## Grants and Contracts Details

### Description

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.

Status | Finished |
---|---|

Effective start/end date | 5/2/06 → 10/30/08 |

### Funding

- National Security Agency: $62,650.00

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