On the (co)homology of the partition lattice and the free Lie algebra

Abstract We explore the well-known relationship between the representations of the symmetric group on the homology of the partition lattice and on the l n component of the free Lie algebra. We give two new combinatorial proofs of the sign twisted isomorphism between the two modules. One involves a bijection between bases and the other involves a bijection between generating sets which takes relations to relations.

[1]  Hélène Barcelo,et al.  On some submodules of the action of the symmetric group on the free Lie algebra , 1993 .

[2]  P. Hanlon,et al.  The fixed-point partition lattices , 1981 .

[3]  A. Brandt.  The free Lie ring and Lie representations of the full linear group , 1944 .

[4]  Hélène Barcelo,et al.  On the action of the symmetric group on the Free Lie Algebra and the partition lattice , 1990, J. Comb. Theory, Ser. A.

[5]  Richard P. Stanley,et al.  Some Aspects of Groups Acting on Finite Posets , 1982, J. Comb. Theory, Ser. A.

[6]  Michelle L. Wachs,et al.  Cohomology of Dowling Lattices and Lie (Super)Algebras , 2000, Adv. Appl. Math..

[7]  Kenneth Baclawski,et al.  Whitney Numbers of Geometric Lattices , 1975 .

[8]  Eduard Zehnder,et al.  Analysis, Et Cetera: Research Papers Published in Honor of Jurgen Moser's 60th Birthday , 1990 .

[9]  Kenneth Baclawski,et al.  Combinatorial decompositions of a class of rings , 1981 .

[10]  E. Witt,et al.  Treue Darstellung Liescher Ringe. , 1937 .

[11]  Nantel Bergeron,et al.  The Orlik-Solomon algebra on the partition lattice and the free Lie algebra , 1990, J. Comb. Theory, Ser. A.

[12]  P. Orlik,et al.  Combinatorics and topology of complements of hyperplanes , 1980 .

[13]  Anders Björner,et al.  On the homology of geometric lattices , 1982 .

[14]  Michelle L. Wachs.  A Basis for the Homology of thed-Divisible Partition Lattice , 1996 .

[15]  C. Reutenauer,et al.  Free Lie Superalgebras, Trees and Chains of Partitions , 1996 .

[16]  Michelle L. Wachs,et al.  The homology representations of the k-equal partition lattice , 1997 .

[17]  Adriano M. Garsia,et al.  Combinatorics of the Free Lie Algebra and the Symmetric Group , 1990 .

[18]  A. Joyal.  Foncteurs analytiques et espèces de structures , 1986 .

[19]  Ira M. Gessel,et al.  A coloring problem , 1991 .

[20]  Louis Solomon,et al.  On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes , 1986 .

[21]  P. Hanlon,et al.  On lie k-algebras , 1995 .

[22]  Jon Folkman,et al.  THE HOMOLOGY GROUPS OF A LATTICE , 1964 .