Affine descents and the Steinberg torus

Let $W\ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.Comment: 24 pages, 2 figure

Resonance in orbits of plane partitions

International audienceWe introduce a new concept of resonance on discrete dynamical systems. Our main result is an equivariant bijection between plane partitions in a box under rowmotion and increasing tableaux under K-promotion, using a generalization of the equivariance of promotion and rowmotion [J. Striker–N. Williams '12] to higher dimensional lattices. This theorem implies new results for K-promotion and new proofs of previous results on plane partitions

Affine descents and the Steinberg torus

Abstract. Let W ⋉ L be an irreducible affine Weyl group with Coxeter complex Σ, where W denotes the associated finite Weyl group and L the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of Σ by the lattice L. We show that the ordinary and flag h-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over W for a descent-like statistic first studied by Cellini. We also show that the ordinary h-polynomial has a nonnegative γ-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the h-polynomials of Steinberg tori. Résumé. Nous considérons un groupe de Weyl affine irréductible W ⋉ L avec complexe de Coxeter Σ, où W désigne le groupe de Weyl fini associé et L le sous-groupe des translations. Le tore de Steinberg est le complexe cellulaire Booléen obtenu comme le quotient de Σ par L. Nous montrons que les h-polynômes, ordinaires et de drapeaux, du tore de Steinberg (sans la face vide) sont des fonctions génératrices sur W pour une statistique de type descente, étudiée en premier lieu par Cellini. Nous montrons également qu’un h-polynôme ordinaire possède un γ-vecteur positif, et par conséquent, a des coéfficients symétriques et unimodaux. Dans les cas classiques, nous donnons également des développements, des identités et des fonctions génératrices pour les h-polynômes des tores de Steinberg

Sensitivity of fluvial sediment source apportionment to mixing model assumptions: A Bayesian model comparison

Mixing models have become increasingly common tools for apportioning fluvial sediment load to various sediment sources across catchments using a wide variety of Bayesian and frequentist modeling approaches. In this study, we demonstrate how different model setups can impact upon resulting source apportionment estimates in a Bayesian framework via a one-factor-at-a-time (OFAT) sensitivity analysis. We formulate 13 versions of a mixing model, each with different error assumptions and model structural choices, and apply them to sediment geochemistry data from the River Blackwater, Norfolk, UK, to apportion suspended particulate matter (SPM) contributions from three sources (arable topsoils, road verges, and subsurface material) under base flow conditions between August 2012 and August 2013. Whilst all 13 models estimate subsurface sources to be the largest contributor of SPM (median ∼76%), comparison of apportionment estimates reveal varying degrees of sensitivity to changing priors, inclusion of covariance terms, incorporation of time-variant distributions, and methods of proportion characterization. We also demonstrate differences in apportionment results between a full and an empirical Bayesian setup, and between a Bayesian and a frequentist optimization approach. This OFAT sensitivity analysis reveals that mixing model structural choices and error assumptions can significantly impact upon sediment source apportionment results, with estimated median contributions in this study varying by up to 21% between model versions. Users of mixing models are therefore strongly advised to carefully consider and justify their choice of model structure prior to conducting sediment source apportionment investigations

Genetic Factors Leading to Chronic Epstein–Barr Virus Infection and Nasopharyngeal Carcinoma in South East China: Study Design, Methods and Feasibility

Nasopharyngeal carcinoma (NPC) is a complex disease caused by a combination of Epstein-Barr virus chronic infection, the environment and host genes in a multi-step process of carcinogenesis. The identity of genetic factors involved in the development of chronic Epstein-Barr virus infection and NPC remains elusive, however. Here, we describe a two-phase, population-based, case-control study of Han Chinese from Guangxi province, where the NPC incidence rate rises to a high of 25-50 per 100,000 individuals. Phase I, powered to detect single gene associations, enrolled 984 subjects to determine feasibility, to develop infrastructure and logistics and to determine error rates in sample handling. A microsatellite screen of Phase I study participants, genotyped for 319 alleles from 34 microsatellites spanning an 18-megabase region of chromosome 4 (4p15.1-q12), previously implicated by a linkage analysis of familial NPC, found 14 alleles marginally associated with developing NPC or chronic immunoglobulin A production (p = 0.001-0.03). These associations lost significance after applying a correction for multiple tests. Although the present results await confirmation, the Phase II study population has tripled patient enrolment and has included environmental covariates, offering the potential to validate this and other genomic regions that influence the onset of NPC

Involutions on Baxter Objects

Baxter numbers are known to count several families of combinatorial objects, all of which come equipped with natural involutions. In this paper, we add a combinatorial family to the list, and show that the known bijections between these objects respect these involutions. We also give a formula for the number of objects fixed under this involution, showing that it is an instance of Stembridge’s “q = −1 phenomenon”

Resonance in orbits of plane partitions

We introduce a new concept of resonance on discrete dynamical systems. Our main result is an equivariant bijection between plane partitions in a box under rowmotion and increasing tableaux under K-promotion, using a generalization of the equivariance of promotion and rowmotion [J. Striker–N. Williams '12] to higher dimensional lattices. This theorem implies new results for K-promotion and new proofs of previous results on plane partitions