arXiv subjects

Oliver Pechenik

Publications and source records attributed to Oliver Pechenik.

44 records · Page 3Linked to original sources

arXiv2016

Genomic Tableaux

We explain how genomic tableaux [Pechenik-Yong '15] are a semistandard complement to increasing tableaux [Thomas-Yong '09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender-Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood-Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch '02]) and maximal orthogonal Grassmannians (after [Clifford-Thomas-Yong '14], [Buch-Ravikumar '12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.

math.CO↗

arXiv2015

Equivariant K-theory of Grassmannians II: The Knutson-Vakil conjecture

In 2005, A. Knutson--R. Vakil conjectured a puzzle rule for equivariant K-theory of Grassmannians. We resolve this conjecture. After giving a correction, we establish a modified rule by combinatorially connecting it to the authors' recently proved tableau rule for the same Schubert calculus problem.

math.CO↗

arXiv2015

Proofs and generalizations of a homomesy conjecture of Propp and Roby

Let $G$ be a group acting on a set $X$ of combinatorial objects, with finite orbits, and consider a statistic $ξ: X \to \mathbb{C}$. Propp and Roby defined the triple $(X, G, ξ)$ to be \emph{homomesic} if for any orbits $\mathcal{O}_1, \mathcal{O}_2$, the average value of the statistic $ξ$ is the same, that is \[\frac{1}{{|\mathcal{O}_1|}}\sum_{x \in \mathcal{O}_1} ξ(x) = \frac{1}{|\mathcal{O}_2|}\sum_{y \in \mathcal{O}_2} ξ(y).\] In 2013 Propp and Roby conjectured the following instance of homomesy. Let $\mathrm{SSYT}_k(m \times n)$ denote the set of semistandard Young tableaux of shape $m \times n$ with entries bounded by $k$. Let $S$ be any set of boxes in the $m \times n$ rectangle fixed under $180^\circ$ rotation. For $T \in \mathrm{SSYT}_k(m \times n)$, define $σ_S(T)$ to be the sum of the entries of $T$ in the boxes of $S$. Let $\langle \mathcal{P} \rangle$ be a cyclic group of order $k$ where $\mathcal{P}$ acts on $\mathrm{SSYT}_k(m \times n)$ by promotion. Then $(\mathrm{SSYT}_k(m \times n), \langle \mathcal{P} \rangle, σ_S)$ is homomesic. We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the K-promotion of Thomas and Yong, and prove limited results in that direction.

math.CO↗

arXiv2015

Equivariant K-theory of Grassmannians

We address a unification of the Schubert calculus problems solved by [A. Buch '02] and [A. Knutson-T. Tao '03]. That is, we prove a combinatorial rule for the structure coefficients in the torus-equivariant K-theory of Grassmannians with respect to the basis of Schubert structure sheaves. We thereby deduce the conjectural rule of [H. Thomas-A. Yong '13] for the same coefficients. Both rules are positive in the sense of [D. Anderson-S. Griffeth-E. Miller '11] (and moreover in a stronger form). Our work is based on the combinatorics of genomic tableaux and a generalization of [M.-P. Schutzenberger '77]'s jeu de taquin.

math.CO↗

arXiv2014

Deformed Cohomology of Flag Varieties

This paper introduces a two-parameter deformation of the cohomology of generalized flag varieties. One special case is the Belkale-Kumar deformation (used to study eigencones of Lie groups). Another picks out intersections of Schubert varieties that behave nicely under projections. Our construction yields a new proof that the Belkale-Kumar product is well-defined. This proof is shorter and more elementary than earlier proofs.

math.AG↗

arXiv2014

Cyclic Sieving of Increasing Tableaux and small Schröder Paths

An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.

math.CO↗

arXiv2010

A Midsummer Knot's Dream

In this paper, we introduce playing games on shadows of knots. We demonstrate two novel games, namely, To Knot or Not to Knot and Much Ado about Knotting. We also discuss winning strategies for these games on certain families of knot shadows. Finally, we suggest variations of these games for further study.

math.GT↗

arXiv2010

Classical and Virtual Pseudodiagram Theory and New Bounds on Unknotting Numbers and Genus

A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by R. Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we investigate how much crossing information must be known to conclude that a diagram is a diagram of the unknot (the trivializing number). We also consider how much information is necessary to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.

math.GT↗