Transition Matrices between Shifted $t$-Schur Bases and Cyclotomic Schur $Q$-Positivity

Abstract

For a strict partition $\lambda$, let ${\mathcal{Q}}_{\lambda }(X;t)=Q_{\lambda }[X-tX]$ be the shifted $t$-Schur function arising from the modified Greaves--Jing--Zhu operator on the odd power-sum ring. We study transition matrices between the shifted bases with parameters $t$ and $s$. The relative scaling operator is diagonal in the odd power-sum basis, leading to explicit spectral data, determinant and trace formulas, weighted symmetry, a spin-character formula, and a transition Cauchy identity. For the cyclotomic specialization $C_{\lambda \mu }^{[M]}(t)=C_{\lambda \mu }(t^M,t)$, the relative operator becomes plethystic substitution by $1+t+\cdots +t^{M-1}$. We prove Schur $Q$-positivity and reciprocity, derive factorization and root-of-unity rank formulas, and give an exact computation method. For $M=2$, all one-row transitions are computed explicitly, and the nonzero coefficients are unimodal.