Polytopes: Inequalities for Flag Vectors, Poset Tranformations and Complex Polytopes

The PI proposes four projects in the broad area of algebraic combinatorics. The first project is a continuation of the PI's successful research program of determining linear inequalities for flag I-vectors of convex polytopes. The PI has introduced a new lifting technique which. when given an inequality for k-dimensional polytopes, produces inequalities in higher dimensions. As these new inequalities are not believed to be sharp, Project Ia is a conjectured sharpening of these inequalities using techniques of the PI. Project Ib is to find a similar lifting technique for flag I-vector inequalities for zonotopes. Finally, Project Ic concerns issues of proving that classes of inequalities for polytopes (and zonotopes) are not implied by previously known inequalities. The last two subprojects are suitable for graduate research assistants. Project II concerns poset transformations that produce new families of Eulerian posets. One importance of these transformations is that their image will serve as a testing ground for Stanley's Gorenstein' conjecture. Another main issue is to understand how a given poset transform changes the flag I -vector. Poset transformations that the PI has studied so far are the intersection lattice of a hyperplane arrangement to zonotope map (with Billera and Readdy), the r-signed Birkhoff transform, and the Tchebyshev transform (with Readdy). Project IIa concerns finding more poset transformations that fit into the coalgebraic framework of these three transformations. Project lIb is to study the extended Birkhoff transformation due to Hetyei and Hsiao, and Project IIc concerns the antiprisrn and the E,-construction by Paffenholz and Ziegler. Convex polytopes lie in real Euclidean space. Project III is to find a suitable definition of complex zonotopes and explore these structures. The PI conjectures that the flag I-vector of a complex zonotope is given by the r-signed Birkhoff transform. Project IV is to explore the Stanley chromatic symmetric function of a graph and its underlying coalgebraic properties.