arXiv subjects

J. Parkinson

Publications and source records attributed to J. Parkinson.

8 recordsLinked to original sources

arXiv2025

Bounded weight functions on regular languages and groups

We introduce the notion of a bounded weight function on a language, and show that the set of bounded weight functions on a regular language is a rational polyhedral cone. We study the cell recognised by a bounded weight function (that is, the set of elements of the language where the bound is attained), and show that if the language is regular then this cell is regular. The related notion of a weight function on a finitely generated group is introduced, and the case of Coxeter groups is studied in detail. Applications to the representation theory of weighted Hecke algebras are given.

math.GR↗

arXiv2019

Patterns in sets of positive density in trees and affine buildings

We prove an analogue for homogeneous trees and certain affine buildings of a result of Bourgain on pinned distances in sets of positive density in Euclidean spaces. Furthermore, we construct an example of a non-homogeneous tree with positive Hausdorff dimension, and a subset with positive density thereof, in which not all sufficiently large (even) distances are realised.

math.CO↗

arXiv2019

Opposition diagrams for automorphisms of small spherical buildings

An automorphism $θ$ of a spherical building $Δ$ is called \textit{capped} if it satisfies the following property: if there exist both type $J_1$ and $J_2$ simplices of $Δ$ mapped onto opposite simplices by $θ$ then there exists a type $J_1\cup J_2$ simplex of $Δ$ mapped onto an opposite simplex by $θ$. In previous work we showed that if $Δ$ is a thick irreducible spherical building of rank at least $3$ with no Fano plane residues then every automorphism of $Δ$ is capped. In the present work we consider the spherical buildings with Fano plane residues (the \textit{small buildings}). We show that uncapped automorphisms exist in these buildings and develop an enhanced notion of "opposition diagrams" to capture the structure of these automorphisms. Moreover we provide applications to the theory of "domesticity" in spherical buildings, including the complete classification of domestic automorphisms of small buildings of types $\mathsf{F}_4$ and $\mathsf{E}_6$.

math.CO↗

arXiv2018

Balanced representations, the asymptotic Plancherel formula, and Lusztig's conjectures for $\tilde{C}_2$

We prove Lusztig's conjectures ${\bf P1}$-${\bf P15}$ for the affine Weyl group of type $\tilde{C}_2$ for all choices of positive weight function. Our approach to computing Lusztig's $\mathbf{a}$-function is based on the notion of a `balanced system of cell representations'. Once this system is established roughly half of the conjectures ${\bf P1}$-${\bf P15}$ follow. Next we establish an `asymptotic Plancherel Theorem' for type $\tilde{C}_2$, from which the remaining conjectures follow. Combined with existing results in the literature this completes the proof of Lusztig's conjectures for all rank $1$ and $2$ affine Weyl groups for all choices of parameters.

math.RT↗

arXiv2018

A proof of Lusztig's conjectures for affine type $G_2$ with arbitrary parameters

We prove Lusztig's conjectures ${\bf P1}$--${\bf P15}$ for the affine Weyl group of type $\tilde{G}_2$ for all choices of parameters. Our approach to compute Lusztig's $\mathbf{a}$-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the $\mathbf{a}$-function and we explicitly construct such a system in type $\tilde{G}_2$ for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type $\tilde{G}_2$, allowing us to prove ${\bf P1}$ and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types $G_2$ and $A_1$, along with some explicit computations for the finite cells.

math.RT↗

arXiv2018

Opposition diagrams for automorphisms of large spherical buildings

Let $θ$ be an automorphism of a thick irreducible spherical building $Δ$ of rank at least $3$ with no Fano plane residues. We prove that if there exist both type $J_1$ and $J_2$ simplices of $Δ$ mapped onto opposite simplices by $θ$, then there exists a type $J_1\cup J_2$ simplex of $Δ$ mapped onto an opposite simplex by $θ$. This property is called "cappedness". We give applications of cappedness to opposition diagrams, domesticity, and the calculation of displacement in spherical buildings. In a companion piece to this paper we study the thick irreducible spherical buildings containing Fano plane residues. In these buildings automorphisms are not necessarily capped.

math.CO↗

arXiv2018

Buildings, groups of Lie type, and random walks

In this paper we survey the theory of random walks on buildings and associated groups of Lie type and Kac-Moody groups. We begin with an introduction to the theory of Coxeter systems and buildings, taking a largely combinatorial perspective. We then survey the theory of random walks on buildings, and show how this theory leads to limit theorems for random walks on the associated groups.

math.PR↗

arXiv2014

Limit theorems for random walks on Fuchsian buildings and Kac-Moody groups

In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac-Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in [21]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building.

math.PR↗