Multiple Schramm-Loewner Evolutions (SLE) are conformally invariant random processes of several curves, whose construction by growth processes relies on partition functions -Möbius covariant solutions to a system of second order partial differential equations. In this article, we use a quantum group technique to construct a distinguished basis of solutions, which conjecturally correspond to the extremal points of the convex set of probability measures of multiple SLEs. A link formed by the pair {a, b} of indices will be denoted by [â, b]. A link pattern will be denoted byThe non-crossing condition, i.e., the planar property of the pair partition, can be expressed as theThe set of link patterns of N links is denoted by LP N , and we recall that the number of these is a Catalan number,1 These PDEs also arise in conformal field theory -see, e.g., [BBK05,FK15a]. The conformal weight h = h 1,2 appears in the Kac table, and the PDEs are the null-field equations associated with the degeneracy at level two of the boundary changing operators at the 2N marked boundary points. © ∈ LP 4 , we obtain α/[3, 4] = ¶ [1, 6], [2, 3], [4, 5]© ∈ LP 3 after relabeling the remaining endpoints.By convention, we include the empty link pattern ∅ ∈ LP 0 in the case N = 0. The set of link patterns of any possible size is denoted by LP = N ≥0 LP N , and for α ∈ LP N , we denote |α| = N.We seek pure geometries of multiple SLEs corresponding to each link pattern, so in view of the theorem above, the task is to construct their corresponding partition functions (Z α ) α∈LP . Each Z α must solve the system (1.2) -(1.3). This system, which is the same for all link patterns α of the same number of links, is supplemented by boundary conditions which depend on α. The quantum group methodIn this section, we present the quantum group method in the form it will be used for the solution of the problem (1.2) -(1.4). The method was developed more generally in [KP14].The relevant quantum group is a q-deformation U q (sl 2 ) of the Lie algebra sl 2 (C), and the deformation parameter q is related to κ by q = e i4π/κ . We assume that κ ∈ (0, 8) \ Q, so that q is not a root of unity. The method associates functions of n variables to vectors in a tensor product of n irreducible representations of this quantum group.

Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm-Loewner evolutions (SLE). In this article, we introduce a correspondence and establish its fundamental properties, which are used in the companion articles [JJK16, KP16] for explicitly solving two such problems. The correspondence associates Coulomb gas type integrals to vectors in a tensor product representation of a quantum group, a q-deformation of the Lie algebra sl 2 . We show that desired properties of the functions are guaranteed by natural representation theoretical properties of the vectors. denote the vector obtained by identifying v as a vector in an (n − 1)-fold tensor product representation. Then, as |x j+1 − x j | → 0, the function F[v] has the asymptotic behaviorwhere the constant B = B dj ,dj+1 d and the exponent ∆ = ∆ dj ,dj+1 d are explicit. The analogous statement holds also for F (x0) [v].In practice, this theorem is applied as follows. In specific problems, we are looking for particular solutions to systems of PDEs of conformal field theory. Typically, the sought solution has specific Möbius covariance properties and specific boundary conditions as the distance of some of its arguments tend to zero. By our correspondence, the task of finding a function with these properties is translated to the problem of finding a corresponding vector in the tensor product representation. The different parts (PDE), (COV), and (ASY) of the theorem state that a careful choice of the vector would ensure the desired properties of the function -even the most delicate boundary conditions for the function can be guaranteed by (ASY) if the vector has appropriate projections to certain subrepresentations. All of the requirements are explicit linear conditions on the vector living in a finite dimensional vector space, and we moreover have a variety of representation theoretical tools at our disposal to solve for such a vector. By outlining a few case studies in Sections 1.4 and 1.5, we exemplify how the correspondence thus allows us to translate the original, possibly rather complicated problem to an explicitly solvable one, and to eventually express the function of interest explicitly as a linear combination of integral form functions.Before example applications, we make a few further observations about the interpretation of the constructed correspondence, and comparisons to related research.

We find explicit formulas for the probabilities of general boundary visit events for planar loop-erased random walks, as well as connectivity events for branches in the uniform spanning tree. We show that both probabilities, when suitably renormalized, converge in the scaling limit to conformally covariant functions which satisfy partial differential equations of second and third order, as predicted by conformal field theory. The scaling limit connectivity probabilities also provide formulas for the pure partition functions of multiple SLEκ at κ = 2. e inê

This article concerns a generalization of the Temperley-Lieb algebra, important in applications to conformal field theory. We call this algebra the valenced Temperley-Lieb algebra. We prove salient facts concerning this algebra and its representation theory, which are both of independent interest and used in our subsequent work [FP18b + , FP18c + , FP18d + ], where we uniquely and explicitly characterize the monodromy invariant correlation functions of certain conformal field theories. Diagram algebrasA. Valenced tangles and link states B. Basic combinatorial properties C. Composition of valenced tangles and valenced link states D. Valenced Temperley-Lieb algebra 3. Standard modules A. Networks and the link state bilinear form B. Standard modules and their radicals C. Faithfulness of the link state representations 4. Trivalent link states and Gram matrix A. Definition of the trivalent link states B. Properties of the trivalent link states C. Determinant of the Gram matrix D. Recursion formulas for the Gram determinant 5. Radical of the link state bilinear form A. Radical at roots of unity B. Valenced radical at roots of unity C. Nondegenerate cases D. Totally degenerate cases 6. Semisimplicity of the valenced Temperley-Lieb algebra A. Perspective from general representation theory of algebras B. Simple modules of the valenced Temperley-Lieb algebra C. Semisimplicity of the valenced Temperley-Lieb algebra Appendix A. Diagram simplifications Appendix B. Jones-Wenzl algebra Appendix C. Trivalent link states at roots of unity