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

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

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

Age structure, dispersion and diet of a population of stoats (Mustela erminea) in southern Fiordland during the decline phase of the beechmast cycle

The dispersion, age structure and diet of stoats (Mustela erminea) in beech forest in the Borland and Grebe Valleys, Fiordland National Park, were examined during December and January 2000/01, 20 months after a heavy seed-fall in 1999. Thirty trap stations were set along a 38-km transect through almost continuous beech forest, at least 1 km apart. Mice were very scarce (nights, C/100TN) along two standard index lines placed at either end of the transect, compared with November 1999 (>60/100TN), but mice were detected (from footprints in stoat tunnels) along an 8 km central section of the transect (stations 14-22). Live trapping with one trap per station (total 317.5 trap nights) in December 2000 caught 2 female and 23 male stoats, of which 10 (including both females) were radio collared. The minimum range lengths of the two females along the transect represented by the trap line were 2.2 and 6.0 km; those of eight radio-tracked males averaged 2.9 ± 1.7 km. Stations 14-22 tended to be visited more often, by more marked individual stoats, than the other 21 stations. Fenn trapping at the same 30 sites, but with multiple traps per station (1333.5 trap nights), in late January 2001 collected carcasses of 35 males and 28 females (including 12 of the marked live-trapped ones). Another two marked males were recovered dead. The stoat population showed no sign of chronic nutritional stress (average fat reserve index = 2.8 on a scale of 1-4 where 4 = highest fat content); and only one of 63 guts analysed was empty. Nevertheless, all 76 stoats handled were adults with 1-3 cementum annuli in their teeth, showing that reproduction had failed that season. Prey categories recorded in descending frequency of occurrence were birds, carabid beetle (ground beetle), weta, possum, rat, and mouse. The frequencies of occurrence of mice and birds in the diet of these stoats (10% and 48%, respectively) were quite different from those in stoats collected in Pig Creek, a tributary of the Borland River (87%, 5%), 12 months previously when mice were still abundant. Five of the six stoat guts containing mice were collected within 1 km of stations 14-22

Observation of Beam Spin Asymmetries in the Process ep → e\u27π⁺π⁻ X with CLAS 12

The observation of beam spin asymmetries in two-pion production in semi-inclusive deep inelastic scattering off an unpolarized proton target is reported. The data presented here were taken in the fall of 2018 with the CLAS12 spectrometer using a 10.6 GeV longitudinally spin-polarized electron beam delivered by CEBAF at JLab. The measured asymmetries provide the first opportunity to extract the parton distribution function e(x), which provides information about the interaction between gluons and quarks, in a collinear framework that offers cleaner access than previous measurements. The asymmetries also constitute the first ever signal sensitive to the helicity-dependent two-pion fragmentation function G⊥1. A clear sign change is observed around the ρ mass that appears in model calculations and is indicative of the dependence of the produced pions on the helicity of the fragmenting quark

Measurements of the transverse-momentum-dependent cross sections of J /ψ production at mid-rapidity in proton+proton collisions at s =510 and 500 GeV with the STAR detector

We present measurements of the differential cross sections of inclusive J/ψ meson production as a function of transverse momentum (pTJ/ψ) using the μ+μ- and e+e- decay channels in proton+proton collisions at center-of-mass energies of 510 and 500 GeV, respectively, recorded by the STAR detector at the Relativistic Heavy Ion Collider. The measurement from the μ+μ- channel is for

Charge-dependent pair correlations relative to a third particle in p + Au and d + Au collisions at RHIC

Quark interactions with topological gluon configurations can induce chirality imbalance and local parity violation in quantum chromodynamics. This can lead to electric charge separation along the strong magnetic field in relativistic heavy-ion collisions – the chiral magnetic effect (CME). We report measurements by the STAR collaboration of a CME-sensitive observable in p+Au and d+Au collisions at 200 GeV, where the CME is not expected, using charge-dependent pair correlations relative to a third particle. We observe strong charge-dependent correlations similar to those measured in heavy-ion collisions. This bears important implications for the interpretation of the heavy-ion data

Measurement of inclusive J/ψ suppression in Au+Au collisions at sNN=200 GeV through the dimuon channel at STAR

J/ψ suppression has long been considered a sensitive signature of the formation of the Quark-Gluon Plasma (QGP) in relativistic heavy-ion collisions. In this letter, we present the first measurement of inclusive J/ψ production at mid-rapidity through the dimuon decay channel in Au+Au collisions at sNN=200 GeV with the STAR experiment. These measurements became possible after the installation of the Muon Telescope Detector was completed in 2014. The J/ψ yields are measured in a wide transverse momentum (pT) range of 0.15 GeV/c to 12 GeV/c from central to peripheral collisions. They extend the kinematic reach of previous measurements at RHIC with improved precision. In the 0-10% most central collisions, the J/ψ yield is suppressed by a factor of approximately 3 for pT>5 GeV/c relative to that in p+p collisions scaled by the number of binary nucleon-nucleon collisions. The J/ψ nuclear modification factor displays little dependence on pT in all centrality bins. Model calculations can qualitatively describe the data, providing further evidence for the color-screening effect experienced by J/ψ mesons in the QGP

Observation of Excess J/ψ Yield at Very Low Transverse Momenta in Au+Au Collisions at sqrt[s_{NN}]=200 GeV and U+U Collisions at sqrt[s_{NN}]=193 GeV.

We report on the first measurements of J/ψ production at very low transverse momentum (p_{T}<0.2  GeV/c) in hadronic Au+Au collisions at sqrt[s_{NN}]=200  GeV and U+U collisions at sqrt[s_{NN}]=193  GeV. Remarkably, the inferred nuclear modification factor of J/ψ at midrapidity in Au+Au (U+U) collisions reaches about 24 (52) for p_{T}<0.05  GeV/c in the 60%-80% collision centrality class. This noteworthy enhancement cannot be explained by hadronic production accompanied by cold and hot medium effects. In addition, the dN/dt distribution of J/ψ for the very low p_{T} range is presented for the first time. The distribution is consistent with that expected from the Au nucleus and shows a hint of interference. Comparison of the measurements to theoretical calculations of coherent production shows that the excess yield can be described reasonably well and reveals a partial disruption of coherent production in semicentral collisions, perhaps due to the violent hadronic interactions. Incorporating theoretical calculations, the results strongly suggest that the dramatic enhancement of J/ψ yield observed at extremely low p_{T} originates from coherent photon-nucleus interactions. In particular, coherently produced J/ψ's in violent hadronic collisions may provide a novel probe of the quark-gluon plasma