We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalizatio...We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.展开更多
In[Frobenius Categories Versus Brauer Blocks,Progress in Math.,274]and in[Ordinary Grothendieck groups of a Frobenius P-category,Algebra Colloq.18(2011)1-76],we consider suitable inverse limits of Grothendieck groups ...In[Frobenius Categories Versus Brauer Blocks,Progress in Math.,274]and in[Ordinary Grothendieck groups of a Frobenius P-category,Algebra Colloq.18(2011)1-76],we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics p and zero,obtained from a folded Frobenius P-category(F,aut_(F)sc),which covers the case of the Frobenius P-categories associated with blocks;moreover,in[Beyond a question of Markus Linckelmann,arxiv.org/abs/1507.04278]we show that a folded Frobenius P-category is actually equivalent to the choice of a regular central k^(*)-extension F^(SC) of F^(SC) c.Here,taking advantage of the existence of a perfect F^(SC) -locality P^(SC) we exhibit those inverse limits as the true Grothendieck groups of the categories of K_(*)G-and k_(*)G-modules for a suitable k_(*)-group G associated to the k^(*)-category P^(SC) obtained from P^(SC) and F^(SC) .It depends on a vanishing cohomology result,given with more generality intheAppendix.展开更多
文摘We introduce a new avatar of a Frobenius P-category F under the form of a suitable subring HF of the double Burnside ring of P -- called the Hecke algebra of F- where we are able to formulate： （i） the generalization to a Frobenius P-category of the Alperin Fusion Theorem, （ii） the ＂canonical decomposition＂ of the morphisms in the exterior quotient of a Frobenius P-category restricted to the selfcentralizing objects as developed in chapter 6 of [4], and （iii） the ＂basic P × P-sets＂ in chapter 21 of [4] with its generalization by Kari Ragnarsson and Radu Stancu to the virtual P ×P-sets in [6]. We also explain the relationship with the usual Hecke algebra of a finite group.
文摘In[Frobenius Categories Versus Brauer Blocks,Progress in Math.,274]and in[Ordinary Grothendieck groups of a Frobenius P-category,Algebra Colloq.18(2011)1-76],we consider suitable inverse limits of Grothendieck groups of categories of modules in characteristics p and zero,obtained from a folded Frobenius P-category(F,aut_(F)sc),which covers the case of the Frobenius P-categories associated with blocks;moreover,in[Beyond a question of Markus Linckelmann,arxiv.org/abs/1507.04278]we show that a folded Frobenius P-category is actually equivalent to the choice of a regular central k^(*)-extension F^(SC) of F^(SC) c.Here,taking advantage of the existence of a perfect F^(SC) -locality P^(SC) we exhibit those inverse limits as the true Grothendieck groups of the categories of K_(*)G-and k_(*)G-modules for a suitable k_(*)-group G associated to the k^(*)-category P^(SC) obtained from P^(SC) and F^(SC) .It depends on a vanishing cohomology result,given with more generality intheAppendix.
