Opposite-parabolic Ringel duality on a finite ideal

The two matrices. Finite Levi Weyl cancellation with $W_J=\{e,s_1\}$ and $W_{\bar{J}}=\{e,s_2\}$, together with all ordinary $S_3$ Kazhdan-Lusztig values at $1$ being $1$, gives the target standard/simple matrix $D$ and the opposite matrix $N$ independently on the displayed labels; subtracting the scalar $(2,2,2)$ reduces the comparison to the actual regular $S_3$ example of the block, and the target dominant labels appear in the reverse tableau order while the reverse-side dominant labels appear in the displayed order.

The semiregular shift and the functor values. Applying the explicit block-offset shift of [F1] to each of the three highest weights verifies the label computation: the shifted coordinates return the original column readings, so $F\bar{M}(A)=\mathrm{\nabla }(A)$ and the central-character blocks match. Opposite BGG then gives the entrywise tilting-standard matrix $N^t$ through $F\bar{P}(A)=T(A)$.

The proper ideal and the quotient. Restricting to $\mathrm{\Gamma }=\{A_0,A_1\}$, the invariant tiltings have rows $\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right)$, while the projective-corner $\mathrm{Hom}$ quotient of [F3] has standard/simple rows $\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$; this is the opposite highest-weight Serre quotient by the complementary label, not the same opposite down-set subcategory. Simples and canonical classes are unchanged ambient ones, while projectives and quasi-canonical classes use the standard-flag truncation of the ambient projectives.

Conclusion. In this finite ${\mathfrak{g}\mathfrak{l}}_3$ example the semiregular label shift, the functor values $F\bar{M}=\mathrm{\nabla }$ and $F\bar{P}=T$, the opposite BGG tilting matrix $N^t$, the distinction from the target projective rows $D^t$, and the two-label ideal quotient data are all verified; the three Fock families correspond to simples, projectives and tiltings exactly as stated.