Searcharxiv⌕ Search

arXiv subjects

Joel Brewster Lewis

Publications and source records attributed to Joel Brewster Lewis.

At least 37 records · Page 2Linked to original sources
arXiv2022

The tree search game for two players

We consider a two-player search game on a tree $T$. One vertex (unknown to the players) is randomly selected as the target. The players alternately guess vertices. If a guess $v$ is not the target, then both players are informed in which subtree of $T \smallsetminus v$ the target lies. The winner is the player who guesses the target. When both players play optimally, we show that each of them wins with probability approximately $1/2$. When one player plays optimally and the other plays randomly, we show that the player with the optimal strategy wins with probability between $9/16$ and $2/3$ (asymptotically). When both players play randomly, we show that each wins with probability between $13/30$ and $17/30$ (asymptotically).

math.PR↗
arXiv2021

Hurwitz numbers for reflection groups I: Generatingfunctionology

The classical Hurwitz numbers count the fixed-length transitive transposition factorizations of a permutation, with a remarkable product formula for the case of minimum length (genus $0$). We study the analogue of these numbers for reflection groups with the following generalization of transitivity: say that a reflection factorization of an element in a reflection group $W$ is full if the factors generate the whole group $W$. We compute the generating function for full factorizations of arbitrary length for an arbitrary element in a group in the combinatorial family $G(m, p, n)$ of complex reflection groups in terms of the generating functions of the symmetric group $\mathfrak{S}_n$ and the cyclic group of order $m/p$. As a corollary, we obtain leading-term formulas which count minimum-length full reflection factorizations of an arbitrary element in $G(m,p,n)$ in terms of the Hurwitz numbers of genus $0$ and $1$ and number-theoretic functions. We also study the structural properties of such generating functions for any complex reflection group; in particular, we show via representation-theoretic methods that they can by expressed as finite sums of exponentials of the variable.

math.CO↗
arXiv2021

The Hurwitz action in complex reflection groups

We enumerate Hurwitz orbits of shortest reflection factorizations of an arbitrary element in the infinite family $G(m, p, n)$ of complex reflection groups. As a consequence, we characterize the elements for which the action is transitive and give a simple criterion to tell when two shortest reflection factorizations belong to the same Hurwitz orbit. We also characterize the quasi-Coxeter elements (those with a shortest reflection factorization that generates the whole group) in $G(m, p, n)$.

math.CO↗
arXiv2020

Enriched set-valued P-partitions and shifted stable Grothendieck polynomials

We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse's shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a "$K$-theoretic" Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new explicit formulas for the involution $ω$ on the ring of symmetric functions.

math.CO↗
arXiv2020

Factorization problems in complex reflection groups

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our approach is fully combinatorial. It gives results analogous to those of Jackson in the symmetric group and can be refined to encode a notion of cycle type. As one application of our results, we give a previously overlooked characterization of the poset of $W$-noncrossing partitions.

math.CO↗
arXiv2017

Computing reflection length in an affine Coxeter group

In any Coxeter group, the conjugates of elements in its Coxeter generating set are called reflections and the reflection length of an element is its length with respect to this expanded generating set. In this article we give a simple formula that computes the reflection length of any element in any affine Coxeter group and we provide a simple uniform proof.

math.CO↗
arXiv2017

Rook theory of the finite general linear group

Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue of rook placements on a board. We develop this $q$-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding theory, we show that these $q$-hit and $q$-rook numbers obey a variety of identities analogous to the classical case. We also explore connections to earlier $q$-analogues of rook theory, as well as settling a polynomiality conjecture and finding a counterexample of a positivity conjecture of the authors and Klein.

math.CO↗
arXiv2017

Monodromy in Kazhdan-Lusztig cells in affine type A

We use the affine Robinson-Schensted correspondence to describe the structure of bidirected edges in the Kazhdan-Lusztig cells in affine type A. Equivalently, we give a comprehensive description of the Knuth equivalence classes of affine permutations.

math.CO↗
arXiv2016

Circuits and Hurwitz action in finite root systems

In a finite real reflection group, two factorizations of a Coxeter element into an arbitrary number of reflections are shown to lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. The proof makes use of a surprising lemma, derived from a classification of the minimal linear dependences (matroid circuits) in finite root systems: any set of roots forming a minimal linear dependence with positive coefficients has a disconnected graph of pairwise acuteness.

math.CO↗
arXiv2015

GL_n(F_q)-analogues of factorization problems in the symmetric group

We consider GL_n(F_q)-analogues of certain factorization problems in the symmetric group S_n: rather than counting factorizations of the long cycle (1, 2, ..., n) given the number of cycles of each factor, we count factorizations of a regular elliptic element given the fixed space dimension of each factor. We show that, as in S_n, the generating function counting these factorizations has attractive coefficients after an appropriate change of basis. Our work generalizes several recent results on factorizations in GL_n(F_q) and also uses a character-based approach. As an application of our results, we compute the asymptotic growth rate of the number of factorizations of fixed genus of a regular elliptic element in GL_n(F_q) into two factors as n goes to infinity. We end with a number of open questions.

math.CO↗
arXiv2015

Combinatorics of diagrams of permutations

There are numerous combinatorial objects associated to a Grassmannian permutation $w_λ$ that index cells of the totally nonnegative Grassmannian. We study several of these objects and their $q$-analogues in the case of permutations $w$ that are not necessarily Grassmannian. We give two main results: first, we show that certain acyclic orientations, rook placements avoiding a diagram of $w$, and fillings of a diagram of $w$ are equinumerous for all permutations $w$. Second, we give a $q$-analogue of a result of Hultman-Linusson-Shareshian-Sjöstrand by showing that under a certain pattern condition the Poincaré polynomial for the Bruhat interval of $w$ essentially counts invertible matrices avoiding a diagram of $w$ over a finite field. In addition to our main results, we include at the end a number of open questions.

math.CO↗
arXiv2015

Absolute order in general linear groups

This paper studies a partial order on the general linear group GL(V) called the absolute order, derived from viewing GL(V) as a group generated by reflections, that is, elements whose fixed space has codimension one. The absolute order on GL(V) is shown to have two equivalent descriptions, one via additivity of length for factorizations into reflections, the other via additivity of fixed space codimensions. Other general properties of the order are derived, including self-duality of its intervals. Working over a finite field F_q, it is shown via a complex character computation that the poset interval from the identity to a Singer cycle (or any regular elliptic element) in GL_n(F_q) has a strikingly simple formula for the number of chains passing through a prescribed set of ranks.

math.CO↗
arXiv2014

Reflection factorizations of Singer cycles

The number of shortest factorizations into reflections for a Singer cycle in GL_n(F_q) is shown to be (q^n-1)^(n - 1). Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given. The method is a standard character-theory technique, requiring the compilation of irreducible character values for Singer cycles, semisimple reflections, and transvections. The results suggest several open problems and questions, which are discussed at the end.

math.CO↗
arXiv2013

Flashcard games

We study a certain family of discrete dynamical processes introduced by Novikoff, Kleinberg and Strogatz that we call flashcard games. We prove a number of results on the evolution of these games, an in particular we settle a conjecture of NKS on the frequency with which a given card appears. We introduce a number of generalizations and variations that we believe are of interest, and provide a large number of open questions and problems.

math.CO↗
arXiv2013

Skyscraper Numbers

We introduce numbers depending on three parameters which we call skyscraper numbers. We discuss properties of these numbers and their relationship with Stirling numbers of the first kind, and we also introduce a skyscraper sequence.

math.CO↗