## Universality of Rank 6 Plücker relations and Grassmann cone preserving maps

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- by Alex Kasman, Kathryn Pedings, Amy Reiszl and Takahiro Shiota PDF
- Proc. Amer. Math. Soc.
**136**(2008), 77-87 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

## 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
- © 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