The critical group of the Kneser graph on 22-subsets of an nn-element set

In this paper we compute the critical group of the Kneser graph $KG(n,2)$. 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 $0 \leq \ell \leq k_{r} \leq k_{c} \leq n$, we give a description for the Smith group of the incidence matrix with rows (columns) indexed by the size $k_r$ ($k_c$, respectively) subsets of an $n$-element set, where incidence means intersection in a set of size $\ell$. 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 $\mathbb{Z}$-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 $k$ parameters, but is independent of $n$ and $\ell$, 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 $3$, 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 11-intersection incidence matrices: 22-subsets vs. nn-subsets

Let $W_{k,n}^{i}(m)$ denote a matrix with rows and columns indexed by the $k$-subsets and $n$-subsets, respectively, of an $m$-element set. The row $S$, column $T$ entry of $W_{k,n}^{i}(m)$ is $1$ if $|S \cap T| = i$, and is $0$ otherwise. We compute the rank of the matrix $W_{2,n}^{1}(m)$ over any field by making use of the representation theory of the symmetric group. We also give a simple condition under which $W_{k,n}^{i}(m)$ has large $p$-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