We analyze the structure of reduced expressions in the Coxeter groups An, Bn and Dn. Several special classes of elements are singled out for their connections with symmetric functions or the theory of P -partitions. Membership in these special classes is characterized in a variety of ways, including forbidden patterns, forbidden subwords, and by the form of canonically chosen reduced words.

We introduce and study a class of simplicial complexes, the orphan complexes, associated to simple graphs whose family of (open or closed) vertex-neighborhoods are anti-Sperner. Under suitable restrictions, we show that orphan complexes of such graphs are always shellable and provide a characterization of graphs in terms of induced forbidden subgraphs contained in this restricted subfamily.

In this paper we demonstrate that several theorems from [1] and [3] do not hold as they are stated. These are the theorems regarding forbidden subposet characterizations of certain classes of coverincomparability graphs. In this paper we correct a mistake in the main theorem of [1], reformulate the corresponding statements and present corrected proofs. We further characterize posets whose cover...

We characterize 2–dimensional complexes associated canonically with basis graphs of matroids as simply connected triangle-square complexes satisfying some local conditions. This proves a version of a (disproved) conjecture by Stephen Maurer (Conjecture 3 of S. Maurer, Matroid basis graphs I, JCTB 14 (1973), 216–240). We also establish Conjecture 1 from the same paper about the redundancy of the...

While the 3-dimensional analogue of the Sperner problem in the plane was known to be PPAD-complete, the complexity of the 2DSPERNER itself is not known to be PPAD-complete or not. In this paper, we settle this open problem proposed by Papadimitriou [7] fifteen years ago. This also allows us to derive the computational complexity characterization of a discrete version of the 2-dimensional Brouwe...

We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich [7] for R3 and B. Morel [16] for S3 and H3. The main argument is the interpretation of the energy-momentum tensor of a genra...

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties $X$ dimension 3 Picard rank 1. prove that if scheme has 0, then sheaf is $\mu$-stable whenever tangent bundle $TX$ stable, apply this fact to characterization certain irreducible components moduli space 2 reflexive $\mathbb{P}^3$ a quadric hypersurface $Q_3\subset\mathbb{P}^4$. Finally,...

AND GENERIC RIGIDITY IN THE PLANE SACHIN PATKAR, BRIGITTE SERVATIUS, AND K. V. SUBRAHMANYAM Abstract. We consider the concept of abstract 2–dimensional rigidity and provide necessary and sufficient conditions for a matroid to be an abstract rigidity matroid of a complete graph. This characterization is a natural extenWe consider the concept of abstract 2–dimensional rigidity and provide necessa...

We discuss the dimensional characterization of the solutions space of a formally integrable system of partial differential equations and provide certain formulas for calculations of these dimensional quantities.