Macdonald Polynomials from Sklyanin Algebras: A Conceptual Basis for the pp-Adics-Quantum Group Connection

We establish a previously conjectured connection between $p$-adics and quantum groups. We find in Sklyanin's two parameter elliptic quantum algebra and its generalizations, the conceptual basis for the Macdonald polynomials, which ``interpolate'' between the zonal spherical functions of related real and $p$\--adic symmetric spaces. The elliptic quantum algebras underlie the $Z_n$\--Baxter models. We show that in the n \air \infty limit, the Jost function for the scattering of {\em first} level excitations in the $Z_n$\--Baxter model coincides with the Harish\--Chandra\--like $c$\--function constructed from the Macdonald polynomials associated to the root system $A_1$. The partition function of the $Z_2$\--Baxter model itself is also expressed in terms of this Macdonald\--Harish\--Chandra\ $c$\--function, albeit in a less simple way. We relate the two parameters $q$ and $t$ of the Macdonald polynomials to the anisotropy and modular parameters of the Baxter model. In particular the $p$\--adic ``regimes'' in the Macdonald polynomials correspond to a discrete sequence of XXZ models. We also discuss the possibility of ``$q$\--deforming'' Euler products.Comment: 25 page

Simple does not mean poor: grasslands and forests harbor similar ant species richness and distinct composition in highlands of southern Brazil

Several studies addressed ant communities in the dense Atlantic Forest that runs along the Brazilian coast. However, comparatively little is known about the mixed forests and grasslands that occur in the southern range of the Atlantic Forest domain. In this study we performed the first standardized assessment of ants in the forest- grassland mosaic found in the highlands of the state of Santa Catarina. We aimed to investigate and compare ant richness and composition between mixed forests and grasslands in the main mountain range of south Brazil. Ants were collected in two years with ground pitfalls, tree pitfalls and litter samples. Sixty ant species were recorded, resulting in 22 new records for “Planalto Serrano” region and three for the state of Santa Catarina: Eurhopalothrix depressa , Pheidole radoszkowskii and Wasmannia williamsoni . There was significant dissimilarity in ant species composition between grasslands and forests, but no difference in ant species richness, even considering the higher number of strata in mixed forests. Similar richness and low number of arboreal species suggest that this ant community is structured similarly to temperate ones. Both habitats presented a large proportion of exclusive species. The fact that species composition between grassland and forest areas differed, coupled with the similarity in species richness between habitats and the record of new ant species for the region, calls for strong conservation efforts in grasslands of southern Brazil, which still are little protected by conservation areas