Cauchy Type Integrals of Algebraic Functions

We consider Cauchy type integrals $I(t)={1\over 2\pi i}\int_{\gamma} {g(z)dz\over z-t}$ with $g(z)$ an algebraic function. The main goal is to give constructive (at least, in principle) conditions for $I(t)$ to be an algebraic function, a rational function, and ultimately an identical zero near infinity. This is done by relating the Monodromy group of the algebraic function $g$, the geometry of the integration curve $\gamma$, and the analytic properties of the Cauchy type integrals. The motivation for the study of these conditions is provided by the fact that certain Cauchy type integrals of algebraic functions appear in the infinitesimal versions of two classical open questions in Analytic Theory of Differential Equations: the Poincar\'e Center-Focus problem and the second part of the Hilbert 16-th problem.Comment: 58 pages, 19 figure

Obstructions to combinatorial formulas for plethysm

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples; v3: final version as in Electronic Journal of Combinatoric

An Invitation to the Generalized Saturation Conjecture

We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and ``saturation'' properties of several generalizations of the Littlewood--Richardson numbers.Comment: 79 pages, new sections, new results and example

Combinatorial representation theory of Lie algebras. Richard Stanley's work and the way it was continued

Richard Stanley played a crucial role, through his work and his students, in the development of the relatively new area known as combinatorial representation theory. In the early stages, he has the merit to have pointed out to combinatorialists the potential that representation theory has for applications of combinatorial methods. Throughout his distinguished career, he wrote significant articles which touch upon various combinatorial aspects related to representation theory (of Lie algebras, the symmetric group, etc.). I describe some of Richard's contributions involving Lie algebras, as well as recent developments inspired by them (including some open problems), which attest the lasting impact of his work.Comment: 11 page