Combinatorial Hopf algebras and Towers of Algebras

Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$.Comment: 7 page

Secondary Power Diagram, Dual of Secondary Polytope

An ingenious construction of Gel’fand et al. (Discriminants, Resultants, and Multidimensional Determinants. Birkhäuser, Basel, 1994) geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with non-empty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a non-degenerated coherent triangulation to another non-degenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are non-degenerated and coherent