The Computational Complexity to Evaluate Representations of General Linear Groups

Abstract

We describe a fast algorithm to evaluate irreducible matrix representations of complex general linear groups ${\rm GL}_{m}$ with respect to a symmetry adapted basis (Gelfand--Tsetlin basis). This is complemented by a lower bound, which shows that our algorithm is optimal up to a factor $m^2$ with regard to nonscalar complexity. Our algorithm can be used for the fast evaluation of special functions: for instance, we obtain an $O(\ell\log\ell)$ algorithm to evaluate all associated Legendre functions of degree $\ell$. As a further application we obtain an algorithm to evaluate immanants, which is faster than previous algorithms due to Hartmann and Barvinok.