The cyclotomic Murphy basis and cellularity

For every $\lambda$, the componentwise and shift lemmas [F1]–[F3] give two-sided ideals $N^{⊵\lambda },N^{▹\lambda }$ and show that right multiplication by any algebra generator preserves shape dominance. They also give the fixed-initial-row law for $m_{t^{\lambda },t}$: modulo $N^{▹\lambda }$ it is a linear combination of $m_{t^{\lambda },v}$, with coefficients independent of the first index after left multiplication. To see closure under the full generator set, use the finite generators $T_i$ and $X_1$ from [F1]; the cyclotomic polynomial has unit constant coefficient, so $X_1^{-1}$ is a polynomial in $X_1$, while $T_i^{-1}=Q^{-1}(T_i-(Q-1))$ and the recursion expresses every $X_j$ as a word in $T_i,X_1$.

Let $U_{\lambda }$ be the span of $m_{t^{\lambda },v}$ for standard $v$ of shape $\lambda$. The fixed-initial-row law shows $U_{\lambda }+N^{▹\lambda }$ is a right ideal. For any $h\in H_{r,n}$ there are coefficients $r_{t,v}(h)$ such that$m_{t^{\lambda },t}h\equiv {\sum }_v{r}_{t,v}(h)m_{t^{\lambda },v}(\mathrm{m}\mathrm{o}\mathrm{d}{N}^{▹\lambda })\mathrm{.}$ Since $N^{▹\lambda }$ is a two-sided ideal, multiplying by $T_{d(s)}^{\ast }$ gives the displayed cell law for every $s$, and the coefficients remain independent of $s$. The generator-fixing anti-involution gives $m_{st}^{\ast }=m_{ts}$.

The row-filled multipartition ${\lambda }_0=(\mathrm{\varnothing },\dots ,\mathrm{\varnothing },(1^n))$ is dominated by every multipartition. Before the final component its cumulative counts are zero. At the first $k$ rows of the final component its count is $\min (k,n)$. If a multipartition has $a$ boxes in earlier components and final component $\gamma$, then $\mathrm{\mid }\gamma \mathrm{\mid }=n-a$ and the first $k$ rows of the partition $\gamma$ contain at least $\min (k,n-a)$ boxes; thus its cumulative count is at least $a+\min (k,n-a)\ge \min (k,n)$. Its row subgroup is trivial and its cyclotomic factor $u_{{\lambda }_0}$ is empty, so $m_{t^{{\lambda }_0},t^{{\lambda }_0}}=1$. Taking $\lambda ={\lambda }_0$ in step 2.1 expresses every $h\in H_{r,n}$ as a linear combination of the one initial-row layer elements plus elements of $N^{▹{\lambda }_0}$, which by definition are in the span of the standard Murphy family. Hence the family spans $H_{r,n}$.

By [F5] the spanning family has $r^{n}n!$ elements, and by [F4] the algebra is free of rank $r^{n}n!$ over $R$. Its square coordinate matrix is surjective, so [F6] gives a right inverse; taking determinants shows its determinant is a unit. Therefore the family is an $R$-basis. Together with step 2.1 this gives a cellular basis and the stated dominance ideals by [F7]. Tensoring the universal basis and its multiplication congruences along any commutative base change gives the same basis and cell law after specialization.