Conformal blocks, $q$-combinatorics, and quantum group symmetry

In this article, we find a $q$-analogue for Fomin's formulas. The original Fomin's formulas relate determinants of random walk excursion kernels to loop-erased random walk partition functions, and our formulas analogously relate conformal block functions of conformal field theories to pure partition functions of multiple SLE random curves. We also provide a construction of the conformal block functions by a method based on a quantum group, the $q$-deformation of $\mathfrak{sl}_2$. The construction both highlights the representation theoretic origin of conformal block functions and explains the appearance of $q$-combinatorial formulas.
Comment: 24 pages, 7 figures. v3: minor improvements. Accepted for publication in Annales de l'Institut Henri Poincar\'e D