A Two-Color Lift of the Shifted $t$-Schur Measure

At the specialization $t=-q$, $q\ge 0$, the shifted $t$-Schur function associated with the modified odd Greaves–Jing–Zhu operator is $Q_{\lambda }[X+qX]$. Instead of merging the two alphabets $X$ and $qX$, we insert an intermediate strict partition between the two corresponding half-vertex operators. This gives a two-color lift of the shifted Schur measure on pairs $\mu \subseteq \lambda$ with weight $$Q_{\mu }(qX)Q_{\lambda \mathrm{/}\mu }(X)P_{\lambda }(Y)\mathrm{.}$$ We compute the normalization and both marginals, identify an explicit Markov transition kernel, prove a semigroup property, and show that the two color volumes $\mathrm{\mid }\mu \mathrm{\mid }$ and $\mathrm{\mid }\lambda \mathrm{\mid }-\mathrm{\mid }\mu \mathrm{\mid }$ are independent. We also realize the model as a two-time shifted Schur process and write its Pfaffian correlation kernel in Vuletić’s convention. Rectangular specializations give closed formulas and Gaussian limits for the color volumes.