9 research outputs found
The critical group of the Kneser graph on -subsets of an -element set
In this paper we compute the critical group of the Kneser graph .
This is equivalent to computing the Smith normal form of a Laplacian matrix of
this graph.Comment: 16 pages, minor change
The Elementary Divisors of the Incidence Matrix of Skew Lines in PG(3,q)
The elementary divisors of the incidence matrix of lines in PG(3,q) are
computed, where two lines are incident if and only if they are skew.Comment: 13 pages. The results of this paper supersede those in the paper
arXiv:math/1001.2551 V2. Minor correction
Integer diagonal forms for subset intersection relations
For integers , we give a
description for the Smith group of the incidence matrix with rows (columns)
indexed by the size (, respectively) subsets of an -element set,
where incidence means intersection in a set of size . This generalizes
work of Wilson and Bier from the 1990s which dealt only with the case where
incidence meant inclusion. Our approach also describes the Smith group of any
matrix in the -linear span of these matrices so includes all
integer matrices in the Bose-Mesner algebra of the Johnson association scheme:
for example, the association matrices themselves as well as the Laplacian,
signless Laplacian, Seidel adjacency matrix, etc. of the associated graphs. In
particular, we describe the critical (also known as sandpile) groups of these
graphs. The complexity of our formula grows with the parameters, but is
independent of and , which often leads to an efficient algorithm for
computing these groups. We illustrate our techniques to give diagonal forms of
matrices attached to the Kneser and Johnson graphs for subsets of size ,
whose invariants have never before been described, and recover results from a
variety of papers in the literature in a unified way.Comment: 28 page
A representation-theoretic computation of the rank of -intersection incidence matrices: -subsets vs. -subsets
Let denote a matrix with rows and columns indexed by the
-subsets and -subsets, respectively, of an -element set. The row ,
column entry of is if , and is
otherwise. We compute the rank of the matrix over any field by
making use of the representation theory of the symmetric group. We also give a
simple condition under which has large -rank.Comment: 13 pages. Corrected the statement of Theorem 2.1, part (v), and
consequently needed to modify the proof of the case "char(F)=2, m odd