Toric Cubes

A toric cube is a subset of the standard cube defined by binomial inequalities. These basic semialgebraic sets are precisely the images of standard cubes under monomial maps. We study toric cubes from the perspective of topological combinatorics. Explicit decompositions as CW-complexes are constructed. Their open cells are interiors of toric cubes and their boundaries are subcomplexes. The motivating example of a toric cube is the edge-product space in phylogenetics, and our work generalizes results known for that space.Comment: to appear in Rendiconti del Circolo Matematico di Palermo (special issue on Algebraic Geometry

Nearest Points on Toric Varieties

We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the $A$-discriminant in terms of Euler obstructions. Our primary goal is the development of reliable algorithmic tools for computing the points on a real toric variety that are closest to a given data point.Comment: 20 page

Inhomogeneous lattice paths, generalized Kostka polynomials and An−1_{n-1} supernomials

Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials. One of these provides a generalization of the Kostka polynomials while the other, which we name the A$_{n-1}$ supernomial, is a $q$-deformation of the expansion coefficients of products of Schur polynomials. Many well-known results for Kostka polynomials are extended leading to representations of our polynomials in terms of a charge statistic on Littlewood-Richardson tableaux and in terms of fermionic configuration sums. Several identities for the generalized Kostka polynomials and the A$_{n-1}$ supernomials are proven or conjectured. Finally, a connection between the supernomials and Bailey's lemma is made.Comment: 44 pages, figures, AMS-latex; improved version to appear in Commun. Math. Phys., references added, some statements clarified, relation to other work specifie

Dominance Over ℵ

This paper provides an overview of the b-dominance order over the natural numbers, ℵ, using the base b expansion of natural numbers. The b-dominance order is an accessible partially-ordered set that is less complex than the divisor relation but more complex than ≤; thus, it supplies a good medium through which an undergraduate can be exposed to the subject of order theory. Here we discuss many ideas in order theory, including the Poincare polynomial and the Mobius function

Dominance Over N

Abstract.This paper provides an overview of the b-dominance order over the natural numbers, N, using the base b expansion of natural numbers. The b-dominance order is an accessible partially-ordered set that is less complex than the divisor relation but more complex than ≤; thus, it supplies a good medium through which an undergraduate can be exposed to the subject of order theory. Here we discuss many ideas in order theory, including the Poincaré polynomial and the Möbius function. Acknowledgements: The authors thank the M.J. Murdock Charitable Trust and the Pacific Lutheran University Division of Natural Sciences for their generous support. They would also like to extend their thanks to Dr. Tom Edgar for the project idea and all his help throughout their summer program. Page 24 RHIT Undergrad. Math. J., Vol. 14, no. 2

An affine generalization of evacuation

We establish the existence of an involution on tabloids that is analogous to Schutzenberger's evacuation map on standard Young tableaux. We find that the number of its fixed points is given by evaluating a certain Green's polynomial at $q = -1$, and satisfies a "domino-like" recurrence relation.Comment: 32 pages, 7 figure