Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Weak order on complete quadrics


Authors: Mahir Bilen Can and Michael Joyce
Journal: Trans. Amer. Math. Soc. 365 (2013), 6269-6282
MSC (2010): Primary 14M17; Secondary 14L30
DOI: https://doi.org/10.1090/S0002-9947-2013-05813-8
Published electronically: July 10, 2013
MathSciNet review: 3105751
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using an action of the Richardson-Springer monoid on involutions, we study the weak order on the variety of complete quadrics. Maximal chains in the poset are explicitly determined. Applying results of Brion, our calculations describe certain cohomology classes in the complete flag variety.


References [Enhancements On Off] (What's this?)

  • 1. E. Bagno and Y. Cherniavsky, Congruence $ B$-orbits and the Bruhat poset of involutions of the symmetric group, Discrete Math. 312 (2012), no. 6, 1289-1299. MR 2876381
  • 2. M. Brion, The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), no. 1, 137-174. MR 1610599 (99b:14049)
  • 3. -, On orbit closures of spherical subgroups in flag varities, Comment. Math. Helv. 76 (2001), no. 2, 263-299. MR 1839347 (2002e:14084)
  • 4. M.B. Can and M. Joyce, Ordered Bell numbers, Hermite polynomials, skew Young tableaux, and Borel orbits, Submitted for publication.
  • 5. M.B. Can and L.E. Renner, Bruhat-Chevalley order on the rook monoid, Turkish J. Math. 35 (2011), no. 2, 1-21.
  • 6. C. De Concini and C. Procesi, Complete symmetric varieties, Invariant theory (Montecatini, 1982) (Berlin), Lecture Notes in Math., vol. 996, Springer, 1983, pp. 1-44. MR 718125 (85e:14070)
  • 7. C. Ehresmann, Sur la topologie de certains espaces homogènes, Ann. of Math. (2) 35 (1934), no. 2, 396-443. MR 1503170
  • 8. F. Incitti, The Bruhat order on the involutions of the symmetric group, J. Algebraic Combin. 20 (2004), no. 3, 243-261. MR 2106960 (2005h:06003)
  • 9. F. Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995), no. 2, 285-309. MR 1324631 (96c:14039)
  • 10. Edwin A. Pennell, Mohan S. Putcha, and Lex E. Renner, Analogue of the Bruhat-Chevalley order for reductive monoids, J. Algebra 196 (1997), no. 2, 339-368. MR 1475115 (98j:20090)
  • 11. Mohan S. Putcha, Shellability in reductive monoids, Trans. Amer. Math. Soc. 354 (2002), no. 1, 413-426 (electronic). MR 1859281 (2002i:20095)
  • 12. M.S. Putcha, Bruhat-Chevalley order in reductive monoids, J. Algebraic Combin. 20 (2004), no. 1, 34-53. MR 2104819 (2005h:20154)
  • 13. L.E. Renner, Analogue of the Bruhat decomposition for algebraic monoids, J. Algebra 101 (1986), no. 2, 303-338. MR 847163 (87f:20066)
  • 14. -, Linear algebraic monoids, Encyclopaedia of Mathematical Sciences, vol. 134, Springer-Verlag, Berlin, 2005, Invariant Theory and Algebraic Transformation Groups, V. MR 2134980 (2006a:20002)
  • 15. R.W. Richardson and T.A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1-3, 389-436. MR 1066573 (92e:20032)
  • 16. R.W. Richardson and T.A Springer, Complements to:``The Bruhat order on symmetric varieties'', Geom. Dedicata 49 (1994), no. 2, 231-238. MR 1266276 (95f:20074)
  • 17. T.A. Springer, Combinatorics of $ B$-orbits in a wonderful compactification, Algebraic groups and arithmetic, Tata Inst. Fund. Res., 2004, pp. 99-117. MR 2094109 (2006b:14090)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 14M17, 14L30

Retrieve articles in all journals with MSC (2010): 14M17, 14L30


Additional Information

Mahir Bilen Can
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70130

Michael Joyce
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70130

DOI: https://doi.org/10.1090/S0002-9947-2013-05813-8
Received by editor(s): January 11, 2012
Received by editor(s) in revised form: February 17, 2012
Published electronically: July 10, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society