Speaker
Colby Sherwood
(University of Delaware)
Description
Let $W^i_{k,n}(m)$ 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 $W^i_{k,n}(m)$ is 1 if $|S \cap T|= i$, and is 0 otherwise. When $i=k$ the matrix $W^k_{k,n}(m)$ is the subset inclusion matrix for which Wilson found a diagonal form, solving the $p$-rank problem for any prime $p$. This diagonal form was used to calculate the Smith group of the hypercube graph.
We compute the rank of the matrix $W^1_{2,n}(m)$ over any field by making use of the representation theory of the symmetric group. We also give a simple condition under which $W^i_{k,n}(m)$ has large $p$-rank.
Authors
Colby Sherwood
(University of Delaware)
Dr
Joshua Ducey