Universality of Rank 6 Plücker relations and Grassmann cone preserving maps
Authors:
Alex Kasman, Kathryn Pedings, Amy Reiszl and Takahiro Shiota
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
Published electronically:
October 11, 2007
MathSciNet review:
2350391
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- Silvana Abeasis, On the Plücker relations for the Grassmann varieties, Adv. in Math. 36 (1980), no. 3, 277–282. MR 577305, DOI https://doi.org/10.1016/0001-8708%2880%2990017-1
- 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
- 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
- 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 https://doi.org/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 https://doi.org/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
- 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
- S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, DOI https://doi.org/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 https://doi.org/10.1515/crll.2000.064
- Hiroshi Fujita, Peter D. Lax, and Gilbert Strang (eds.), Nonlinear partial differential equations in applied science, Lecture Notes in Numerical and Applied Analysis, vol. 5, Kinokuniya Book Store Co., Ltd., Tokyo; Kinokuniya Company Ltd., Tokyo, 1983. North-Holland Mathematics Studies, 81. MR 730231
- 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
- Kanehisa Takasaki and Takashi Takebe, Integrable hierarchies and dispersionless limit, Rev. Math. Phys. 7 (1995), no. 5, 743–808. MR 1346289, DOI https://doi.org/10.1142/S0129055X9500030X
- R. Westwick, Linear transformations on Grassmann spaces. III, Linear and Multilinear Algebra 2 (1974/75), 257–268. MR 429961, DOI https://doi.org/10.1080/03081087408817069
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Additional 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
Article copyright:
© Copyright 2007
American Mathematical Society
