| In this manuscript we study combinatorial Hopf algebras and supercharacter theories of certain finite groups. More specifically, we study the group of unitriangular matrices of Lie type A,B,C,D with entries in a finite field Fq. Given that the representation theory of these groups over C is wild, we study combinatorially supercharacter theories for these groups which are a coarser version of their representation theory. We make emphasis on type D, showing two different supercharacter theories and endowing them with a combinatorial Hopf algebra structure. Our combinatorial approach allows us, for example, to give an explicit formula for the antipode in one of these supercharacter theories. Finally we address some further directions on the study of supercharacters and combinatorial Hopf algebras. |
