Three-dimensional reformulation of the transfer-matrix superposition

Develop a reformulation of the alternating sum over transfer matrices associated with all fundamental representations of the quantum affine algebra U_t(fsl_{n+1}) through three-dimensional integrability.

Background

The paper expresses the inhomogeneous n-species t-PushTASEP Markov matrix as an alternating sum of commuting transfer matrices whose auxiliary spaces are the fundamental antisymmetric representations of U_t(fsl_{n+1}). The authors observe that the sum over these representations may indicate a more structural reformulation.

They specifically point to three-dimensional integrability as a possible framework for such a reformulation, but do not derive one. The problem is therefore to identify and establish that reformulation while preserving the inclusion-exclusion structure that produces the PushTASEP dynamics.

References

The summation over the fundamental representations corresponds to the dimension n1 0˘ n1 1˘ ¨ ¨ ¨ n1n1˘ “ 2n`1. It indicates a further reformulation, possibly through three-dimensional integrability (cf. [K22, Chap. 18]), which is left, however, as a problem for future investigation.

— Multispecies inhomogeneous $t$-PushTASEP from antisymmetric fusion  (2503.00829 - Ayyer et al., 2 Mar 2025) in Section 1, page 2