The Ariki–Koike PBW basis and generic splitness

There are $r$ choices for each exponent $a_i$ and $n!$ choices for $w$, so $\mathrm{\mid }{\mathcal{B}}_{r,n}\mathrm{\mid }=r^{n}n!$. By [F1], these elements span $H_{r,n}$ and their images span $H_{r,n}{\otimes }_{R}K$, whose dimension is at most $r^{n}n!$.

Let $\mathcal{T}={\coprod }_{\lambda \in {\mathcal{P}}_r(n)}\mathrm{Std}(\lambda )$ and, for each $j$, let $C_j=\{{\mathrm{res}}_s(j):s\in \mathcal{T}\}$. On $V={⨁}_{\lambda }{V}^{\lambda }$, the commuting $L_j$ are diagonal. For each $t\in \mathcal{T}$ the element $e_t={\prod }_{j=1}^n{\prod }_{\begin{array}{c}c\in C_j\\ c\mathrm{\ne }{\mathrm{res}}_t(j)\end{array}}\frac{L_j-c}{{\mathrm{res}}_t(j)-c}$ is defined over $K$ by [F2]. It acts as $1$ on $v_t$ and $0$ on every other basis vector: a different tableau has a different content sequence, so at some coordinate its content contributes a zero factor. These are explicit joint spectral projectors in the image of $H_{r,n}{\otimes }_{R}K$.

If $s_{i}t$ is standard, then $e_{s_{i}t}{T}_i{e}_t$ acts as $B_i(t)$ times the matrix unit carrying $v_t$ to $v_{s_{i}t}$ and all other basis vectors to $0$. The coefficient is nonzero by [F2], so division by it gives that matrix unit in the image. Adjacent swaps connect all linear extensions of a fixed shape; products of these edge matrix units give every matrix unit between two tableaux of that shape. The projectors separate different shapes, so the action homomorphism $H_{r,n}{\otimes }_{R}K⟶{\prod }_{\lambda \in {\mathcal{P}}_r(n)}{\mathrm{End}}_K(V^{\lambda })$ is surjective. Its target dimension is ${\sum }_{\lambda }(f^{\lambda }{)}^2=r^{n}n!$ by [F3], hence ${\dim }_K(H_{r,n}{\otimes }_{R}K)\ge r^{n}n!$.

Steps 1.1 and 2.1 give opposite dimension bounds, so $H_{r,n}{\otimes }_{R}K$ has dimension exactly $r^{n}n!$ and its $r^{n}n!$ spanning monomials are a $K$-basis. The surjective action map in step 2.1 is now an isomorphism, proving the displayed generic split decomposition.

Let $F_R$ be the free $R$-module on the symbols in ${\mathcal{B}}_{r,n}$ and map it onto $H_{r,n}$ by the spanning result [F1]. If $x$ is in its kernel, then $x\otimes 1$ maps to zero in $H_{r,n}{\otimes }_{R}K$; the images of the symbols are a $K$-basis by step 3.1, so $x\otimes 1=0$ in $K^{r^{n}n!}$. Since $R$ embeds in $K$ by [F4], the map $R^{r^{n}n!}\to K^{r^{n}n!}$ is injective and $x=0$. Thus the spanning family is $R$-linearly independent and is an $R$-basis.

For any commutative $R$-algebra $S$, [F5] identifies $S{\otimes }_R{H}_{r,n}$ with the quotient of the free associative $S$-algebra by the same relations with coefficients mapped to $S$. Indeed the image of the two-sided relation ideal is generated by the images of its defining relators. Tensoring the free $R$-basis from step 4.1 gives the corresponding free $S$-basis, without requiring $S$ to be flat over $R$.

Steps 3.1–4.1 establish the universal basis, the split generic algebra, and all stated specializations. The independence proof uses the explicitly constructed generic modules and the proved square count, not the PBW theorem as its own premise.