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

Abstract

At the specialization $t=-q$, $q\geq0$, the shifted $t$-Schur function associated with the modified odd Greaves--Jing--Zhu operator is $Q_λ[X+qX]$. 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 $μ\subseteqλ$ with weight \[ Q_μ(qX)Q_{λ/μ}(X)P_λ(Y). \] We compute the normalization and both marginals, identify an explicit Markov transition kernel, prove a semigroup property, and show that the two color volumes $|μ|$ and $|λ|-|μ|$ 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.