Representations of reductive groups over finite rings and extended Deligne-Lusztig varieties

In a previous paper it was shown that a certain family of varieties suggested by Lusztig, is not enough to construct all irreducible complex representations of reductive groups over finite rings coming from the ring of integers in a local field, modulo a power of the maximal ideal. In this paper we define a generalisation of Lusztig's varieties, corresponding to an extension of the maximal unramified extension of the local field. We show in a particular case that all irreducible representations appear in the cohomology of some extended variety. We conclude with a discussion about reformulation of Lusztig's conjecture.Comment: 10 page

Assessing Plant and Lichen Diversity Using Reflectance Spectra

Biodiversity is changing and it is imperative that we continually assess it in order to preserve ecological services that we rely on. Spectral platforms are increasingly being used to assess biodiversity due to the fact that light reflected from an organism’s surface carries much of information about that organism. Despite the promise spectroscopy shows, two gaps in our knowledge remain. First, we do not know how well reflectance spectra can be used to estimate fine-scale diversity – intraspecific genetic and phenotypic diversity – that is fundamental to ecological and evolutionary processes. Second, spectral libraries, used to construct models to estimate diversity, have largely been built from plant spectra and have neglected other ecologically important organisms such as lichens. To investigate the first problem, my colleagues and I tested the utility of reflectance spectra for distinguishing genomically defined populations. We collected spectra (400–2400 nm) and samples from co-occurring Dryas alaskensis, Dryas ajanensis, and hybrid individuals from six different mountaintops in the interior of Alaska, United States. We used partial least squares discriminant analysis (PLS-DA) to classify leaf reflectance spectra into six populations defined by STRUCTURE and PCA analyses using genomic data. We also estimated the phylogenetic signal carried by the spectra, and we used PLS beta regression to estimate the proportion of ancestry for each individual from the reflectance spectra. We found that the two species and their six populations could be distinguished with 99.7% and 98.9% overall accuracy, respectively. A significant phylogenetic signal was found for all regions of the spectrum, and the model for estimating the proportion of ancestry explained 91% of the variation with an RMSE of 0.13. Hybrids were classified with 80% accuracy, and this is thought to be due to a lack of strong trait correlations. These findings suggest that fine-scale diversity can be retrieved from reflectance spectra and this should be considered in future spectrally-based biodiversity assessments. To address the second problem, I investigated whether herbarium specimens would be valuable for building a spectral library for lichens. Specifically, I investigated whether lichen specimens were altered by the long-term desiccation inherent with herbarium storage and if that influenced the classification of herbarium specimens. I used a spectral dataset of 30 lichen species that covered an age range of 126 years, and used linear mixed-effects models and PLS-DA to determine 1) how reflectance changed with age, and 2) the influence of age on classification accuracy. I found that the reflectance for wavelengths between 700 and 1900 nm decreased by less than 0.2% reflectance per year, but wavelengths outside this range did not clearly respond to aging. This implies a gradual change in thallus structure over time in herbarium storage. Species, families, orders, and classes were classified with 77.0 to 94.5% accuracy, and these models were only marginally influenced by specimen age. These results indicate that lichen specimens do change over time, but these changes do not negate their utility for building spectral libraries for lichens

The smooth representations of GL_2(o)

We give a classification of the smooth (complex) representations of GL2(

Rationality of representation zeta functions of compact pp-adic analytic groups

We prove that for any FAb compact $p$-adic analytic group $G$, its representation zeta function is a finite sum of terms $n_{i}^{-s}f_{i}(p^{-s})$, where $n_{i}$ are natural numbers and $f_{i}(t)\in\mathbb{Q}(t)$ are rational functions. Meromorphic continuation and rationality of the abscissa of the zeta function follow as corollaries. If $G$ is moreover a pro-$p$ group, we prove that its representation zeta function is rational in $p^{-s}$. These results were proved by Jaikin-Zapirain for $p>2$ or for $G$ uniform and pro-$2$, respectively. We give a new proof which avoids the Kirillov orbit method and works for all $p$. Moreover, we prove analogous results for twist zeta functions of compact $p$-adic analytic groups, which enumerate irreducible representations up to one-dimensional twists. In the course of the proof, we develop a Clifford theory for twist isoclasses of representations, including a new cohomological invariant of a twist isoclass

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

For a principal ideal domain A, the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in Mn(A) with irreducible characteristic polynomial f(x) and the ideal classes of the order A[x]/(f(x)). We prove that when A[x]/(f(x)) is maximal (i.e. integrally closed, i.e. a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when A[x]/(f(x)) is maximal, every ideal class contains an ideal of degree one

The regular representations of GLN over finite local principal ideal rings

Let o o be the ring of integers in a non-Archimedean local field with finite residue field, p p its maximal ideal, and r ⩾ 2 r⩾2 an integer. An irreducible representation of the finite group G r = GL N ( o / p r ) Gr=GLN(o/pr), for an integer N ⩾ 2 N⩾2, is called regular if its restriction to the principal congruence kernel K r − 1 = 1 + p r − 1 M N ( o / p r ) Kr−1=1+pr−1MN(o/pr) consists of representations whose stabilisers modulo K 1 K1 are centralisers of regular elements in M N ( o / p ) MN(o/p). The regular representations form the largest class of representations of G r Gr which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of G r Gr

Estratégias de comercialização do feijão no Rio Grande do Sul.

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