It is well known that for a group G and a subgroup A of G it is impossible to cover G with the conjugates of A. Thus, instead of the conjugate of A we take the conjugate of the coset Ax in G and check if the union of (Ax)^g covers G-{1} for g in G. Moreover, if (Ax)^g covers G for all Ax in Cos(G:A), we say that (G,A) is CCI. We are aiming to classify all such pairs. It has been proven by Baumeister-Kaplan-Levy that this can be reduced down to the case where A is maximal in G and so that the action of G on Cos(G:A) is primitive. and they showed that (G,A) is CCI if G is 2-transitive. By O'Nan-Scott Theorem and CFSG, we see that G is either an affine group or almost simple. In the paper of Baumeister-Kaplan-Levy, it has been shown that affine CCI groups are 2-transitive. Thus, it remains to consider the almost simple groups.\\ By employing the knowledge of buildings, representation theory and Aschbacher-Dynkin theorem, we prove that, apart from finitely many small cases, the CCI almost simple groups are 2-transitive.

This talk assumes no prior knowledge for any of the terms mentioned in this abstract. Matroids are a combinatorial structure that abstracts linear independence. They have a deep connection with Grassmannians, the variety of subspaces of a vector space cut out by the Plücker equations. Grassmannians and matroids are inherently 'type A' objects and have generalizations to other root systems giving rise to what is known as Coxeter matroids. As such, one can generalize (through representation theory) these Grassmannians and Plücker equations to other Coxeter types using miniscule varieties and equations that define the quotient G/P of a simply connected complex Lie Group G by a maximal parabolic subgroup P with a miniscule fundamental representation. In addition, Borovik, Gelfand and White give a description of a strong exchange property for Coxeter matroids. It turns out there is a strong link between the strong exchange property and the tropicalization of the equations that define G/P. In this talk, we describe this link through representation theory, tropical geometry and polytope theory. (This is joint work with Kieran Calvert, Alex Fink and Ben Smith).