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.
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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
Ding, K.: Rook placements and cellular decompositions of partition varieties. Discrete Math. 170, 107–151 (1997)
Ding, K.: Rook placements and classification of partition varieties \(B\backslash M\sb \lambda\). Commun. Contemp. Math. 3, 495–500 (2001)
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)
Garsia, A.M., Remmel, J.B.: Q-counting rook configurations and a formula of Frobenius. J. Comb. Theory Ser. A 41, 246–275 (1986)
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)
Kaplansky, I., Riordan, J.: The problem of the rooks and its applications. Duke Math. J. 13, 259–268 (1946)
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Develin, M. Rook Poset Equivalence of Ferrers Boards. Order 23, 179–195 (2006). https://doi.org/10.1007/s11083-006-9039-8
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DOI: https://doi.org/10.1007/s11083-006-9039-8