Universal cyclotomic Hecke ordered monomials form a basis

Spanning over $R$. By [F1] the family $\mathcal{B}$ spans $H_n$ over the universal ring $R$: every element is an $R$-combination of the bounded ordered monomials with $0\le a_r<\mathrm{\ell }$ and $w\in S_n$. The calculation uses no field division and therefore holds over $R$; for $n=0$ the singleton empty word gives $H_0=R$, and for $\mathrm{\ell }=0$, $n>0$ the quotient is zero with the empty family.

Independence by finite-field specialization. Suppose ${\sum }_b{f}_{b}b=0$ is a nontrivial relation in $H_n$ with $f_b\in R$, not all zero. Fix $b_0$ with $f:=f_{b_0}\mathrm{\ne }0$ and multiply the relation by a Laurent monomial so that $f\in \mathbb{Z}[z,U_1,\dots ,U_{\mathrm{\ell }}]$; choose a prime $p$ for which some nonzero integer coefficient of $f$ survives modulo $p$, and view $f$ as a polynomial in $U_1,\dots ,U_{\mathrm{\ell }}$ with coefficients in ${\mathbb{F}}_p[z]$; let $g(z)\mathrm{\ne }0$ be one of those coefficients. By [F3] the number of primitive elements of ${\mathbb{F}}_{p^d}^{\times }$ is $\phi (p^d-1)$, which is unbounded and satisfies $\phi (N)\ge \sqrt{N\mathrm{/}2}$ (factor $N$ over prime powers: each factor $\phi (r^a{)}^2\mathrm{/}{r}^a=r^{a-2}(r-1{)}^2\ge 1$ except $r=2$, $a=1$, where it is $1\mathrm{/}2$). Choose $d$ with $\phi (p^d-1)>\mathrm{deg}g$ and $p^d-1$ larger than every $U$-degree of $f$; then some primitive $z\in {\mathbb{F}}_{p^d}^{\times }$ avoids the roots of $g$. Repeated one-variable root bounds on the nonzero-coordinate grid (each $U_a$ is constrained to be a power $z^{c_a}\mathrm{\ne }0$, and the remaining polynomial in the next variable has degree bounded by the $U$-degree of $f$) supply $U_a=z^{c_a}\mathrm{\ne }0$ with $f(z,U_1,\dots ,U_{\mathrm{\ell }})\mathrm{\ne }0$ in ${\mathbb{F}}_{p^d}$. This specialization is exactly an allowed finite cyclic type-A field parameter: $z$ has order $e=p^d-1\ge 2$, and the cyclotomic charges $c_a$ lie in $\mathbb{Z}\mathrm{/}e\mathbb{Z}$.

The specialized basis forces a contradiction. Over the field ${\mathbb{F}}_{p^d}$ with the parameter and charges of step 1.2, the early explicit BK maps identify the direct sum of the Hecke central sectors with the standard homogeneous cyclic KLR algebra, whose total dimension is ${\mathrm{\ell }}^{n}n!$ by [F2]. The bounded spanning of step 1.1 provides exactly ${\mathrm{\ell }}^{n}n!$ ordered words $X^a{T}_w$, so over the specialized field the family $\mathcal{B}$ is a basis; specializing the assumed relation forces $f_b(z,U)=0$ for every $b$, and in particular $f(z,U)=0$, contradicting the choice of $z,U$ in step 1.2. Hence the relation was trivial, the family $\mathcal{B}$ is $R$-linearly independent, and with step 1.1 it is an $R$-basis of $H_n$.

Base change and boundary cases. Formation of the presented algebra commutes with arbitrary base change: the presentation is a quotient of a free algebra and its specialized ideal is precisely the specialized relation ideal, so the $R$-basis $\mathcal{B}$ specializes to a basis of $H_n{\otimes }_R{R}^{\mathrm{\prime }}$ for every commutative $R$-algebra $R^{\mathrm{\prime }}$ (in particular for the LM deformation DVR with $z=q_{H}t$ and $U_a=Q_a{t}^{na}$). No integral freeness was inferred from generic rank, no flatness was assumed, and no Schur or primitive-block classification was used.