A Shifted $t$-Schur Weight from the Modified Odd Operator

Abstract

We study the one-time weight on strict partitions obtained from the modified odd Greaves--Jing--Zhu operator. The shifted $t$-Schur functions generated by this operator are obtained from the classical Schur $Q$-functions by the plethystic substitution $X\mapsto X-tX$. Thus the corresponding weight \[ λ\longmapsto \mathcal Q_λ(X;t)P_λ(Y) \] is a shifted Schur weight with a virtual first alphabet. We give its normalization, its Pfaffian correlation kernel, its Fredholm Pfaffian for the largest part, and its size cumulants. For $t=-q$ with $q\ge 0$ the virtual alphabet becomes the positive alphabet $X+qX$, giving a genuine probability measure. This positive specialization is the one-time marginal of the two-color lift considered in a companion note.