Two-parameter Littlewood identities and half-space Yang--Baxter random fields for stable spin Hall--Littlewood symmetric functions
We prove a two-parameter skew Littlewood identity for stable spin Hall--Littlewood symmetric functions, generalizing Warnaar's identity. This identity yields a half-space extension of the Yang--Baxter random field of Bufetov and Petrov. Using their stochastic Yang--Baxter move together with the skew Littlewood identity, we construct explicit bulk and boundary sampling rules and characterize the two boundary regimes in which these rules admit autonomous projections onto the first $R$ column lengths for every $R\ge1$. In these regimes, the partition-length fields agree, after explicit coordinate and parameter changes, with the half-space stochastic six-vertex model of Barraquand, Borodin, Corwin and Wheeler and a subfamily of He's model. The two-column projections retain spin dependence and converge, under the respective continuous-time scalings, to the same two-layer exclusion process whose first layer is open ASEP. We also prove that the joint distributions of partition lengths in ascending processes are independent of spin throughout the nonnegative parameter range. Using this spin independence and the distributional comparisons with half-space six-vertex heights, we transfer He's asymptotic results to diagonal partition lengths, obtaining Tracy--Widom GSE and GOE limits, Gaussian limits, and a GSE--GOE crossover under boundary tuning. The GOE limit also holds at the even-column specialization. Finally, the first-layer particle count of the continuous-time process has GOE fluctuations.