Determinant condition for Taylor–Wiles Galois lifts

Let $Q$ be an auxiliary Taylor–Wiles set, let $\overline{\rho}_{\mathfrak m}$ be the residual representation, let $\mu$ be the specified character, and let $r_{{\mathfrak m},Q}$ be the determinant-$\mu$ twist of the conjectural lift. Define

$${\mathcal S}_Q=(\overline{\rho}_{\mathfrak m},\mu,\{\Lambda_v\}_{v\in S}\cup\{{\mathcal O}[\Delta_v]\}_{v\in Q},\{{\mathcal D}_v^\triangle\}_{v\in S_p}\cup\{{\mathcal D}_v^\mathrm{TW}\}_{v\in Q})$$

with $\Lambda_v={\mathcal O}\llbracket{\mathcal O}_{F_v}^{\times}(p)^{n-1}\rrbracket$ for $v\in S_p$, and with $\Lambda_S=\Lambda$ and $\Lambda_{S\cup Q}=\Lambda[\Delta_Q]$. Determinant-type conjecture. The lifting $r_{{\mathfrak m},Q}$ is of type ${\mathcal S}_Q$. This specifies the local deformation conditions expected of the Taylor–Wiles lift and is part of the deformation-theoretic input for patching; the source gives no resolution status.