Related papers: A pithy look at the Polytope Algebra
The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we first show some properties concerning overlap algebras: we prove that the notion of overlap morphism corresponds…
Volume polynomials form a distinguished class of log-concave polynomials with remarkable analytic and combinatorial properties. I will survey realization problems related to them, review fundamental inequalities they satisfy, and discuss…
This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…
Orthogonal polynomials are of fundamental importance in many fields of mathematics and science, therefore the study of a particular family is always relevant. In this manuscript, we present a survey of some general results of the Hermite…
We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. We study its interplay with the general presentation for any finite-dimensional basic algebra. We give an interpretation of evaluating $f_M$ at a weight vector.…
Problems for the graduate students who want to improve problem-solving skills in geometry. Every problem has a short elegant solution -- this gives a hint which was not available when the problem was discovered.
Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They…
For a given graph $G$, we construct an associated commutative algebra, whose dimension is equal to the number of $t$-labeled forests of $G$. We show that the dimension of the $k$-th graded component of this algebra also has a combinatorial…
Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for…
We generalize the method of combinatorial telescoping to the case of multiple summations. We shall demonstrate this idea by giving combinatorial proofs for two identities of Andrews on parity indices of partitions.
We give combinatorial formulas for F-polynomials in cluster algebras of classical types in terms of the weighted paths in certain directed graphs. As a consequence we prove the positivity of F-polynomials in cluster algebras of classical…
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
The flow polytope $\mathcal{F}_{\widetilde{G}}$ is the set of nonnegative unit flows on the graph $\widetilde{G}$. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope $\mathcal{F}_{\widetilde{G}}$ into…
We define a GL-variety to be a (typically infinite dimensional) algebraic variety equipped with an action of the infinite general linear group under which the coordinate ring forms a polynomial representation. Such varieties have been used…
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…
Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the…
In this work, we derive numerous identities for multivariate q-Euler polynomials by using umbral calculus.
The so called generalized down-up algebras are revisited from a viewpoint of Gr\"obner basis theory. Particularly it is shown explicitly that generalized down-up algebras are solvable polynomial algebras (provided $\lambda\omega\ne 0$), and…
Algebraic datatypes, and among them lists and trees, have attracted a lot of interest in automated reasoning and Satisfiability Modulo Theories (SMT). Since its latest stable version, the SMT-LIB standard defines a theory of algebraic…
We introduce a general class of symmetric polynomials that have saturated Newton polytope and their Newton polytope has integer decomposition property. The class covers numerous previously studied symmetric polynomials.