Wall-sum evaluations for exceptional root systems E6, E7, and E8

Evaluate the wall sums corresponding to the second-pole residues of the normalized Witten zeta functions for the exceptional root systems E6, E7, and E8.

Background

The paper establishes that, for every irreducible reduced crystallographic root system of rank at least two, the first distinct pole below the leading pole occurs at q2=(r-1)/(N-1), is simple, and has a residue expressed as a positive sum of marked projective wall periods. Exact evaluations of these wall sums are obtained for types F4 and D5 using chamber-orbit relations together with Dixon and Selberg integral evaluations. The corresponding wall sums for E6, E7, and E8 are explicitly identified as remaining problems, with no evaluation supplied in the paper.

References

When n is positive and odd, the proof above does not provide further insight about the value \xi_\Phi(-n)\in \mathbb{Q}. It is in general non-zero. Proposition \ref{prop:I-values} gives a consice but inefficient formula for computing it: the formula involves a sum of N! residues, each of which is itself difficult to evaluate. Even for Witten zeta functions, the efficient computation of \xi_\Phi(-n) remains largely open and is currently understood only for root systems of rank two .

— Combinatorics of hyperplane arrangements and Witten zeta function at the origin  (2609.20740 - Au et al., 17 Sep 2026) in Remark (b) following the proof of Theorem \ref{thm:witten_0_der}, Section “Witten zeta function”

Two natural problems remain. The first is to evaluate the corresponding wall sums in types E6, E7, and E8. The second is to determine which weighted restrictions admit a chamber-orbit relation strong enough to reduce all periods to classical beta, Dixon, or Selberg integrals.

— The second pole of Witten zeta functions and exact evaluations in types F4 and D5  (2608.16363 - Matuzas, 17 Aug 2026) in Section 9, Conclusion