[proofplan]
We use the standard basis theorem to identify the Hecke algebra as a flat deformation of the group algebra and to control specialization at $q=1$. The modules $S_{R,\lambda}$ are the integral type A Hecke Specht modules, obtained from the Dipper-James-Murphy construction with left-module conventions fixed by the standard anti-involution. Their tableau bases prove finite freeness and specialize to the ordinary integral Specht modules $S_\lambda$. Finally, over $K=\mathbb Q(q)$, the Dipper-James-Murphy generic irreducibility theorem, together with Tits' deformation principle, identifies these particular generic Specht modules as the complete simple spectrum of the generic Hecke algebra.
[/proofplan]
[step:Identify the specialization of the Hecke algebra with the group algebra]
For $1\le i\le n-1$, let $s_i=(i,i+1)\in S_n$. By the standard basis theorem for type A Hecke algebras [citetheorem:8464], for each $w\in S_n$ the element $T_w\in \mathcal H_R(S_n)$ is well-defined by any reduced expression for $w$, and the set
\begin{align*}
\{T_w:w\in S_n\}
\end{align*}
is an $R$-basis of $\mathcal H_R(S_n)$.
Let
\begin{align*}
\varepsilon:R\to\mathbb Z
\end{align*}
be the ring homomorphism with $\varepsilon(q)=1$, and define
\begin{align*}
\mathcal H_{\mathbb Z}:=\mathcal H_R(S_n)\otimes_{R,\varepsilon}\mathbb Z.
\end{align*}
In $\mathcal H_{\mathbb Z}$, the quadratic relation becomes
\begin{align*}
(T_i-1)(T_i+1)=0,
\end{align*}
hence
\begin{align*}
T_i^2=1
\end{align*}
for every $i$. Together with the braid relations, these are exactly the Coxeter relations for $S_n$. Therefore the assignment $T_i\otimes 1\mapsto s_i$ defines a surjective $\mathbb Z$-algebra homomorphism
\begin{align*}
\Phi:\mathcal H_{\mathbb Z}\to \mathbb Z[S_n].
\end{align*}
The basis theorem gives that $\{T_w\otimes 1:w\in S_n\}$ is a $\mathbb Z$-basis of $\mathcal H_{\mathbb Z}$, while $\{w:w\in S_n\}$ is a $\mathbb Z$-basis of $\mathbb Z[S_n]$. Since $\Phi(T_w\otimes 1)=w$ for every $w\in S_n$, the map $\Phi$ sends one basis bijectively to the other. Thus $\Phi$ is an isomorphism of $\mathbb Z$-algebras.
[/step]
[step:Construct the integral Hecke Specht modules with left-module conventions]
We invoke the Dipper-James-Murphy construction of integral Specht modules for the type A Hecke algebra with quadratic convention
\begin{align*}
(T_i-q)(T_i+1)=0.
\end{align*}
For each partition $\lambda\vdash n$, let $\operatorname{SYT}(\lambda)$ denote the finite set of standard Young tableaux of shape $\lambda$. This construction gives an $R$-module $S_{R,\lambda}$ with an $\mathcal H_R(S_n)$-action and an $R$-basis indexed by $\operatorname{SYT}(\lambda)$.
More explicitly, the usual Murphy construction first produces a right Specht module from the permutation module attached to row tabloids by imposing the Hecke analogues of the classical column antisymmetry and Garnir straightening relations. To obtain the left module used here, use the standard $R$-linear anti-involution
\begin{align*}
\iota:\mathcal H_R(S_n)\to \mathcal H_R(S_n)
\end{align*}
defined on the standard basis by
\begin{align*}
\iota(T_w)=T_{w^{-1}} \quad \text{for } w\in S_n.
\end{align*}
If $M_{R,\lambda}$ denotes the standard right Hecke Specht module, define the corresponding left module $S_{R,\lambda}$ on the same underlying $R$-module by
\begin{align*}
h\cdot m := m\,\iota(h)
\end{align*}
for $h\in\mathcal H_R(S_n)$ and $m\in M_{R,\lambda}$. The anti-involution identity $\iota(h_1h_2)=\iota(h_2)\iota(h_1)$ verifies the left-module law: $h_1\cdot(h_2\cdot m)=h_1\cdot(m\,\iota(h_2))=m\,\iota(h_2)\iota(h_1)=m\,\iota(h_1h_2)=(h_1h_2)\cdot m$. Also $1_{\mathcal H_R(S_n)}\cdot m=m$ because $\iota(1_{\mathcal H_R(S_n)})=1_{\mathcal H_R(S_n)}$.
The standard tableau basis theorem in the Murphy construction says that the standard polytabloids of shape $\lambda$ form an $R$-basis. Hence $S_{R,\lambda}$ is finite free over $R$, with rank
\begin{align*}
f_\lambda:=|\operatorname{SYT}(\lambda)|.
\end{align*}
[guided]
The construction has one convention issue: many references build Hecke Specht modules as right modules, while the theorem asks for left modules. We fix this without changing the representation by using the standard anti-involution. The type A Hecke algebra has an $R$-linear anti-involution
\begin{align*}
\iota:\mathcal H_R(S_n)\to \mathcal H_R(S_n)
\end{align*}
specified on basis elements by
\begin{align*}
\iota(T_w)=T_{w^{-1}}.
\end{align*}
This is compatible with the braid and quadratic relations because inversion reverses reduced words and preserves each simple reflection.
Let $M_{R,\lambda}$ be the right Specht module supplied by the Dipper-James-Murphy construction. We define a left action on the same $R$-module by
\begin{align*}
h\cdot m:=m\,\iota(h)
\end{align*}
for every $h\in\mathcal H_R(S_n)$ and every $m\in M_{R,\lambda}$. The left-module axiom follows from the anti-involution property. For $h_1,h_2\in\mathcal H_R(S_n)$ and $m\in M_{R,\lambda}$,
\begin{align*}
h_1\cdot(h_2\cdot m)=h_1\cdot(m\,\iota(h_2)).
\end{align*}
Applying the definition of the left action again gives
\begin{align*}
h_1\cdot(h_2\cdot m)=m\,\iota(h_2)\iota(h_1).
\end{align*}
Since $\iota$ reverses products,
\begin{align*}
m\,\iota(h_2)\iota(h_1)=m\,\iota(h_1h_2)=(h_1h_2)\cdot m.
\end{align*}
The identity element acts as the identity because $\iota(1_{\mathcal H_R(S_n)})=1_{\mathcal H_R(S_n)}$.
Let $\operatorname{SYT}(\lambda)$ denote the finite set of standard Young tableaux of shape $\lambda$. The Dipper-James-Murphy standard basis theorem for Specht modules then gives a basis indexed by $\operatorname{SYT}(\lambda)$. Therefore $S_{R,\lambda}$ is not merely finitely generated; it is finite free over $R$, with basis cardinality
\begin{align*}
f_\lambda=|\operatorname{SYT}(\lambda)|.
\end{align*}
This freeness is the key integral point: after specialization or extension of scalars, the rank is preserved, so the module has the expected classical dimension.
[/guided]
[/step]
[step:Specialize the Hecke Specht relations to the ordinary Specht relations]
Let
\begin{align*}
S_{\mathbb Z,\lambda}:=S_{R,\lambda}\otimes_{R,\varepsilon}\mathbb Z.
\end{align*}
The module $S_{\mathbb Z,\lambda}$ is naturally a left module over
\begin{align*}
\mathcal H_R(S_n)\otimes_{R,\varepsilon}\mathbb Z.
\end{align*}
Using the algebra isomorphism from the first step, we regard it as a left $\mathbb Z[S_n]$-module.
We use the specialization theorem for the Dipper-James-Murphy integral Specht construction with the quadratic convention
\begin{align*}
(T_i-q)(T_i+1)=0.
\end{align*}
It states that the DJM Specht lattice is compatible with base change in the parameter: after applying a ring homomorphism from the parameter ring, the generators, row relations, column antisymmetry relations, and Garnir straightening relations specialize to the corresponding relations for the specialized Hecke algebra. In the present specialization $\varepsilon(q)=1$, the generator $T_i$ maps to the adjacent transposition $s_i$, the Hecke row symmetrizers specialize to the ordinary row symmetrizers, the Hecke column antisymmetrizers specialize to the ordinary signed column antisymmetrizers, and the $q$-polytabloids specialize to the ordinary polytabloids. Therefore the specialized presentation of $S_{\mathbb Z,\lambda}$ is exactly the ordinary integral Specht presentation constructed from polytabloids, as in the Specht module construction [citetheorem:8438] and the polytabloid realization [citetheorem:8441]. Hence there is an isomorphism of left $\mathbb Z[S_n]$-modules
\begin{align*}
S_{R,\lambda}\otimes_{R,\varepsilon}\mathbb Z\cong S_\lambda.
\end{align*}
[/step]
[step:Pass to the generic field and apply Tits deformation]
Let
\begin{align*}
K:=\operatorname{Frac}(R)=\mathbb Q(q)
\end{align*}
and define the generic Hecke algebra
\begin{align*}
\mathcal H_K(S_n):=\mathcal H_R(S_n)\otimes_R K.
\end{align*}
Also define
\begin{align*}
S_{K,\lambda}:=S_{R,\lambda}\otimes_R K.
\end{align*}
Since $S_{R,\lambda}$ is finite free over $R$, the $K$-dimension of $S_{K,\lambda}$ is the $R$-rank of $S_{R,\lambda}$, namely
\begin{align*}
\dim_K S_{K,\lambda}=f_\lambda.
\end{align*}
We now use the Dipper-James-Murphy generic irreducibility and classification theorem for type A Hecke Specht modules in the following form: if $F$ is a field and $Q\in F^\times$ is not a root of unity, then for the type A Iwahori-Hecke algebra over $F$ with quadratic convention $(T_i-Q)(T_i+1)=0$, the DJM Specht modules labelled by partitions of $n$ are simple, pairwise non-isomorphic, exhaust the simple modules, and hence make the algebra split semisimple. Its hypotheses match the present setting: the ground field is $K=\mathbb Q(q)$, the parameter $q\in K^\times$ is an indeterminate and hence is not a root of unity, and the algebra is the type A Iwahori-Hecke algebra with quadratic convention $(T_i-q)(T_i+1)=0$. Applying the same anti-involution when one passes from the usual right-module convention to left modules, the theorem states that the modules obtained by extending the integral Dipper-James-Murphy Specht lattices to $K$ are simple, pairwise non-isomorphic, and complete among simple left modules. In particular, for each partition $\lambda\vdash n$, the specific left module $S_{K,\lambda}=S_{R,\lambda}\otimes_R K$ is the generic simple module labelled by $\lambda$.
This classification is compatible with the specialization at $q=1$ by Tits' deformation theorem. The required finite-free algebra is supplied by the standard basis theorem [citetheorem:8464], which gives the $R$-basis $\{T_w:w\in S_n\}$ of $\mathcal H_R(S_n)$. The specialization at $q=1$ over $\mathbb Q$ is
\begin{align*}
\mathcal H_R(S_n)\otimes_{R,\varepsilon}\mathbb Q\cong\mathbb Q[S_n],
\end{align*}
because the integral specialization is $\mathbb Z[S_n]$ and tensoring with $\mathbb Q$ preserves this algebra isomorphism. By [Maschke's theorem](/theorems/2409) [citetheorem:8439], $\mathbb Q[S_n]$ is split semisimple, since $\operatorname{char}(\mathbb Q)=0$ does not divide $|S_n|$ and the ordinary Specht modules over $\mathbb Q$ realize all simple modules by the Specht classification [citetheorem:8440]. Tits' theorem therefore confirms that the generic simple labels and dimensions agree with the ordinary Specht labels, while the Dipper-James-Murphy generic classification identifies the actual generic modules as the base changes
\begin{align*}
S_{K,\lambda}=S_{R,\lambda}\otimes_R K.
\end{align*}
[/step]
[step:Conclude simplicity, non-isomorphism, and completeness]
For each partition $\lambda\vdash n$, the preceding step shows that $S_{K,\lambda}$ is a simple left $\mathcal H_K(S_n)$-module. If $\lambda\ne\mu$, then the Dipper-James-Murphy generic classification, equivalently the Tits deformation comparison with the non-isomorphic ordinary Specht modules over $\mathbb Q[S_n]$, gives that $S_{K,\lambda}$ and $S_{K,\mu}$ are non-isomorphic.
Finally, every simple left $\mathcal H_K(S_n)$-module occurs in the Dipper-James-Murphy generic classification from a unique partition of $n$. Hence every simple left $\mathcal H_K(S_n)$-module is isomorphic to exactly one of the modules
\begin{align*}
S_{K,\lambda} \quad \text{with } \lambda\vdash n.
\end{align*}
This proves that the generic Specht modules form a complete set of pairwise non-isomorphic simple left modules for
\begin{align*}
\mathcal H_R(S_n)\otimes_R K.
\end{align*}
[/step]
