Brauer relations for finite groups in the ring of semisimplified modular representations

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for all finite groups, to classify the primitive quotient of the kernel; that is for each $G$, the kernel modulo elements coming from the kernel for proper subquotients of $G$. We are able to identify exactly which groups have non-trivial primitive quotient and we give generators for the primitive quotient in the soluble case

Relations between permutation representations in positive characteristic

Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with wide-ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0, a complete description of K_F(G) is now known. In this paper, we give a similar description of K_F(G) when F is a field of characteristic p>0 in all but the most complicated case, which is when G has a subquotient that is a non-p-hypo-elementary (p,p)-Dress group.Comment: 18 pages; minor corrections and improvements. Final version to appear in Bull. London Math. So

An intrinsic characterization of cofree representations of reductive groups

We formulate and partially verify a conjecture characterizing cofree representations of connected reductive groups. As we explain, this conjecture may be viewed as a natural generalization of the Chevalley-Shepard-Todd theorem from the case of finite groups to the case of connected reductive groups.Comment: This version extends the results of the previous version from SL_n to all simple Lie groups. Spencer Whitehead is now a co-author and new techniques have been used to shorten the proo

Size change, shape change, and the growth space of a community

AbstractMeasures of biodiversity change such as the Living Planet Index describe proportional change in the abundance of a typical species, which can be thought of as change in the size of a community. Here, I discuss the orthogonal concept of change in relative abundances, which I refer to as shape change. To be logically consistent, a measure of the rate of shape change should be scaling invariant (have the same value for all data with the same vector of proportional change over a given time interval), but existing measures do not have this property. I derive a new, scaling invariant measure. I show that this new measure and existing measures of biodiversity change such as the Living Planet Index describe different aspects of dynamics. I show that neither body size nor environmental variability need affect the rate of shape change. I extend the measure to deal with colonizations and extinctions, using the surreal number system. I give examples using data on hoverflies in a garden in Leicester, UK, and the higher plant community of Surtsey. I hypothesize that phylogenetically restricted assemblages will show a higher proportion of size change than diverse communities

The representation theory of Iwahori-Hecke algebras with unequal parameters.

PhDThe Iwahori-Hecke algebras of finite Coxeter groups play an important role in many areas of mathematics. In this thesis we study the representation theory of the Iwahori-Hecke algebras of the Coxeter groups of type Bn and F4, in the unequal parameter case. We denote these algebras HQ and KQ respectively. This follows on from work done by Dipper, James, Murphy and Norton. We are interested in the Iwahori-Hecke algebras of type Bn and F4 since these are the only cases, apart from the dihedral groups, where the Coxeter generators lie in different conjugacy classes, and therefore the Iwahori-Hecke algebra can have unequal parameters. There are two parameters associated with these algebras, Q and q. Norton dealt with the case Q = q = 0, whilst Dipper, James and Murphy addressed the case q 6= 0 in type Bn. In this thesis we look at the case Q 6= 0; q = 0. We begin by constructing the simple modules for HQ, then compute the Ext quiver and find the blocks of HQ. We continue by observing that there is a natural embedding of the algebra of type n 1 in the algebra of type n, and this gives rise to the notion of an induced module. We look at the structure of the induced module associated with a given simple HQ-module. Here we are able to construct a composition series for the induced module and show that in a particular case the induced modules are self-dual. Finally, we look at KQ and find that the representation theory is related to representation theory of the Iwahori-Hecke algebra of type B3. Using this relationship we are able to construct the simple modules for KQ and begin the analysis of the Ext quiver.Engineering and Physical Sciences Research Council

Off Road Autonomous Vehicle Modeling and Repeatability Using Real World Telemetry via Simulation

One approach to autonomous control of high mobility ground vehicle platforms operating on challenging terrain is with the use of predictive simulation. Using a simulated or virtual world, an autonomous system can optimize use of its control systems by predicting interaction between the vehicle and ground as well as the vehicle actuator state. Such a simulation allows the platform to assess multiple possible scenarios before attempting to execute a path. Physically realistic simulations covering all of these domains are currently computationally expensive, and are unable to provide fast execution times when assessing each individual scenario due to the use of high simulation frequencies (\u3e 1000Hz). This work evaluates using an Unreal Engine 4 vehicle model and virtual environment, leveraging its underlying PhysX library to build a simple unmanned vehicle platform. The simulation is demonstrated to run at low simulation frequencies (\u3c 1000Hz) when performing multiple off road driving maneuvers. Real world path telemetry is used as input to drive the unmanned vehicle\u27s integrated Pure Pursuit and PID autonomous driving control algorithms within the simulation. Cross-track-error and vehicle heading error between the simulation and real world telemetry is then observed after each maneuver\u27s execution. It is concluded after running multiple different vehicle maneuvers in real time at low simulation frequencies, a lower threshold frequency of 190Hz was shown to reliably control the virtual vehicle model with minimal average cross-track-error and heading angle deviation. Higher simulation frequencies approaching 400Hz, the recorded sampling frequency of the real world telemetry for each maneuver, had little change in system performance. Setting the simulation to execute at lower frequencies \u3c 190Hz resulted in a point of exponential increase in both the overall average cross-track-error and heading error. Additional simulation failures were also observed when setting the AV to travel at higher velocities with set simulation frequencies \u3c 190Hz