Unbounding Ext

We produce examples in the cohomology of algebraic groups which answer two questions of Parshall and Scott. Specifically, if $G=SL_2$, then we show: (a) \dim \Ext_G^2(L,L) can be arbitrarily large for a simple module $L$; and (b) the sequence $\max_{L-\text{irred}}\dim H^k(G,L)$ grows exponentially fast with $k$.Comment: 14 pages; version to appear in J. Al

On the minimal modules for exceptional Lie algebras: Jordan blocks and stabilisers

Let G be a simple simple-connected exceptional algebraic group of type G_2, F_4, E_6 or E_7 over an algebraically closed field k of characteristic p>0 with \g=Lie(G). For each nilpotent orbit G.e of \g, we list the Jordan blocks of the action of e on the minimal induced module V_min of \g. We also establish when the centralisers G_v of vectors v\in V_min and stabilisers \Stab_G of 1-spaces \subset V_min are smooth; that is, when \dim G_v=\dim\g_v or \dim \Stab_G=\dim\Stab_\g.Comment: This contains corrections and should be used instead of the published versio

A multipath analysis of biswapped networks.

Biswapped networks of the form $Bsw(G)$ have recently been proposed as interconnection networks to be implemented as optical transpose interconnection systems. We provide a systematic construction of $\kappa+1$ vertex-disjoint paths joining any two distinct vertices in $Bsw(G)$, where $\kappa\geq 1$ is the connectivity of $G$. In doing so, we obtain an upper bound of $\max\{2\Delta(G)+5,\Delta_\kappa(G)+\Delta(G)+2\}$ on the $(\kappa+1)$-diameter of $Bsw(G)$, where $\Delta(G)$ is the diameter of $G$ and $\Delta_\kappa(G)$ the $\kappa$-diameter. Suppose that we have a deterministic multipath source routing algorithm in an interconnection network $G$ that finds $\kappa$ mutually vertex-disjoint paths in $G$ joining any $2$ distinct vertices and does this in time polynomial in $\Delta_\kappa(G)$, $\Delta(G)$ and $\kappa$ (and independently of the number of vertices of $G$). Our constructions yield an analogous deterministic multipath source routing algorithm in the interconnection network $Bsw(G)$ that finds $\kappa+1$ mutually vertex-disjoint paths joining any $2$ distinct vertices in $Bsw(G)$ so that these paths all have length bounded as above. Moreover, our algorithm has time complexity polynomial in $\Delta_\kappa(G)$, $\Delta(G)$ and $\kappa$. We also show that if $G$ is Hamiltonian then $Bsw(G)$ is Hamiltonian, and that if $G$ is a Cayley graph then $Bsw(G)$ is a Cayley graph