On the Varieties of Parabolic Subgroups, their Generalizations and Combinatorial Applications

摘要

Investigations of homogeneous varieties T = (G : P) of all cosets of finite Coxeter or Chevalley groups G by their maximal parabolic subgroups P had been conducted at the Kaluznin seminar at Kiev State University since the 1970's, as were investigations of their corresponding permutation groups, geometries and association schemes. In I. A. Faradzev et al. (eds), Investigations in Algebraic Theory of Combinatorial Objects (Kluwer Acad. Publ., 1994), one can find some results on the investigation of noncomplete Galois correspondence between fusion schemes of the orbital scheme for (G, T) and overgroups of (G, T), as well as calculations of the intersectional indices of the Hecke algebra of (G, T). We will discuss additional results on this topic and consider questions related to the following problems: ·embeddings of varieties (G : P) into the Lie algebra corresponding to Chevalley group G; ·interpretations of Lie geometries, small Schubert cells, connections between the geometry of G and its associated Weyl geometry in terms of linear algebra, and applications of these problems to calculations performed in Lie geometries and associations schemes; ·constructions of geometric objects arising from Kac-Moody Lie algebras and superalgebras, and applications of these constructions to investigations of graphs of large girth and large size.