Weight filtration of Hurwitz spaces and quantum shuffle algebras

TL;DR

The work establishes a powerful link between filtrations on primitive bialgebras and factorizable perverse sheaves via an extended Kapranov–Schechtman framework, enabling a transport of filtrations and their associated spectral sequences across algebra and geometry. It identifies the word-length filtration on quantum shuffle algebras with the codimension filtration on factorizable perverse sheaves, and introduces an algebraic weight filtration that corresponds to the geometric weight filtration in the mixed Hodge-theoretic setting. By relating Hurwitz-space cohomology to Ext–Tor computations in quantum shuffle algebras and proving a comparison theorem between weight filtrations over finite fields and the complex numbers, the paper provides explicit weight descriptions for Hurwitz spaces, including concrete bounds in the S3-transpositions case. The results offer a detailed, computationally usable picture of how Frobenius weights distribute in middle cohomology and show that many weights are smaller than naive degree-based expectations, with clear implications for arithmetic statistics in function-field settings. Overall, the framework yields both conceptual unification and quantitative tools for studying weights and cohomology of Hurwitz spaces through braided shuffle algebras and perverse-sheaf formalism.

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.

Figures (6)

• Figure 1: Dimensions of shifted word length filtration for $V=k$ and $q$ third root of unity.
• Figure 2: Dimensions of shifted word length filtration for the usual shuffle algebra with $\dim(V)=3$.
• Figure 3: Dimensions of shifted word length filtration for the $S_3$ transposition rack with cocycle $x=1$ on the left and $x=-1$ on the right.
• Figure 4: $E_0$ page of spectral sequence.
• Figure 5: Dimensions of weight filtration for $V=k$ and $q$ third root of unity.

Theorems & Definitions (55)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: KS20
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Definition 2.2
• Example 2.3