Related papers: Lexicographic Shellability of Partial Involutions

We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new…

We compare the following three families of geometric objects: Schubert varieties in flag manifolds, matrix Schubert varieties, and Borel orbits of 2-nilpotent matrices. The first family is governed by permutations, the second by partial…

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence.…

We notice a remarkable connection between Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four…

We investigate two and three-dimensional shell-structured-inflatable froths, which can be constructed by a recursion procedure adding successive layers of cells around a germ cell. We prove that any froth can be reduced into a system of…

We classify conjugacy classes of involutions in the isometry groups of nondegenerate, symmetric bilinear forms over the field of two elements. The new component of this work focuses on the case of an orthogonal form on an even dimensional…

A geometric approach to immersion formulas for soliton surfaces is provided through new cohomologies on spaces of special types of $\mathfrak{g}$-valued differential forms. This leads us to introduce Poincar\'e-type lemmas for these…

We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…

We define a birational map between labelings of a rectangular poset and its associated trapezoidal poset. This map tropicalizes to a bijection between the plane partitions of these posets of fixed height, giving a new bijective proof of a…

The `Weyl symmetric functions' studied here naturally generalize classical symmetric (polynomial) functions, and `Weyl bialternants,' sometimes also called Weyl characters, analogize the Schur functions. For this generalization, the…

Let $\mathcal A$ be a unital algebra equipped with an involution $(\cdot)^\dagger$, and suppose that the multiplicative set $\mathcal S\subseteq \mathcal A$ generated by the elements of the form $1 + a^\dagger a$ satisfies the Ore…

We explore a somewhat unexpected connection between knot Floer homology and shellable posets, via grid diagrams. Given a grid presentation of a knot K inside S^3, we define a poset which has an associated chain complex whose homology is the…

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different…

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…

Let G be an almost simple group over an algebraically closed field k of characteristic zero, let g be its Lie algebra and let B be a Borel subgroup of G. Then B acts with finitely many orbits on the variety N_2 of the nilpotent elements in…

A key theorem of Yakimov's proves that the torus-invariant prime spectra of De Concini-Kac-Procesi algebras are isomorphic as partially ordered sets to corresponding Bruhat order intervals of Weyl groups. We present examples of more general…

Every partial applicative structure gives rise to an indexed binary relation, that is a contravariant functor from the category of sets to the category of sets endowed with binary relations and maps preserving them. In this paper we…

We study the partially ordered set $P(a_1,\ldots, a_n)$ of all multidegrees $(b_1,\dots,b_n)$ of monomials $x_1^{b_1}\cdots x_n^{b_n}$ which properly divide $x_1^{a_1}\cdots x_n^{a_n}$. We prove that the order complex…

Bj\"orner and Wachs introduced CL-shellability as a technique for studying the topological structure of order complexes of partially ordered sets (posets). They also introduced the notion of recursive atom ordering, and they proved that a…

We examine properties of generic automorphisms of the random poset, with the goal of explicitly characterizing them. We associate to each automorphism an auxiliary first-order structure, consisting of the random poset equipped with an…