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 sl_2 . The construction both highlights the representation theoretic origin of conformal block functions and explains the appearance of q -combinatorial formulas.