Tensor powers of vector representation of U_q(𝔰𝔩_2) at even roots of unity

We study the decomposition of tensor powers of two dimensional irreducible representations of quantum 𝔰𝔩_2 at even roots of unity into direct sums of tilting modules. We derive a combinatorial formula for multiplicity of tilting modules in the N-th tensor power of two dimensional irreducible representations, interpret it in terms of lattice paths and find its asymptotic behavior when N→∞. We also describe the limit of character and Plancherel measures when N→∞. We consider both U_q(𝔰𝔩_2) with divided powers and the small quantum sl_2.