Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G be GL_N or SL_N as reductive linear algebraic group over a field k of positive characteristic p. We prove several results that were previously established only when N 2^N. Let G act rationally on a finitely generated commutative k-algebra A. Assume that A as a G-module has a good filtration or a Schur filtration. Let M be a noetherian A-module with compatible G action. Then M has finite good/Schur filtration dimension, so that there are at most finitely many nonzero H^i(G,M). Moreover these H^i(G,M) are noetherian modules over the ring of invariants A^G. Our main tool is a resolution involving Schur functors of the ideal of the diagonal in a product of Grassmannians.Comment: 22 pages; final versio

Constant terms of powers of a Laurent polynomial

We prove a special case of a conjecture of Mathieu ([Mat]). Conjecture 1 (Mathieu) Let K be a connected real compact Lie group. Let f and g be K-nite functions on K. Assume that for all n 1 the constant term of fn vanishes. Then for all but nitely many n the constant term of fng also vanishes

Frobenius Splittings

We give a gentle introduction to Frobenius splittings. Then we recall a few results that have been obtained with the method.Comment: 21 pages, typos correcte

Steinberg modules and Donkin pairs

We prove that in positive characteristic a module with good filtration for a group of type E6 restricts to a module with good filtration for a subgroup of type F4. (Recall that a filtration of a module for a semisimple algebraic group is called good if its layers are dual Weyl modules.) Our result confirms a conjecture of Brundan for one more case. The method relies on the canonical Frobenius splittings of Mathieu. Next we settle the remaining cases, in characteristic not 2, with a computer-aided variation on the old method of Donkin.Comment: 16 pages; proof of Brundan's conjecture adde

The quotient Unimodular Vector group is nilpotent

Jose-Rao introduced and studied the Special Unimodular Vector group $SUm_r(R)$ and $EUm_r(R)$, its Elementary Unimodular Vector subgroup. They proved that for $r \geq 2$, $EUm_r(R)$ is a normal subgroup of $SUm_r(R)$. The Jose-Rao theorem says that the quotient Unimodular Vector group, $SUm_r(R)/EUm_r(R)$, for $r \geq 2$, is a subgroup of the orthogonal quotient group $SO_{2(r+1)}(R)/EO_{2(r + 1)}(R)$. The latter group is known to be nilpotent by the work of Hazrat-Vavilov, following methods of A. Bak; and so is the former. In this article we give a direct proof, following ideas of A. Bak, to show that the quotient Unimodular Vector group is nilpotent of class $\leq d = \dim(R)$. We also use the Quillen-Suslin theory, inspired by A. Bak's method, to prove that if $R = A[X]$, with $A$ a local ring, then the quotient Unimodular Vector group is abelian

Influence of low back pain and prognostic value of MRI in sciatica patients in relation to back pain

Background: Patients with sciatica frequently complain about associated back pain. It is not known whether there are prognostic relevant differences in Magnetic Resonance Imaging (MRI) findings between sciatica patients with and without disabling back pain. Methods: The study population contained patients with sciatica who underwent a baseline MRI to assess eligibility for a randomized trial designed to compare the efficacy of early surgery with prolonged conservative care for sciatica. Two neuroradiologists and one neurosurgeon independently evaluated all MR images. The MRI readers were blinded to symptom status. The MRI findings were compared between sciatica patients with and without disabling back pain. The presence of disabling back pain at baseline was correlated with perceived recovery at one year. Results: Of 379 included sciatica patients, 158 (42%) had disabling back pain. Of the patients with both sciatica and disabling back pain 68% did reveal a herniated disc with nerve root compression on MRI, compared to 88% of patients with predominantly sciatica (P,0.001). The existence of disabling back pain in sciatica at baseline was negatively associated with perceived recovery at one year (Odds ratio [OR] 0.32, 95% Confidence Interval 0.18-0.56, P,0.001). Sciatica patients with disabling back pain in absence of nerve root compression on MRI at baseline reported less perceived recovery at one year compared to those with predominantly sciatica and nerve root compression on MRI (50% vs 91%, P,0.001). Conclusion: Sciatica patients with disabling low back pain reported an unfavorable outcome at one-year follow-up compared to those with predominantly sciatica. If additionally a clear herniated disc with nerve root compression on MRI was absent, the results were even worse. Copyright

Subcutaneous Adipose Tissue and Systemic Inflammation Are Associated With Peripheral but Not Hepatic Insulin Resistance in Humans

Obesity-related insulin resistance (IR) may develop in multiple organs, representing different etiologies towards cardiometabolic diseases. We identified abdominal subcutaneous adipose tissue (ScAT) transcriptome profiles in relation to liver or muscle IR by means of RNA sequencing in overweight/obese participants of the DiOGenes cohort (n=368). Tissue-specific IR phenotypes were derived from a 5-point oral glucose tolerance test. Hepatic and muscle IR were characterized by distinct abdominal ScAT transcriptome profiles. Genes related to extracellular remodeling were upregulated in individuals with primarily hepatic IR, whilst genes related to inflammation were upregulated in individuals with primarily muscle IR. In line with this, in two independent cohorts, CODAM (n=325) and the Maastricht Study (n=685), an increased systemic low-grade inflammation profile was specifically related to muscle IR, but not to liver IR. We propose that increased ScAT inflammatory gene expression may translate into an increased systemic inflammatory profile, linking ScAT inflammation to the muscle IR phenotype. These distinct IR phenotypes may provide leads for more personalized prevention of cardiometabolic diseases. DiOGenes was registered at clinicaltrials.gov as NCT00390637

Upstream transcription factor 1 (USF1) in risk of type 2 diabetes:Association study in 2000 Dutch Caucasians

Type 2 diabetes shares substantial genetic and phenotypic overlap with familial combined hyperlipidemia. Upstream stimulatory factor 1 (USF7), a well-established susceptibility gene for familial combined hyperlipidemia, is postulated to be such a shared genetic determinant. We evaluated two established variants in familial combined hyperlipidemia (rs2073658 and rs3737787) for association with type 2 diabetes in two Dutch case-control samples (N=2011). The first case-control sample comprised 501 subjects with type 2 diabetes from the Breda cohort and 920 healthy blood bank donors of Dutch Caucasian origin. The second case-control sample included 211 subjects with type 2 diabetes, and 379 normoglycemic controls. SNP rs2073658 and SNP rs3737787 were in perfect linkage disequilibrium. In the first case-control sample, prevalence of the major allele was higher in patients than in controls (75% versus 71%, OR=1.25, p=0.018). A similar effect-size and -direction was observed in the second case-control sample (76% versus 72%, OR=1.22, p=0.16). A combined analysis strengthened the evidence for association (OR=1.23, p=0.006). Notably, the increased risk for type 2 diabetes could be ascribed to the major allele, and its high frequency translated to a substantial population attributable risk of 14.5%. In conclusion, the major allele of rs2073658 in the USF1 gene is associated with a modestly increased risk to develop type 2 diabetes in Dutch Caucasians, with considerable impact at the population level. (c) 2008 Elsevier Inc. All rights reserved