Order dimension, strong Bruhat order and lattice properties for posets

Order

Volumn 19, Issue 1, 2002, Pages 73-100

  Reading, Nathan ^a  

^a UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Bruhat; Clivage; Congruence; Coxeter; Critical complex; Dilworth; Dissective; Distributive lattice; MacNeille completion; Monotone triangle; Order dimension

Indexed keywords

EID: 0141465264     PISSN: 01678094     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1015287106470     Document Type: Article

References (33)

1. Billey, S., Jockusch, W. and Stanley, R. (1993) Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2(4), 345-374.
2. Birkhoff, G. (1973) Lattice Theory, 3rd edn, Amer. Math. Soc. Colloq. Publ. 25, Amer. Math. Soc.
3. Björner, A. and Brenti, F., Combinatorics of Coxeter Groups, Graduate Texts in Math., Springer-Verlag, to appear.
4. Björner, A. and Brenti, F. (1996) An improved tableau criterion for Bruhat order, Electron. J. Combin. 3(1), Research Paper 22.
5. Björner, A. and Wachs, M. (1988) Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc. 308(1), 1-37.
6. Björner, A. and Wachs, M. (1997) Shellable nonpure complexes and posets. II, Trans. Amer. Math. Soc. 349(10), 3945-3975.
7. Chajda, I. and Snášel, V. (1998) Congruences in ordered sets, Math. Bohem. 123(1), 95-100.
8. Deodhar, R. and Srinivasan, M. (2001) A statistic on involutions, J. Algebraic Combin. 13(2), 187-198.
9. Dilworth, R. (1950) A decomposition theorem for partially ordered sets, Ann. of Math. (2)51, 161-165.
10. Elkies, N., Kuperberg, G., Larsen, M. and Propp, J. (1992) Alternating sign matrices and domino tilings (Part I), J. Algebraic Combin. 1(2), 111-132.
11. Fan, K. (1972) On Dilworth's coding theorem, Math Z. 127, 92-94.
12. Felsner, S. and Trotter, W. (2000) Dimension, graph and hypergraph coloring, Order 17(2), 167-177.
13. Flath, S. (1993) The order dimension of multinomial lattices, Order 10(3), 201-219.
14. Garsia, A. and Stanton, D. (1984) Group actions on Stanley-Reisner rings and invariants of permutation groups, Adv. Math. 51(2), 107-201.
15. Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G. (1996) CHEVIE - A system for computing and processing generic character tables for finite groups of Lie type, in Weyl Groups and Hecke Algebras, AAECC 7, pp. 175-210.
16. Geck, M. and Kim, S. (1997) Bases for the Bruhat-Chevalley order on all finite Coxeter groups, J. Algebra 197(1), 278-310.
17. Grätzer, G. (1998) General Lattice Theory, 2nd edn, Birkhauser, Boston.
18. Hoffman, K. and Padberg, M. (2001) Set covering, packing and partitioning problems, in C. A. Floudas and M. Pardalos (eds), Encyclopedia of Optimization, Kluwer Academic Publishers, Dordrecht.
19. Humphreys, J. (1990) Reflection Groups and Coxeter Groups, Cambridge Stud. in Adv. Math. 29, Cambridge Univ. Press.
20. Kung, J., Personal communication.
21. Lascoux, A. and Schützenberger, M.-P. (1996) Treillis et bases des groupes de Coxeter, Electron. J. Combin. 3, #R27.
22. MacNeille, H. (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42(3), 416-460.
23. Markowsky, G. (1992) Primes, irreducibles and extremal lattices, Order 9(3), 265-290.
24. Okada, S. (1993) Alternating sign matrices and some deformations of Weyl's denominator formulas, J. Algebraic Combin. 2(2), 155-176.
25. Rabinovitch, I. and Rival, I. (1979) The rank of a distributive lattice, Discrete Math. 25(3), 275-279.
26. Reading, N. (2002) Lattice and order properties of the poset of regions in a hyperplane arrangement, Preprint.
27. Reading, N. (2002) On the structure of Bruhat order, Ph.D. dissertation, University of Minnesota.
28. Robbins, D. (1991) The story of 1, 2, 7, 42, 429, 7436, ..., Math. Intelligencer 13(2), 12-19.
29. Rozen, V. (1991) Coding of ordered sets, Ordered Sets and Lattices 10, 88-96 (Russian).
30. Schönert, M. et al. (1995) GAP - Groups, Algorithms, and Programming, 5th edn, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany.
31. Simion, R. (1999) A type-B associahedron, Preprint.
32. Stanley, R. (1997) Enumerative Combinatorics, Vol. I, Cambridge Stud. in Adv. Math. 49, Cambridge Univ. Press.
33. Trotter, W. (1992) Combinatorics and Partially Ordered Sets: Dimension Theory, Johns Hopkins Series in the Math. Sci., The Johns Hopkins Univ. Press.