–12. MR 649054
- Shigeru Mukai, Curves and Grassmannians, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 19–40. MR 1285374
- Kristian Ranestad and Frank-Olaf Schreyer, Varieties of sums of powers, J. Reine Angew. Math. 525 (2000), 147–181. MR 1780430, DOI 10.1515/crll.2000.064
- Hiroshi Fujita, Peter D. Lax, and Gilbert Strang (eds.), Nonlinear partial differential equations in applied science, North-Holland Mathematics Studies, vol. 81, Kinokuniya Book Store Co., Ltd., Tokyo; Kinokuniya Company Ltd., Tokyo, 1983. MR 730231, DOI 10.1016/0167-2789(83)90239-7
- Richard P. Stanley, Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62, Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. MR 1676282, DOI 10.1017/CBO9780511609589
- Kanehisa Takasaki and Takashi Takebe, Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), no. 5, 743–808. MR 1346289, DOI 10.1142/S0129055X9500030X
- R. Westwick, Linear transformations on Grassmann spaces. III, Linear and Multilinear Algebra 2 (1974/75), 257–268. MR 429961, DOI 10.1080/03081087408817069
Bibliographic Information
- Alex Kasman
- Affiliation: Department of Mathematics, College of Charleston, 66 George Street, Charleston, South Carolina 29424
- MR Author ID: 366818
- Email: kasman@cofc.edu
- Kathryn Pedings
- Affiliation: Department of Mathematics, College of Charleston, 66 George Street, Charleston, South Carolina 29424
- Email: kepedings@edisto.cofc.edu
- Amy Reiszl
- Affiliation: Department of Mathematics, College of Charleston, 66 George Street, Charleston, South Carolina 29424
- Email: amreiszl@edisto.cofc.edu
- Takahiro Shiota
- Affiliation: Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan
- Received by editor(s): September 30, 2005
- Received by editor(s) in revised form: January 31, 2007
- Published electronically: October 11, 2007
- Communicated by: Michael Stillman
- © Copyright 2007 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 77-87
- MSC (2000): Primary 14M15, 15A75
- DOI: https://doi.org/10.1090/S0002-9939-07-09122-8
- MathSciNet review: 2350391