Tensor powers of vector representation of $U_q({\mathfrak{sl}}_2)$ at even roots of unity

Abstract

We study the decomposition of tensor powers of two dimensional irreducible representations of quantum ${\mathfrak{sl}}_2$ 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 $N\to \infty$. We also describe the limit of character and Plancherel measures when $N\to \infty$. We consider both $U_q({\mathfrak{sl}}_2)$ with divided powers and the small quantum $s{l}_2$.

43 pages