Weight filtration of Hurwitz spaces and quantum shuffle algebras

Abstract

We prove an equivalence between filtrations of primitive bialgebras and filtrations of factorizable perverse sheaves, generalizing the results obtained by Kapranov-Schechtman. Under this equivalence, we find that the word length filtration of quantum shuffle algebras as defined in Ellenberg-Tran-Westerland corresponds to the codimension filtration of factorizable perverse sheaves. Furthermore, we find that the geometric weight filtration of factorizable perverse sheaves corresponds to a filtration on quantum shuffle algebras which has not been previously defined in the literature, and we call this the algebraic weight filtration. To apply this to Hurwitz spaces, we prove a comparison theorem between the weight filtrations for Hurwitz spaces over $\mathbb F_p$ and $\mathbb C$, generalizing the comparison theorem of Ellenberg-Venkatesh-Westerland. This allows us to determine the cohomological weights for Hurwitz spaces explicitly using the algebraic weight filtration of the corresponding quantum shuffle algebra. As a consequence, we find that most weights of Hurwitz spaces are smaller than expected from cohomological degree, and we prove explicit nontrivial upper bounds for weights in some cases, such as when $G=S_3$ and $c$ is the conjugacy class of transpositions.