OIST Representation Theory Seminar

Kay Jin Lim, Nanyang Technological University

Title: Integral Basic Algebras

Abstract: Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. In the recent preprint with David J. Benson, we initiate the study by focusing on the integral basic algebras. That is, we consider a p-modular system \((K,\mathcal{O},k)\) and an \(\mathcal{O}\)-algebra \(A\) where both the algebras \(K\otimes_\mathcal{O} A\) and \(k\otimes_\mathcal{O}A\) are basic. When the algebra satisfies the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. As a case study, we focus on the descent algebras of Coxeter groups. They have been extensively studied since the introduction by Louis Solomon in 1976. We investigate their invariants as mentioned previously, their Ext quivers and representation type. The classification of the representation type in the \(p=0\) case has previously achieved by Manfred Schocker. In a recent preprint, together with Karin Erdmann, we complete the classification in the \(p>0\) case.