The type-A Hecke Specht module and its cell form

Let $R$ be a commutative ring and $q\in R^{\times }$. With the Murphy cellular basis $(\mathcal{M},\mathrm{\Lambda })$ of $H=H_{R,q}(S_n)$ (The Murphy tableau elements form a cellular basis, Cellular bases and cell modules) and the canonical tableau $t^{\lambda }$ of shape $\lambda ⊢n$ (Murphy elements and Murphy tableau elements), the Murphy cell module $S_{\mathrm{M}}^{\lambda }$ is the right cell module $C^{\lambda }$ of the Murphy basis: it is the free $R$-module with basis $\{m_t:=m_{t^{\lambda }t}:t\in \mathrm{S}\mathrm{t}\mathrm{d}(\lambda )\}$ - the classes of the Murphy elements whose first index is the canonical tableau - and $H$ acts by $m_{t}h={\sum }_v{r}_v{m}_v$ whenever $m_{t^{\lambda }t}h\equiv {\sum }_v{r}_v{m}_{t^{\lambda }v}(\mathrm{m}\mathrm{o}\mathrm{d}{H}^{⊳\lambda })$. (The cellular action rule transforms the second index and fixes the first, so the cell module is defined using the fixed-first-index family above; the diagonal span in $H$ need not be stable under multiplication. For $\lambda =(n)$ it is stable, since $m_{\lambda }{T}_i=q{m}_{\lambda }$.) Its cell form is the unique bilinear form $⟨,{⟩}_{\lambda }:S_{\mathrm{M}}^{\lambda }\times S_{\mathrm{M}}^{\lambda }\to R$ with $m_{us}{m}_{tv}\equiv ⟨m_s,m_t{⟩}_{\lambda }{m}_{uv}(\mathrm{m}\mathrm{o}\mathrm{d}{H}^{⊳\lambda })$ for all $s,t,u,v\in \mathrm{S}\mathrm{t}\mathrm{d}(\lambda )$; it is symmetric and associative and $\mathrm{r}\mathrm{a}\mathrm{d}{S}_{\mathrm{M}}^{\lambda }$ is a submodule, so $D_{\mathrm{M}}^{\lambda }:=S_{\mathrm{M}}^{\lambda }\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}{S}_{\mathrm{M}}^{\lambda }$ is a right $H$-module (The cell form is associative and its radical is a submodule).

Convention (Dipper-James signed cellular normalization). Let $\theta :H\to H$ be the involutive algebra automorphism defined by $\theta (T_i)=-q{T}_i^{-1}=(q-1)-T_i$ (The Hecke sign automorphism, pullback and contragredient dual). For a right module $M$, write $M^{\theta }$ for the same $R$-module with action $x\cdot h=x\theta (h)$. The page's ordinary Specht normalization is the signed conjugate Murphy construction $S^{\lambda }:=(S_{\mathrm{M}}^{{\lambda }^{\mathrm{\prime }}}{)}^{\theta },$ where ${\lambda }^{\mathrm{\prime }}$ is the conjugate partition (Partitions, English diagrams, and conjugation). Its form is the form of $S_{\mathrm{M}}^{{\lambda }^{\mathrm{\prime }}}$ on the same underlying module; because $\theta$ commutes with $\ast$ this is associative for the twisted action (The Hecke sign automorphism, pullback and contragredient dual). Over a field put $D^{\lambda }:=S^{\lambda }\mathrm{/}{\mathrm{rad}}_{\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}}{S}^{\lambda }=(D_{\mathrm{M}}^{{\lambda }^{\mathrm{\prime }}}{)}^{\theta }\mathrm{.}$ Thus $S_{\mathrm{M}}^{{\lambda }^{\mathrm{\prime }}}=(S^{\lambda }{)}^{\theta }$ and $D_{\mathrm{M}}^{{\lambda }^{\mathrm{\prime }}}=(D^{\lambda }{)}^{\theta }$; conjugation alone does not identify these modules (Murphy cells are the same-shape contragredient Specht duals, Ordinary Specht modules are the signed conjugate Murphy cells). At $q=1$, $\theta (\sigma )=\mathrm{sgn}(\sigma )\sigma$, so these identities contain tensoring with sign; in characteristic $2$ the sign twist becomes trivial at $q=1$ (Polytabloid specialization and the exact signed multiplicity dictionary). The nonzero quotients in this signed normalization have $e$-regular labels by The simple type-A Hecke modules are indexed by e-regular partitions ↗: that theorem proves the nonzero Murphy quotient criterion in $e$-restricted labels, and conjugation makes these precisely $e$-regular labels (e-regular and e-restricted partitions). No generic root-of-unity Hecke claim is inferred from the published prime-characteristic group-only sign dictionary, and no semisimplicity or choice principle is assumed.