Kazhdan-Lusztig polynomials and drift configurations

The coefficients of the Kazhdan-Lusztig polynomials $P_{v,w}(q)$ are nonnegative integers that are upper semicontinuous on Bruhat order. Conjecturally, the same properties hold for $h$-polynomials $H_{v,w}(q)$ of local rings of Schubert varieties. This suggests a parallel between the two families of polynomials. We prove our conjectures for Grassmannians, and more generally, covexillary Schubert varieties in complete flag varieties, by deriving a combinatorial formula for $H_{v,w}(q)$. We introduce \emph{drift configurations} to formulate a new and compatible combinatorial rule for $P_{v,w}(q)$. From our rules we deduce, for these cases, the coefficient-wise inequality $P_{v,w}(q)\preceq H_{v,w}(q)$.Comment: 26 pages. To appear in Algebra & Number Theor

Competition between s-wave order and d-wave order in holographic superconductors

We study competition between s-wave order and d-wave order through two holographic superconductor models. We find that once the coexisting phase appears, it is always thermodynamically favored, and that the coexistence phase is narrow and one condensate tends to kill the other. The phase diagram is constructed for each model in terms of temperature and the ratio of charges of two orders. We further compare the behaviors of some thermodynamic quantities, and discuss the different aspects and identical ones between two models.Comment: 3 figures added, accepted by JHE