79 research outputs found
Pseudodeterminants and perfect square spanning tree counts
The pseudodeterminant of a square matrix is the last
nonzero coefficient in its characteristic polynomial; for a nonsingular matrix,
this is just the determinant. If is a symmetric or skew-symmetric
matrix then .
Whenever is the boundary map of a self-dual CW-complex ,
this linear-algebraic identity implies that the torsion-weighted generating
function for cellular -trees in is a perfect square. In the case that
is an \emph{antipodally} self-dual CW-sphere of odd dimension, the
pseudodeterminant of its th cellular boundary map can be interpreted
directly as a torsion-weighted generating function both for -trees and for
-trees, complementing the analogous result for even-dimensional spheres
given by the second author. The argument relies on the topological fact that
any self-dual even-dimensional CW-ball can be oriented so that its middle
boundary map is skew-symmetric.Comment: Final version; minor revisions. To appear in Journal of Combinatoric
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces
Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian
subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In
particular, we find out a natural parametrization of maximal elements and
dimension formulas for them. We recover as special cases several results of
Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To
appear in Selecta Mathematic
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