Closed higher-rank multilinear reflection sectors

Determine which $(n,r+2-n)$-type contractions of polylogarithmic and Hurwitz zeta vectors admit systematically closed reflection relations for multiple elliptic Gamma functions of rank $r$.

Background

The natural Mellin kernel for the rank-rr multiple elliptic Gamma function has one polylogarithmic vector and r+1r+1 Hurwitz zeta vectors. The authors note that the underlying linear reflection relations permit more general contractions containing different numbers of the two types of vectors. Classifying the nontrivial contractions that close under reflection is explicitly identified as an unresolved broader problem.

References

A natural broader class consists of $(n,r+2-n)$ contractions containing $n$ polylogarithmic vectors and $(r+2-n)$ Hurwitz zeta vectors. An interesting question is whether there exist nontrivial subclasses for which the reflection relations close systematically, and among them, which admit sufficiently simple inverse Mellin realizations.

— Mellin space reflections, modularity of elliptic Gamma functions and beyond  (2608.24083 - Lei et al., 25 Aug 2026) in Section 4, Discussion and outlook

An important question is then which of the resulting Mellin space reflection structures admit well-defined realizations in the modular variables and whether any of them can be interpreted as partition functions of quantum field theories.

— Mellin space reflections, modularity of elliptic Gamma functions and beyond  (2608.24083 - Lei et al., 25 Aug 2026) in Section 4, Discussion and outlook

It is therefore natural to ask whether this multiple sine representation admits an equally transparent interpretation in terms of Mellin space reflection identities.

— Mellin space reflections, modularity of elliptic Gamma functions and beyond  (2608.24083 - Lei et al., 25 Aug 2026) in Appendix, Section “Multiple elliptic Gamma functions and higher rank structures”