Lexicographic Shellability for Balanced Complexes
Patricia Hersh
DOI: 10.1023/A:1025044720847
Abstract
We introduce a notion of lexicographic shellability for pure, balanced boolean cell complexes, modelled after the CL-shellability criterion of Björner and Wachs ( Adv. in Math. 43 (1982), 87-100) for posets and its generalization by Kozlov ( Ann. of Comp. 1(1) (1997), 67-90) called CC-shellability. We give a lexicographic shelling for the quotient of the order complex of a Boolean algebra of rank 2 n by the action of the wreath product S 2 
S n of symmetric groups, and we provide a partitioning for the quotient complex 
( 
n )/ S n.
S n of symmetric groups, and we provide a partitioning for the quotient complex ( n )/ S n. Stanley asked for a description of the symmetric group representation 
S on the homology of the rank-selected partition lattice n S in Stanley ( J. Combin. Theory Ser. A 32(2) (1982), 132-161), and in particular he asked when the multiplicity b S( n) of the trivial representation in S is 0. One consequence of the partitioning for ( n )/ S n is a (fairly complicated) combinatorial interpretation for b S( n); another is a simple proof of Hanlon”s result ( European J. Combin. 4(2) (1983), 137-141) that b 1, 
, i( n) = 0. Using a result of Garsia and Stanton from ( Adv. in Math. 51(2) (1984), 107-201), we deduce from our shelling for ( B 2 n )/ S 2 S n that the ring of invariants k[ x 1, , x 2 n ] S2 Sn is Cohen-Macaulay over any field k.
S on the homology of the rank-selected partition lattice n S in Stanley ( J. Combin. Theory Ser. A 32(2) (1982), 132-161), and in particular he asked when the multiplicity b S( n) of the trivial representation in S is 0. One consequence of the partitioning for ( n )/ S n is a (fairly complicated) combinatorial interpretation for b S( n); another is a simple proof of Hanlon”s result ( European J. Combin. 4(2) (1983), 137-141) that b 1, , i( n) = 0. Using a result of Garsia and Stanton from ( Adv. in Math. 51(2) (1984), 107-201), we deduce from our shelling for ( B 2 n )/ S 2 S n that the ring of invariants k[ x 1, , x 2 n ] S2 Sn is Cohen-Macaulay over any field k.Pages: 225–254
Keywords: shellability; Boolean cell complex; simplicial poset; partition lattice; wreath product
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References
1. E. Babson and D. Kozlov, “Group actions on posets,” To appear in J. Algebra.
2. A. Bj\ddot orner, “Shellable and Cohen-Macaulay partially ordered sets,” Trans. Amer. Math. Soc. 260(1) (1980), 159-183.
3. A. Bj\ddot orner, “Posets, regular CW complexes and Bruhat order,” European J. Combin. 5(1) (1984), 7-16.
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5. A. Bj\ddot orner and M. Wachs, “On lexicographically shellable posets,” Trans. Amer. Math. Soc. 277(1) (1983), 323-341.
6. A. Bj\ddot orner and M. Wachs, “Bruhat order of Coxeter groups and shellability,” Adv. in Math. 43 (1982), 87-100.
7. A. Duval, “Free resolutions of simplicial posets,” J. Algebra 188 (1997), 363-399.
8. A. Garsia and D. Stanton, “Group actions of Stanley-Reisner rings and invariants of permutation groups,” Adv. in Math. 51(2) (1984), 107-201.
9. P. Hanlon, “A proof of a conjecture of Stanley concerning partitions of a set,” European J. Combin. 4(2) (1983), 137-141.
10. P. Hanlon and P. Hersh, “Multiplicity of the trivial representation in rank-selected homology of the partition lattice,” To appear in J. Algebra.
11. P. Hersh, “A partitioning and related properties for the quotient complex (Blm )/Sl Sm ,” J. Pure and Appl. Alg. 178(3) (2003), 255-272.
12. P. Hersh and R. Kleinberg, “The refinement complex of the poset of partitions of a multiset,” in press.
13. M. Hochster and J.A. Eagon, “Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci,” Amer. J. Math. 93 (1971), 1020-1058.
14. A. Hultman, “Lexicographic shellability and quotient complexes,” J. Algebraic Combinatorics 16 (2002), 83-86.
15. D. Kozlov, “Collapsibility of ( n)/Sn and some related CW complexes,” Proc. Amer. Math. Soc. 128(8) (2000), 2253-2259.
16. D. Kozlov, “General lexicographic shellability and orbit arrangements, Ann. of Comb. 1(1) (1997), 67-90.
17. D. Kozlov, “Rational homology of spaces of complex monic polynomials with multiple roots,” To appear in Mathematika.
18. J. Munkres, “Topological results in combinatorics,” Michigan Math. J. 31, 113-128.
19. V. Reiner, “Quotients of Coxeter complexes and P-partitions,” Memoirs Amer. Math. Soc. 95, January 1992.
20. R. Stanley, “Invariants of finite groups and their applications to combinatorics,” Bull. Amer. Math. Soc. 1 (1979), 475-511.
21. R. Stanley, “Some aspects of groups acting on finite posets,” J. Combin. Theory Ser. A, 32(2) (1982), 132-161.
22. R. Stanley, “ f -vectors and h-vectors of simplicial posets,” J. Pure and Applied Algebra, 71 (1991), 319-331.
23. R. Stanley, Combinatorics and Commutative Algebra, 2nd edition: Birkh\ddot auser, Boston, 1996.
24. S. Sundaram, “The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice,” Adv. Math. 104 (1994), 225-296.
25. M. Wachs, “A basis for the homology of the d-divisible partition lattices,” Adv. in Math. 117(2) (1996) 294-318.
26. G. Ziegler, “On the poset of partitions of an integer,” J. Combin. Theory Ser. A 42(2), (1986) 215-222.
2. A. Bj\ddot orner, “Shellable and Cohen-Macaulay partially ordered sets,” Trans. Amer. Math. Soc. 260(1) (1980), 159-183.
3. A. Bj\ddot orner, “Posets, regular CW complexes and Bruhat order,” European J. Combin. 5(1) (1984), 7-16.
4. A. Bj\ddot orner, “Topological methods,” in Handbook of Combinatorics (R. Graham, M. Gr\ddot otschel, and L. Lovasz, eds.), North-Holland, Amsterdam, 1993.
5. A. Bj\ddot orner and M. Wachs, “On lexicographically shellable posets,” Trans. Amer. Math. Soc. 277(1) (1983), 323-341.
6. A. Bj\ddot orner and M. Wachs, “Bruhat order of Coxeter groups and shellability,” Adv. in Math. 43 (1982), 87-100.
7. A. Duval, “Free resolutions of simplicial posets,” J. Algebra 188 (1997), 363-399.
8. A. Garsia and D. Stanton, “Group actions of Stanley-Reisner rings and invariants of permutation groups,” Adv. in Math. 51(2) (1984), 107-201.
9. P. Hanlon, “A proof of a conjecture of Stanley concerning partitions of a set,” European J. Combin. 4(2) (1983), 137-141.
10. P. Hanlon and P. Hersh, “Multiplicity of the trivial representation in rank-selected homology of the partition lattice,” To appear in J. Algebra.
11. P. Hersh, “A partitioning and related properties for the quotient complex (Blm )/Sl Sm ,” J. Pure and Appl. Alg. 178(3) (2003), 255-272.
12. P. Hersh and R. Kleinberg, “The refinement complex of the poset of partitions of a multiset,” in press.
13. M. Hochster and J.A. Eagon, “Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci,” Amer. J. Math. 93 (1971), 1020-1058.
14. A. Hultman, “Lexicographic shellability and quotient complexes,” J. Algebraic Combinatorics 16 (2002), 83-86.
15. D. Kozlov, “Collapsibility of ( n)/Sn and some related CW complexes,” Proc. Amer. Math. Soc. 128(8) (2000), 2253-2259.
16. D. Kozlov, “General lexicographic shellability and orbit arrangements, Ann. of Comb. 1(1) (1997), 67-90.
17. D. Kozlov, “Rational homology of spaces of complex monic polynomials with multiple roots,” To appear in Mathematika.
18. J. Munkres, “Topological results in combinatorics,” Michigan Math. J. 31, 113-128.
19. V. Reiner, “Quotients of Coxeter complexes and P-partitions,” Memoirs Amer. Math. Soc. 95, January 1992.
20. R. Stanley, “Invariants of finite groups and their applications to combinatorics,” Bull. Amer. Math. Soc. 1 (1979), 475-511.
21. R. Stanley, “Some aspects of groups acting on finite posets,” J. Combin. Theory Ser. A, 32(2) (1982), 132-161.
22. R. Stanley, “ f -vectors and h-vectors of simplicial posets,” J. Pure and Applied Algebra, 71 (1991), 319-331.
23. R. Stanley, Combinatorics and Commutative Algebra, 2nd edition: Birkh\ddot auser, Boston, 1996.
24. S. Sundaram, “The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice,” Adv. Math. 104 (1994), 225-296.
25. M. Wachs, “A basis for the homology of the d-divisible partition lattices,” Adv. in Math. 117(2) (1996) 294-318.
26. G. Ziegler, “On the poset of partitions of an integer,” J. Combin. Theory Ser. A 42(2), (1986) 215-222.
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