Order

, Volume 23, Issue 2–3, pp 179–195 | Cite as

Rook Poset Equivalence of Ferrers Boards

Article

Abstract

A natural construction due to K. Ding yields Schubert varieties from Ferrers boards. The poset structure of the Schubert cells in these varieties is equal to the poset of maximal rook placements on the Ferrers board under the Bruhat order. We determine when two Ferrers boards have isomorphic rook posets. Equivalently, we give an exact categorization of when two Ding Schubert varieties have identical Schubert cell structures. This also produces a complete classification of isomorphism types of lower intervals of 312-avoiding permutations in the Bruhat order.

Key words

Schubert varieties Ferrers boards rook equivalence rook placements 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Develin, M., Martin, J.L., Reiner, V.: Classification of Ding’s Schubert varieties: finer rook equivalence. Can. J. Math. (to appear), http://arxiv.org/math.AG/0403540
  2. 2.
    Ding, K.: Rook placements and cellular decompositions of partition varieties. Discrete Math. 170, 107–151 (1997)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Ding, K.: Rook placements and classification of partition varieties \(B\backslash M\sb \lambda\). Commun. Contemp. Math. 3, 495–500 (2001)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Foata, D., Schützenberger, M.-P.: On the Rook Polynomials of Ferrers Relations. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), pp.413–436. North-Holland, Amsterdam (1970)Google Scholar
  5. 5.
    Garsia, A.M., Remmel, J.B.: Q-counting rook configurations and a formula of Frobenius. J. Comb. Theory Ser. A 41, 246–275 (1986)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Goldman, J.R., Joichi, J.T., White, D.E.: Rook theory l. Rook equivalence of Ferrers boards. Proc. Am. Math. Soc. 52, 485–492 (1975)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Kaplansky, I., Riordan, J.: The problem of the rooks and its applications. Duke Math. J. 13, 259–268 (1946)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.American Institute of MathematicsPalo AltoUSA

Personalised recommendations