2,502 research outputs found
On a curious variant of the -module
We introduce a variant of the much-studied representation of the
symmetric group , which we denote by Our variant gives rise
to a decomposition of the regular representation as a sum of {exterior} powers
of modules This is in contrast to the theorems of
Poincar\'e-Birkhoff-Witt and Thrall which decompose the regular representation
into a sum of symmetrised modules. We show that nearly every known
property of has a counterpart for the module suggesting
connections to the cohomology of configuration spaces via the character
formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber
and Schack, and to the Hodge decomposition of the complex of injective words
arising from Hochschild homology, due to Hanlon and Hersh.Comment: 26 pages, 2 tables. To appear in Algebraic Combinatorics. Parts of
this paper are included in arXiv:1803.0936
Geometrically constructed bases for homology of partition lattices of types A, B and D
We use the theory of hyperplane arrangements to construct natural bases for
the homology of partition lattices of types A, B and D. This extends and
explains the "splitting basis" for the homology of the partition lattice given
in [Wa96], thus answering a question asked by R. Stanley. More explicitly, the
following general technique is presented and utilized. Let A be a central and
essential hyperplane arrangement in R^d. Let R_1,...,R_k be the bounded regions
of a generic hyperplane section of A. We show that there are induced polytopal
cycles \rho_{R_i} in the homology of the proper part \bar{L_A} of the
intersection lattice such that {\rho_{R_i}}_{i=1,...,k} is a basis for \tilde
H_{d-2}(\bar{L_A}). This geometric method for constructing combinatorial
homology bases is applied to the Coxeter arrangements of types A, B and D, and
to some interpolating arrangements.Comment: 29 pages, 4 figure
Complexes of not -connected graphs
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not -connected -hypergraphs on vertices. We
show that the complex of not -connected graphs has the homotopy type of a
wedge of spheres of dimension . This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the -action on the homology of the complex is also determined. For
complexes of not -connected -hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not -connected graphs we provide a formula
for the generating function of the Euler characteristic
Partition complexes, duality and integral tree representations
We show that the poset of non-trivial partitions of 1,2,...,n has a
fundamental homology class with coefficients in a Lie superalgebra. Homological
duality then rapidly yields a range of known results concerning the integral
representations of the symmetric groups S_n and S_{n+1} on the homology and
cohomology of this partially-ordered set.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-41.abs.htm
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