Lecturer: Tyrrell McAllister
|Many problems in pure and applied mathematics can be solved by enumerating the lattice points in rational polyhedra. Recent applications to representation theory include the first polynomial time algorithms for computing tensor product multiplicities for representations of the classical Lie algebras. Computational experiments employing these algorithms reveal an apparent positivity property of these representations that, if true in general, would generalize the saturation theorem of Knutson and Tao. This positivity property reveals itself in coefficients and periods of the Ehrhart quasi-polynomials of certain families of rational polyhedra.
Thereby motivated to study the coefficients and periods of Ehrhart quasi-polynomials, we turn to the Ehrhart quasi-polynomials of rational polygons in the plane. We describe work in progress examining the conditions under which the period of the Ehrhart quasi-polynomial of a rational polygon collapses to a polynomial. We give a characterization of all of the polynomial functions that can arise in this manner. Period collapse of Ehrhart quasi-polynomials is difficult to predict in general, seemingly arising as an inexplicable coincidence of cancelations. We show that this collapse is frequently due to the rational polygon being in fact a signed union of "disguised integral polygon" in a sense we make precise. We conjecture that this phenomenon explains all period collapse in Ehrhart quasi-polynomials of rational polyhedra.
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