Lagrangian turbulence in the Adriatic Sea as computed from drifter data: effects of inhomogeneity and nonstationarity

The properties of mesoscale Lagrangian turbulence in the Adriatic Sea are studied from a drifter data set spanning 1990-1999, focusing on the role of inhomogeneity and nonstationarity. A preliminary study is performed on the dependence of the turbulent velocity statistics on bin averaging, and a preferential bin scale of 0.25 is chosen. Comparison with independent estimates obtained using an optimized spline technique confirms this choice. Three main regions are identified where the velocity statistics are approximately homogeneous: the two boundary currents, West (East) Adriatic Current, WAC (EAC), and the southern central gyre, CG. The CG region is found to be characterized by symmetric probability density function of velocity, approximately exponential autocorrelations and well defined integral quantities such as di usivity and time scale. The boundary regions, instead, are significantly asymmetric with skewness indicating preferential events in the direction of the mean flow. The autocorrelation in the along mean flow direction is characterized by two time scales, with a secondary exponential with slow decay time of 11-12 days particularly evident in the EAC region. Seasonal partitioning of the data shows that this secondary scale is especially prominent in the summer-fall season. Possible physical explanations for the secondary scale are discussed in terms of low frequency fluctuations of forcings and in terms of mean flow curvature inducing fluctuations in the particle trajectories. Consequences of the results for transport modelling in the Adriatic Sea are discussed.Comment: 45 pages, 18 figure

Affine Hecke algebras of type D and generalisations of quiver Hecke algebras

We define and study cyclotomic quotients of affine Hecke algebras of type D. We establish an isomorphism between (direct sums of blocks of) these cyclotomic quotients and a generalisation of cyclotomic quiver Hecke algebras which are a family of Z-graded algebras closely related to algebras introduced by Shan, Varagnolo and Vasserot. To achieve this, we first complete the study of cyclotomic quotients of affine Hecke algebras of type B by considering the situation when a deformation parameter p squares to 1. We then relate the two generalisations of quiver Hecke algebras showing that the one for type D can be seen as fixed point subalgebras of their analogues for type B, and we carefully study how far this relation remains valid for cyclotomic quotients. This allows us to obtain the desired isomorphism. This isomorphism completes the family of isomorphisms relating affine Hecke algebras of classical types to (generalisations of) quiver Hecke algebras, originating in the famous result of Brundan and Kleshchev for the type A.Comment: 26 page

Affine Hecke algebras and generalisations of quiver Hecke algebras for type B

We define and study cyclotomic quotients of affine Hecke algebras of type B. We establish an isomorphism between direct sums of blocks of these algebras and a generalisation, for type B, of cyclotomic quiver Hecke algebras which are a family of graded algebras closely related to algebras introduced by Varagnolo and Vasserot. Inspired by the work of Brundan and Kleshchev we first give a family of isomorphisms for the corresponding result in type A which includes their original isomorphism. We then select a particular isomorphism from this family and use it to prove our result.Comment: 37 page

Markov traces on affine and cyclotomic Yokonuma-Hecke algebras

In this article, we define and study the affine and cyclotomic Yokonuma-Hecke algebras. These algebras generalise at the same time the Ariki-Koike and affine Hecke algebras and the Yokonuma-Hecke algebras. We study the representation theory of these algebras and construct several bases for them. We then show how we can define Markov traces on them, which we in turn use to construct invariants for framed and classical knots in the solid torus. Finally, we study the Markov trace with zero parameters on the cyclotomic Yokonuma-Hecke algebras and determine the Schur elements with respect to that trace.Comment: 37 page

Representation theory of the Yokonuma-Hecke algebra

We develop an inductive approach to the representation theory of the Yokonuma-Hecke algebra ${\rm Y}_{d,n}(q)$, based on the study of the spectrum of its Jucys-Murphy elements which are defined here. We give explicit formulas for the irreducible representations of ${\rm Y}_{d,n}(q)$ in terms of standard $d$-tableaux; we then use them to obtain a semisimplicity criterion. Finally, we prove the existence of a canonical symmetrising form on ${\rm Y}_{d,n}(q)$ and calculate the Schur elements with respect to that form.Comment: 28 page