110 research outputs found

    Gauss decomposition for Chevalley groups, revisited

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    In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups G=G(Φ,R)G=G(\Phi,R) over a semilocal ring admit remarkable Gauss decomposition G=TUUUG=TUU^-U, where T=T(Φ,R)T=T(\Phi,R) is a split maximal torus, whereas U=U(Φ,R)U=U(\Phi,R) and U=U(Φ,R)U^-=U^-(\Phi,R) are unipotent radicals of two opposite Borel subgroups B=B(Φ,R)B=B(\Phi,R) and B=B(Φ,R)B^-=B^-(\Phi,R) containing TT. It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as \sr(R)=1 or \asr(R)=1. Later the second author noticed that condition \sr(R)=1 is necessary for Gauss decomposition. Here, we show that a slight variation of Tavgen's rank reduction theorem implies that for the elementary group E(Φ,R)E(\Phi,R) condition \sr(R)=1 is also sufficient for Gauss decomposition. In other words, E=HUUUE=HUU^-U, where H=H(Φ,R)=TEH=H(\Phi,R)=T\cap E. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, \asr(R)=1, \sr(R,\Lambda)=1, etc., were only needed to guarantee that for simply connected groups G=EG=E, rather than to verify the Gauss decomposition itself

    Death-associated protein 3 is overexpressed in human thyroid oncocytic tumours

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    Background: The human death-associated protein 3 (hDAP3) is a GTP-binding constituent of the small subunit of the mitochondrial ribosome with a pro-apoptotic function.Methods: A search through publicly available microarray data sets showed 337 genes potentially coregulated with the DAP3 gene. The promoter sequences of these 337 genes and 70 out of 85 mitochondrial ribosome genes were analysed in silico with the DAP3 gene promoter sequence. The mitochondrial role of DAP3 was also investigated in the thyroid tumours presenting various mitochondrial contents. Results: The study revealed nine transcription factors presenting enriched motifs for these gene promoters, five of which are implicated in cellular growth (ELK1, ELK4, RUNX1, HOX11-CTF1, TAL1-ternary complex factor 3) and four in mitochondrial biogenesis (nuclear respiratory factor-1 (NRF-1), GABPA, PPARG-RXRA and estrogen-related receptor alpha (ESRRA)). An independent microarray data set showed the overexpression of ELK1, RUNX1 and ESRRA in the thyroid oncocytic tumours. Exploring the thyroid tumours, we found that DAP3 mRNA and protein expression is upregulated in tumours presenting a mitochondrial biogenesis compared with the normal tissue. ELK1 and ESRRA were also showed upregulated with DAP3. Conclusion: ELK1 and ESRRA may be considered as potential regulators of the DAP3 gene expression. DAP3 may participate in mitochondrial maintenance and play a role in the balance between mitochondrial homoeostasis and tumourigenesis
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