Theta transports the Murphy cellular structure to the signed cell modules

The $\theta$-image of the Murphy basis. Since $\theta$ is an $R$-algebra automorphism of $H$ and $\{m_{st}^{\mu }\}$ is an $R$-basis by [F2], the family $\{\theta (m_{st}^{\mu })\}$ is again an $R$-basis, and $\theta (m_{st})=\theta (T_{d(s)}^{\ast })\theta (x_{\lambda })\theta (T_{d(t)})$ by multiplicativity, with $\theta (x_{\lambda })=q^{\mathrm{\ell }(w_0(S_{\lambda }))}{y}_{\lambda }$ by [F1]; also $\theta (m_{st}{)}^{\ast }=\theta (m_{st}^{\ast })=\theta (m_{ts})$ because $\theta$ commutes with $\ast$.

The transported ideals and cellular structure. For a partition $\rho$, the ideal $I^{⊵\rho }$ is spanned by the $m_{uv}^{\mu }$ with $\mu ⊵\rho$ by [F3], so its image $\theta (I^{⊵\rho })$ is spanned by the corresponding $\theta (m_{uv}^{\mu })$; as the image of a two-sided ideal under an automorphism it is a two-sided ideal, and likewise for $\theta (I^{⊳\rho })$. Moreover $\theta (I^{⊵\rho })\subseteq \theta (I^{⊵{\rho }^{\mathrm{\prime }}})$ whenever $\rho ⊵{\rho }^{\mathrm{\prime }}$, and the transported cell law holds: for $h\in H$, $\theta (m_{st})\theta (h)=\theta (m_{st}h)\equiv {\sum }_v{r}_v\theta (m_{sv})(\mathrm{m}\mathrm{o}\mathrm{d}\theta (I^{⊳\rho }))$ with the same coefficients $r_v$ as in [F2], independent of the first index. Hence $(\{\theta (m_{st})\},⊵)$ is a cellular basis with the same dominance poset and the $\theta$-image filtration.

The ideal subspaces move. Take $n=2$ and $\lambda =(2)$. Here $x_{(2)}=1+T_1$ spans $I^{⊵(2)}=R(1+T_1)$ (the only Murphy shape dominating $(2)$ is $(2)$ itself, with a single standard tableau), while $\theta (x_{(2)})=1+\theta (T_1)=1+(q-1)-T_1=q-T_1$. A relation $q-T_1=c(1+T_1)$ with $c\in R$ would compare the coefficients of $1$ and of $T_1$ in the standard basis and force $c=q$ and $c=-1$, hence $q=-1$, which fails over the generic Laurent ring (and over any $R$ with $q\mathrm{\ne }-1$). Conversely, at $q=-1$ one has $q-T_1=-(1+T_1)$, so these ideals coincide. Thus they differ in general, as claimed, without asserting that they differ at every parameter. By step 1.2 both are two-sided ideals with the same shape index, so the poset is preserved but the ideal subspaces are not.

The module isomorphism. Put $K:=M(\lambda )\cap I^{⊳\lambda }$, so that $C_{\mathrm{M}}(\lambda )=M(\lambda )\mathrm{/}K$ by [F2]. The assignment $m+K\mapsto \theta (m)+\theta (K)$ on representatives $m\in M(\lambda )$ is well defined and bijective, since $\theta$ is a bijection carrying $K$ onto $\theta (M(\lambda ))\cap \theta (I^{⊳\lambda })$; it is $H$-linear because for $m\in M(\lambda )$ and $h\in H$ one has $\theta (m\theta (h))=\theta (m){\theta }^2(h)=\theta (m)h$ using ${\theta }^2=\mathrm{i}\mathrm{d}$ by [F1]. Its target is by construction the quotient of the transported row by the transported higher ideal, i.e. the cell module of the transported basis of step 1.2. Hence $C_{\mathrm{M}}(\lambda {)}^{\theta }$ (the same underlying module with action $m\cdot h=m\theta (h)$) is isomorphic to that transported cell row, is free of rank $f^{\lambda }$ with basis $\{\theta (m_t)\}$ and satisfies the transported cell law.

The transported form. Define $G_{\lambda }^{\theta }$ on the underlying module of $C_{\mathrm{M}}(\lambda {)}^{\theta }$ by $G_{\lambda }^{\theta }(x,y):=G_{\lambda }(x,y)$. Symmetry is immediate from [F2], and associativity for the twisted action follows from the ordinary associativity: $G_{\lambda }^{\theta }(x\cdot h,y)=G_{\lambda }(x\theta (h),y)=G_{\lambda }(x,y\theta (h{)}^{\ast })=G_{\lambda }(x,y\theta (h^{\ast }))=G_{\lambda }^{\theta }(x,y\cdot h^{\ast })$, using that $\theta$ commutes with $\ast$ and ${\theta }^2=\mathrm{i}\mathrm{d}$ by [F1]. Its radical is $\{x:G_{\lambda }(x,y)=0\mathrm{\forall }y\}=\mathrm{rad}{G}_{\lambda }$, which is a submodule for the twisted action as well. Under the identification $x\mapsto \theta (x)$ of step 2.2 the form becomes $G_{\lambda }^{\theta }(\theta u,\theta v)=G_{\lambda }(u,v)$ for $u,v\in C_{\mathrm{M}}(\lambda )$, which is exactly the Murphy form of the transported basis, because multiplying two transported basis elements gives $\theta (m_{us})\theta (m_{tv})=\theta (m_{us}{m}_{tv})\equiv G_{\lambda }(m_s,m_t)\theta (m_{uv})(\mathrm{m}\mathrm{o}\mathrm{d}\theta (I^{⊳\lambda }))$ by [F2]; thus $G_{\lambda }^{\theta }$ is the transported form of clause 3 and $x\mapsto \theta (x)$ is an isometry.

Base change and the conjugate clause. Every element, ideal, quotient and form above is defined by formulas with coefficients in $\mathbb{Z}[q^{\pm 1}]$ and the identities are identities of such formulas, so they specialize along any ring map $A=\mathbb{Z}[v^{\pm 1}]\to R$, $v\mapsto q$, with $q\in R^{\times }$, by [F4]; in particular the transported basis stays a basis and the cell law and form identities are preserved. Applying clauses 1-3 to the conjugate partition ${\lambda }^{\mathrm{\prime }}$, which satisfies $f^{{\lambda }^{\mathrm{\prime }}}=f^{\lambda }$ by [F2], gives clause 4. Only the presentation of $H$, the automorphism $\theta$, the Murphy cellular structure and its form are used, so no choice principle is used.