Three-dimensional reformulation of the transfer-matrix inclusion–exclusion formula

Develop a reformulation of the alternating sum over the fundamental-representation transfer matrices for the inhomogeneous multispecies t-PushTASEP, potentially using three-dimensional integrability.

Background

The paper expresses the Markov matrix of the inhomogeneous n-species t-PushTASEP as an alternating sum of commuting transfer matrices associated with all fundamental representations of the quantum affine algebra U_t(\widehat{\mathfrak{sl}}_{n+1}). The authors interpret this alternating sum as an inclusion–exclusion mechanism that retains the permitted sequential particle transitions while canceling forbidden channels.

Because the fundamental representations contribute a total of 2{n+1} terms, the authors suggest that the construction may admit a further reformulation through three-dimensional integrability. Such a reformulation is not developed in the paper and is explicitly left for future investigation.

References

The summation over the fundamental representations corresponds to the dimension \binom{n+1}{0} + \binom{n+1}{1} + \cdots + \binom{n+1}{n+1} = 2{n+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, Introduction, p. 2