arXiv subjects
Publications and source records attributed to G. A. Koshevoy.
We present a list of ``local'' axioms and an explicit combinatorial construction for the regular $B_2$-crystals (crystal graphs of highest weight integrable modules over $U_q(sp_4)$). Also a new combinatorial model for these crystals is developed.
This paper (written in Russian) presents a survey of new and earlier results on fine zonotopal tilings (briefly, cubillages) of cyclic zonotopes. The combinatorial theory of these objects is of interest in its own right and also has a connection to higher Bruhat orders, triangulations of cyclic polytopes, and Tamari-Stasheff posets applied in the study of Kadomtsev--Petviashvily equations, and etc.
We study certain structural properties of fine zonotopal tilings, or cubillages, on cyclic zonotopes $Z(n,d)$ of an arbitrary dimension $d$ and their relations to $(d-1)$-separated collections of subsets of a set $\{1,2,\ldots,n\}$. (Collections of this sort are well known as strongly separated ones when $d=2$, and as chord separated ones when $d=3$.)
We consider three types of set-systems that have interesting applications in algebraic combinatorics and representation theory: maximal collections of the so-called strongly separated, weakly separated, and chord separated subsets of a set $[n]=\{1,2,\ldots,n\}$. These collections are known to admit nice geometric interpretations; namely, they are bijective, respectively, to rhombus tilings on the zonogon $Z(n,2)$, combined tilings on $Z(n,2)$, and fine zonotopal tilings (or `cubillages') on the 3-dimensional zonotope $Z(n,3)$. We describe interrelations between these three types of set-systems in $2^{[n]}$, by studying interrelations between their geometric models. In particular, we completely characterize the sets of rhombus and combined tilings properly embeddable in a fixed cubillage, explain that they form distributive lattices, give efficient methods of extending a given rhombus or combined tiling to a cubillage, and etc.
A regular $A_n$-crystal is an edge-colored directed graph, with $n$ colors, related to an irreducible highest weight integrable module over $U_q(sl_{n+1})$. Based on Stembridge's local axioms for regular simply-laced crystals and a structural characterization of regular $A_2$-crystals in \cite{DKK-07}, we present a new combinatorial construction, the so-called {\em crossing model}, and prove that this model generates precisely the set of regular $A_n$-crystals. Using the model, we obtain a series of results on the combinatorial structure of such crystals and properties of their subcrystals.
We start with an ``algebraic'' RSK-correspondence due to Noumi and Yamada. Given a matrix $X$, we consider a pyramidal array of solid minors of $X$. It turns out that this array satisfies an algebraic variant of octahedron recurrence. The main observation is that this array can also be constructed with the help of some square `genetic' array. Next we tropicalize this algebraic construction and consider $T$-{\em polarized} pyramidal arrays (that is arrays satisfying octahedral relations). As a result we get several bijections, viz: a) a linear bijection between non-negative arrays and supermodular functions; b) a piecewise linear bijection between supermodular functions and the so called infra-modular functions; c) a linear bijection between infra-modular functions and plane partitions. A composition of these bijections yields a bijection between non-negative arrays and plane partitions coinciding with the modified RSK-correspondence.
For simply-laced Kac-Moody algebras $\frak g$, Stembridge (2003) proposed a `local' axiomatization of crystal graphs of representations of $U_q(\frak g)$. In this paper we propose axioms for edge-2-colored graphs which characterize the crystals of integrable representations of $U_q(sp(4))$, regular crystal graphs of $B_2$-type. An edge-colored directed graph which obeys our Axioms (K0)--(K5) is called an R-{\em graph} (for brevity), and our main result is that the regular crystals of $B_2$-type are R-graphs and vice versa. We give a direct combinatorial construction for the crystals in question. On this way we introduce a new, so-called {\em crossing model}, which does not exploit Young tableaux. This combinatorial model consists of a two-component graph of a rather simple form and of a certain set of integer-valued functions on its vertices.
We show that a connected regular $A_2$-crystal (the crystal graph of an irreducible representation of $sl_3$) can be produced from two half-grids by replicating them and glying together in a certain way. Also some extensions and related aspects are discussed.
In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property seems indisputable: X should coincide with the set of all integer points of its convex hull co(X) (in the ambient vector space V). However, this is a first approximation to a proper discrete convexity, because such non-intersecting sets need not be separated by a hyperplane. This issue is closely related to the question when the intersection of two integer polyhedra is an integer polyhedron. We show that unimodular systems (or more generally, pure systems) are in one-to-one correspondence with the classes of discrete convexity. For example, the well-known class of g-polymatroids corresponds to the class of discrete convexity associated to the unimodular system A_n:={\pm e_i, e_i-ej} in Z^n.
This paper is devoted to a study of mathematical structures arising from choice functions satisfying the path independence property (Plott functions). We broaden the notion of a choice function by allowing of empty choice. This enables us to define a lattice structure on the set of Plott functions. Moreover, this lattice is functorially dependent on its base. We introduce a natural convex structure on the set of linear orders (or words) and show that Plott functions are in one-to-one correspondence with convex subsets in this set of linear orders. That correspondence is compatible with both lattice structures. Keywords: Convex geometries, shuffle, linear orders, lattices, direct image, path independence, convex structure