In this notes we shall describe the relation of a certain class of simple random curves arising in 2D statistical mechanics models in the scaling limit, which can be described dynamically by Stochastic Lœwner Evolutions (SLE), and the equivalent Renormalisation Group (RG) theoretic interpretation in Conformal Field Theory, as a fixed point of the RG flow. Further, we shall recall the relation o...

In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker-Akhiezer function Ψ is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators....

The Harish-Chandra homomorphism is due to [Harish-Chandra 1951]. Attention to universal modules with highest weights is in ]Harish-Chandra 1951], [Cartier 1955], as well as [Verma 1968], [BernsteinGelfand-Gelfand 1971a], [Bernstein-Gelfand-Gelfand 1971b], [Bernstein-Gelfand-Gelfand 1975]. See also [Jantzen 1979]. [1] We treat sl(2) in as simple a style as possible, to highlight ideas. Then sl(3...

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators Ln. We calculate explicitly the matrix elements of Ln with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack...