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Matthew Spencer

Publications and source records attributed to Matthew Spencer.

7 recordsLinked to original sources
arXiv2018

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.

math.RT↗
arXiv2018

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.

math.RT↗
arXiv2017

Equivalence relations on ecosystems

In abstract terms, ecosystem ecology is about determining when two ecosystems, superficially different, are alike in some deeper way. An external observer can choose any ecosystem property as being important. In contrast, two ecosystems are equivalent from the point of view of the organisms they contain if and only if for each species, the proportional population growth rate does not differ between the ecosystems. Comparative studies of ecosystems should therefore focus on patterns in proportional population growth rates, rather than patterns in other properties such as relative abundances. Popular activities such as measuring dissimilarity, and representing dissimilarity via ordination, can then be done from the point of view of the organisms in ecosystems. Summarizing the state of an ecosystem under this approach remains challenging. In general, the dynamics on equivalence classes of ecosystems defined in this way are structurally different from the dynamics of ecosystems as seen by an external observer. This may limit the extent to which natural selection can act on ecosystem structure.

q-bio.PE↗
arXiv2017

A note on Green functors with inflation

This note is motivated by the problem to understand, given a commutative ring F, which G-sets X, Y give rise to isomorphic F[G]-representations F[X]\cong F[Y]. A typical step in such investigations is an argument that uses induction theorems to give very general sufficient conditions for all such relations to come from proper subquotients of G. In the present paper we axiomatise the situation, and prove such a result in the generality of Mackey functors and Green functors with inflation. Our result includes, as special cases, a result of Deligne on monomial relations, a result of the first author and Tim Dokchitser on Brauer relations in characteristic 0, and a new result on Brauer relations in characteristic p>0. We will need the new result in a forthcoming paper on Brauer relations in positive characteristic.

math.RA↗
arXiv2016

Among-site variability in the stochastic dynamics of East African coral reefs

Coral reefs are dynamic systems whose composition is highly influenced by unpredictable biotic and abiotic factors. Understanding the spatial scale at which long-term predictions of reef composition can be made will be crucial for guiding conservation efforts. Using a 22-year time series of benthic composition data from 20 reefs on the Kenyan and Tanzanian coast, we studied the long-term behaviour of Bayesian vector autoregressive state-space models for reef dynamics, incorporating among-site variability. We estimate that if there were no among-site variability, the total long-term variability would be approximately one third of its current value. Thus among-site variability contributes more to long-term variability in reef composition than does temporal variability. Individual sites are more predictable than previously thought, and predictions based on current snapshots are informative about long-term properties. Our approach allowed us to identify a subset of possible climate refugia sites with high conservation value, where the long-term probability of coral cover <= 0.1 was very low. Analytical results show that this probability is most strongly influenced by among-site variability and by interactions among benthic components within sites. These findings suggest that conservation initiatives might be successful at the site scale as well as the regional scale.

q-bio.PE↗
arXiv2014

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

Measures 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.

q-bio.PE↗