arXiv subjects

Brendon Rhoades

Publications and source records attributed to Brendon Rhoades.

61 records · Page 4Linked to original sources

arXiv2012

Parking Structures: Fuss Analogs

For any irreducible real reflection group $W$ with Coxeter number $h$, Armstrong, Reiner, and the author introduced a pair of $W \times \ZZ_h$-modules which deserve to be called {\sf $W$-parking spaces} which generalize the type A notion of parking functions and conjectured a relationship between them. In this paper we give a Fuss analog of their constructions. For a Fuss parameter $k \geq 1$, we define a pair of $W \times \ZZ_{kh}$-modules which deserve to be called {\sf $k$-$W$-parking spaces} and conjecture a relationship between them. We prove the weakest version of our conjectures for each of the infinite families ABCDI of finite reflection groups, together with proofs of stronger versions in special cases. Whenever our weakest conjecture holds for $W$, we have the following corollaries. First, there is a simple formula for the character of either $k$-$W$-parking space. Second, we recover a cyclic sieving result due to Krattenthaler and Müller which gives the cycle structure of a generalized rotation action on $k$-$W$-noncrossing partitions. Finally, when $W$ is crystallographic, the restriction of either $k$-$W$-parking space to $W$ isomorphic to the action of $W$ on the finite torus $Q / (kh+1)Q$, where $Q$ is the root lattice.

math.CO↗

arXiv2010

The Shi arrangement and the Ish arrangement

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

math.CO↗

arXiv2010

Enumeration of connected Catalan objects by type

Noncrossing set partitions, nonnesting set partitions, Dyck paths, and rooted plane trees are four classes of Catalan objects which carry a notion of type. There exists a product formula which enumerates these objects according to type. We define a notion of `connectivity' for these objects and prove an analogous product formula which counts connected objects by type. Our proof of this product formula is combinatorial and bijective. We extend this to a product formula which counts objects with a fixed type and number of connected components. We relate our product formulas to symmetric functions arising from parking functions. We close by presenting an alternative proof of our product formulas communicated to us by Christian Krattenthaler which uses generating functions and Lagrange inversion.

math.CO↗

arXiv2010

Cyclic sieving and cluster multicomplexes

Reiner, Stanton, and White \cite{RSWCSP} proved results regarding the enumeration of polygon dissections up to rotational symmetry. Eu and Fu \cite{EuFu} generalized these results to Cartan-Killing types other than A by means of actions of deformed Coxeter elements on cluster complexes of Fomin and Zelevinsky \cite{FZY}. The Reiner-Stanton-White and Eu-Fu results were proven using direct counting arguments. We give representation theoretic proofs of closely related results using the notion of noncrossing and semi-noncrossing tableaux due to Pylyavskyy \cite{PN} as well as some geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky \cite{FZClusterII}.

math.CO↗

arXiv2010

Hall-Littlewood polynomials and fixed point enumeration

We resolve affirmatively some conjectures of Reiner, Stanton, and White \cite{ReinerComm} regarding enumeration of transportation matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms of the bicyclic sieving phenomenon introduced by Barcelo, Reiner, and Stanton \cite{BRSBiD}. The proofs of our results use various tools from symmetric function theory such as the Stanton-White rim hook correspondence \cite{SW} and results concerning the specialization of Hall-Littlewood polynomials due to Lascoux, Leclerc, and Thibon \cite{LLTUnity} \cite{LLTRibbon}.

math.CO↗

arXiv2010

Cyclic sieving, promotion, and representation theory

We prove a collection of conjectures of D. White \cite{WComm}, as well as some related conjectures of Abuzzahab-Korson-Li-Meyer \cite{AKLM} and of Reiner and White \cite{ReinerComm}, \cite{WComm}, regarding the cyclic sieving phenomenon of Reiner, Stanton, and White \cite{RSWCSP} as it applies to jeu-de-taquin promotion on rectangular tableaux. To do this, we use Kazhdan-Lusztig theory and a characterization of the dual canonical basis of $\mathbb{C}[x_{11}, ..., x_{nn}]$ due to Skandera \cite{SkanNNDCB}. Afterwards, we extend our results to analyzing the fixed points of a dihedral action on rectangular tableaux generated by promotion and evacuation, suggesting a possible sieving phenomenon for dihedral groups. Finally, we give applications of this theory to cyclic sieving phenomena involving reduced words for the long elements of hyperoctohedral groups and noncrossing partitions.

math.CO↗

arXiv2008

Promotion and cyclic sieving via webs

We show that Schützenberger's promotion on two and three row rectangular Young tableaux can be realized as cyclic rotation of certain planar graphs introduced by Kuperberg. Moreover, following work of the third author, we show that this action admits the cyclic sieving phenomenon.

math.CO↗