Combinatorics and geometry of Littlewood-Richardson cones

We present several direct bijections between different combinatorial interpretations of the Littlewood-Richardson coefficients. The bijections are defined by explicit linear maps which have other applications.Comment: 15 pages, 9 figures. To be published in the special issue on "Combinatorics and Representation Theory" of the European Journal of Combinatoric

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

Generalized Stability of Heisenberg Coefficients

Stembridge introduced the notion of stability for Kronecker triples which generalize Murnaghan's classical stability result for Kronecker coefficients. Sam and Snowden proved a conjecture of Stembridge concerning stable Kronecker triple, and they also showed an analogous result for Littlewood--Richardson coefficients. Heisenberg coefficients are Schur structure constants of the Heisenberg product which generalize both Littlewood--Richardson coefficients and Kronecker coefficients. We show that any stable triple for Kronecker coefficients or Littlewood--Richardson coefficients also stabilizes Heisenberg coefficients, and we classify the triples stabilizing Heisenberg coefficients. We also follow Vallejo's idea of using matrix additivity to generate Heisenberg stable triples.Comment: 13 page

Geometry and complexity of O'Hara's algorithm

In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general. Finally, we prove that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.Comment: 20 pages, 4 figure