for all $\mathfrak{s}, t \in T(\lambda)$ and $\lambda \in \mathcal{P}$. Recall that definitions of the map $\sigma_n^\Lambda$ from Theorem 3.13 and the maps $\sigma_{\mathcal{P}}$ and $\sigma_{\text{Std}}$ from Definition 4.5. The graded cellular structure that we consider on $\mathcal{R}_n^\Lambda$ is the one described in Section 3.4, with the graded cellular basis obtained from Proposition 5.3.

**Theorem 5.4.** The triple of maps $\sigma = (\sigma_n^\Lambda, \sigma_{\mathcal{P}}, \sigma_{\text{Std}})$ is a shift automorphism of $\mathcal{R}_n^\Lambda$.

*Proof.* First, note that by Corollary 4.10 we know that $\sigma_{\mathcal{P}}$ is a poset automorphism of $(\mathcal{P}_n^\ell, \boxplus_{\rho})$. By Definition 2.22, we need to show that:

a) If $\mathfrak{s} \in \text{Std}(\lambda)$ then $\sigma_{\text{Std}}(\mathfrak{s}) \in \text{Std}(\sigma_{\mathcal{P}}(\lambda))$ and $\text{deg}(\sigma_{\text{Std}}(\mathfrak{s})) = \text{deg}(\mathfrak{s})$.

b) If $\mathfrak{s}, t \in \text{Std}(\lambda)$ then $\sigma_{\mathcal{R}_n^\Lambda}(c_{\text{Std}}) = c_{\sigma_{\text{Std}}(\mathfrak{s})\sigma_{\text{Std}}(t)}$.

c) If $\mathfrak{s}, t \in \text{Std}(\lambda)$, for $\lambda \in \mathcal{P}_n^\ell$, then $\sigma_{\text{Std}}^k(t) = t$ if and only if $\sigma_{\text{Std}}^k(\mathfrak{s}) = \mathfrak{s}$, for $k \in \mathbb{Z}$.

The first requirement in part (a) is immediate from Definition 4.5 and the second requirement, that the fact that $\sigma_{\text{Std}}$ is homogeneous follows from Lemma 4.12. Part (c) follows because all tableaux have order $p$ under the action of $\sigma_{\text{Std}}$. It remains to check part (b). That is, we need to show that

$$ \sigma(C_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}) = C_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}, \quad \text{for any } \lambda \in \mathcal{P}_{\sigma,n}, \mathfrak{s}, t \in \text{Std}_{\sigma}(\lambda) \text{ and } k, l \in \mathbb{Z}/p\mathbb{Z}, $$

Equivalently, we need to show that $\sigma(C_{\sigma^{k}\mathfrak{s}}^* C_{\sigma^{l}t}) = C_{\sigma^{k+1}\mathfrak{s}}^* C_{\sigma^{l+1}t}$. By Definition 5.1 and Lemma 4.14,

$$ \begin{aligned} C_{\sigma^{k}\mathfrak{s}}^{*} &= \tilde{\sigma}^{k} (B_{\mathfrak{s}}^{\text{sing}})^{*} (C_{\sigma^{k}\mathfrak{s}}^{\text{reg}})^{*}, & C_{\sigma^{l}t}^{*} &= C_{\sigma^{l}t}^{\text{reg}} \tilde{\sigma}^{l} (B_{t}^{\text{sing}}), \\ C_{\sigma^{k+1}\mathfrak{s}}^{*} &= \tilde{\sigma}^{k+1} (B_{\mathfrak{s}}^{\text{sing}})^{*} (C_{\sigma^{k+1}\mathfrak{s}}^{\text{reg}})^{*}, & C_{\sigma^{l+1}t}^{*} &= C_{\sigma^{l+1}t}^{\text{reg}} \tilde{\sigma}^{l+1} (B_{t}^{\text{sing}}). \end{aligned} $$

By Lemma 4.30 and Lemma 4.37, any string in $(C_{\mathfrak{s}}^{\text{reg}})^{*} C_{t}^{\text{reg}} \in \mathcal{W}_{\rho}(\hat{\nu}_{\mathfrak{s}}, \hat{\nu}_{t})$ stays inside a single region. Moreover, by Lemma 4.25, any solid or ghost string of $(C_{\mathfrak{s}}^{\text{reg}})^{*} C_{t}^{\text{reg}} \in \mathcal{W}_{\rho}(\hat{\nu}_{\mathfrak{s}}, \hat{\nu}_{t})$ in the region $X_{\rho}^{a,0}$ starts (and ends) at the left of all the red strings in $X_{\rho}^{a,0}$, for $0 \le a < d$. By Definition 5.1 and (4.40), this is also true in the two diagrams $(C_{\sigma^{k}\mathfrak{s}}^{\text{reg}})^{*} C_{\sigma^{l}t}^{\text{reg}}$ and $(C_{\sigma^{k+1}\mathfrak{s}}^{\text{reg}})^{*} C_{\sigma^{l+1}t}^{\text{reg}}$. As both of these diagrams are regular by Lemma 5.2, we can apply Proposition 4.2 to find two regular Webster diagrams $D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}}$ and $D_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}^{\text{reg}}$ that do not contain any generalised double crossings such that in $A_{\rho}^{\rho}(\hat{\nu}_{\sigma^{k}\mathfrak{s}}, \hat{\nu}_{\sigma^{l}t})$

$$ D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}} = (C_{\sigma^{k}\mathfrak{s}}^{\text{reg}})^{*} C_{\sigma^{l}t}^{\text{reg}} \quad \text{and} \quad D_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}^{\text{reg}} = (C_{\sigma^{k+1}\mathfrak{s}}^{\text{reg}})^{*} C_{\sigma^{l+1}t}^{\text{reg}} $$

and all of the solid and ghost strings in any region $X_{\rho}^{a,0}$ are to the left of all of the red strings in $X_{\rho}^{a,0}$. Since $D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}}$ is a regular diagram that has no crossings involving red strings, the diagram $\tilde{\sigma}(D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}})$ is also regular. Hence, the diagrams $\tilde{\sigma}(D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}})$ and $D_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}^{\text{reg}}$ satisfy the assumptions of Corollary 4.3, and so $\tilde{\sigma}(D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}}) = D_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}^{\text{reg}}$ in $A_{\rho}^{\rho}(\hat{\nu}_{\sigma^{k}\mathfrak{s}}, \hat{\nu}_{\sigma^{l}t})$. We have proved so far that

$$ \begin{aligned} C_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{*} &= \tilde{\sigma}^{k} (B_{\mathfrak{s}}^{\text{sing}})^{*} D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}} \tilde{\sigma}^{l} (B_{t}^{\text{sing}}), \\ C_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}^{*} &= \tilde{\sigma}^{k+1} (B_{\mathfrak{s}}^{\text{sing}})^{*} \tilde{\sigma} (D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}}) \tilde{\sigma}^{l+1} (B_{t}^{\text{sing}}). \end{aligned} $$

Consequently,

$$ (5.5) \qquad \tilde{\sigma}(C_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}) = C_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}. $$

Recall that in $D_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{reg}}$, all of the solid and ghost strings that start or, equivalently, end in $X_{\rho}^{a,0}$ are to the left of all of the red strings in $X_{\rho}^{a,0}$. Similarly, since $(B_{\mathfrak{s}}^{\text{sing}})^{*} B_{t}^{\text{sing}} \in \mathcal{W}_{\rho}(\omega, \omega)$, all the solid strings begin and end to the left of all red strings. Moreover by Proposition 4.31(b), in this diagram any solid string that crosses a red string in $X_{\rho}^{a,0}$ must cross all of the red strings in $X_{\rho}^{a,0}$, for $0 \le a < d$. Thus, we can apply Proposition 4.19, which gives

$$ \tilde{\sigma}(C_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{cyc}}) = \tilde{\sigma}(C_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}^{\text{cyc}})^{\text{cyc}}. $$

Combining (5.5) with Lemma 4.16 and Proposition 4.21, we find that

$$ \sigma(C_{\sigma^{k}\mathfrak{s}, \sigma^{l}t}) = C_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}^{\text{cyc}} = C_{\sigma^{k+1}\mathfrak{s}, \sigma^{l+1}t}, $$

as desired.
**Remark 5.6.** A very particular case of Theorem 5.4 can be found in [3, Example 7.5]. Namely, for $\ell = e = 2$ and the 2-charge obtained from (0, 1), the basis elements are of the form

$$
\begin{array}{l@{\qquad}l}
c_{\mathfrak{s},\mathfrak{s}} = e(\mathbf{i}) & c_{\sigma_{\mathfrak{s}},\sigma_{\mathfrak{s}}} = e(-\mathbf{i}) \\[1.5ex]
c_{\mathbf{t},\mathbf{t}} = y_2 e(\mathbf{i}) & c_{\sigma_{\mathbf{t}},\sigma_{\mathbf{t}}} = y_2 e(-\mathbf{i}) \\[1.5ex]
c_{\sigma_{\mathbf{t}},\mathbf{t}} = \psi_1 e(\mathbf{i}) & c_{\mathbf{t},\sigma_{\mathbf{t}}} = \psi_1 e(-\mathbf{i}) \\[1.5ex]
c_{\mathbf{u},\mathbf{u}} = y_2^2 e(\mathbf{i}) & c_{\sigma_{\mathbf{u}},\sigma_{\mathbf{u}}} = y_2^2 e(-\mathbf{i})
\end{array}
$$

where $\mathfrak{s}$ (resp. $\mathbf{u}$) is the only element in $\text{Std}(1^2|0)$ (resp. $\text{Std}(2|0))$, $\mathbf{i} \in \{(1, -1), (-1, 1)\}$ and the tableau $\mathbf{t}$ is a particular element of $\text{Std}(1|1)$. Recall that $\sigma$ fixes $y_1, y_2, \psi_1$ and that $\sigma(e(\mathbf{j})) = e(-\mathbf{j})$.

We can now prove our Main Theorem from the introduction.

**Corollary 5.7.** Assume that $R$ contains a primitive $p$-th root of unity and $p \cdot 1_R$ is invertible in $R$. The algebra $\mathcal{R}_{p,n}^{\Lambda}$ is a graded skew cellular algebra. In particular, if $2 \cdot 1_R \in R^\times$ then $\mathcal{R}_{2d,2,n}^{\Lambda}$ is graded cellular.

*Proof.* This is an immediate consequence of Theorem 5.4, Corollary 2.32 and Theorem 2.28 applied for the graded cellular algebra $\mathcal{R}_n^{\Lambda}$. Note that $\sigma_n^{\Lambda}$ has order $p$ indeed. $\square$

Combining the last result with Proposition 2.21 yields:

**Corollary 5.8.** The graded decomposition matrix of $\mathcal{R}_{p,n}^{\Lambda}$ is unitriangular.

We now explicitly describe the graded skew cellular datum of $\mathcal{R}_{p,n}^{\Lambda}$, following the construction given in Theorem 2.28. Recall that $\mathcal{P}_{\sigma,n}^{\ell}$ is any set of representatives for the action of $\langle\sigma_{\mathcal{P}}\rangle$ on $\mathcal{P}_n^{\ell}$. We endow $\mathcal{P}_{\sigma,n}^{\ell}$ with the partial order $\supseteq_{\rho}$ so that $a \triangleright_{\rho} b$ for $a, b \in \mathcal{P}_{\sigma,n}^{\ell}$ if and only if there exists $\lambda \in a$ and $\mu \in b$ such that $\lambda \triangleright_{\rho} \mu$. Define

$$ (5.9) \qquad \mathcal{P}_{\sigma,p,n}^{\ell} := \mathcal{P}_{\sigma,n}^{\ell} \times \mathbb{Z}/\mathcal{o}_{\lambda}\mathbb{Z}, $$

where $\mathcal{o}_{\lambda}$ is the size of the orbit of $\lambda$ under the action of $\langle\sigma_{\mathcal{P}}\rangle$. The order $\supseteq_{\rho}$ extends to an order of $\mathcal{P}_{\sigma,p,n}^{\ell}$ with the rule $(\lambda, k) \triangleright_{\rho} (\mu, l) \iff \lambda \triangleright_{\rho} \mu$. Let $\iota$ be the involution on $\mathcal{P}_{\sigma,p,n}^{\ell}$ given by $(\lambda, k) \mapsto (\lambda, -k)$. For $(\lambda, l) \in \mathcal{P}_{\sigma,p,n}^{\ell}$ let $\text{Std}_{\sigma}(\lambda, k) := \text{Std}_{\sigma}(\lambda)$. For $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$ and $\mathfrak{s}, \mathbf{t} \in \text{Std}_{\sigma}(\lambda, k)$, define

$$ D_{\mathfrak{st}}^{(k)} := \sum_{j=0}^{p_{\lambda}-1} \varepsilon^{kj\sigma_{\lambda}} \bar{\sigma}_{A}(C_{\mathfrak{s},\sigma^{j\sigma_{\lambda}}\mathbf{t}}), $$

where

$$ \bar{\sigma}_A := \sum_{k=0}^{p-1} \sigma^k \quad \text{and} \quad p_{\lambda} = \frac{p}{\mathcal{o}_{\lambda}}. $$

In particular, note that $\mathcal{o}_T(\lambda) = p_{\lambda}$, since $\mathbf{p} = \mathbf{p}_{\mathcal{P}} = p$, so $\frac{p}{\mathcal{o}_T(\lambda)} = \frac{p}{p_{\lambda}} = \mathcal{o}_{\lambda}$ and $\varepsilon_{\lambda} = \varepsilon^{\mathcal{o}_{\lambda}}$. Finally, define the map

$$ D: \text{Std}_{\sigma}(\lambda, k) \times \text{Std}_{\sigma}(\lambda, k) \to \mathcal{R}_{p,n}^{\Lambda}; (\mathfrak{s}, \mathbf{t}) \mapsto D_{\mathfrak{st}}^{(k)}, $$

and let $\iota_{\lambda}^{(k)}$ be the identity map on $\text{Std}_{\sigma}(\lambda, k)$. Then

$$ (5.10) \qquad (\mathcal{P}_{\sigma,p,n}^{\ell}, \iota, \text{Std}_{\sigma}, D, \text{deg}), $$

is a graded skew cell datum for $\mathcal{R}_{p,n}^{\Lambda}$. In particular, $\mathcal{R}_n^{\Lambda}$ has cell modules $C_{\lambda}^{(k)}$, and simple modules the non-zero quotients $D_{\lambda}^{(k)} = C_{\lambda}^{(k)}/\text{rad}C_{\lambda}^{(k)}$, for $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$.

Note that the graded skew cell datum for $\mathcal{R}_{p,n}^{\Lambda}$ restricts to give a graded skew cell data for the blocks $\mathcal{R}_{p,\alpha}^{\Lambda}$ and that the graded decomposition matrix of $\mathcal{R}_{p,\alpha}^{\Lambda}$ is unitriangular, for $\alpha \in \mathcal{I}_{\sigma}^{n}$.

Recall from Definition 3.7 that $\mathcal{I} = \{\varepsilon^j q^i | 0 \le j < p \text{ and } 0 \le i < e\}$. Define

$$ z := \sum_{\mathbf{i} \in \mathcal{I}^n} i_1^{-1} e(\mathbf{i}) \in \mathcal{R}_n^{\Lambda}. $$
**Lemma 5.11.** The element $z \in \mathcal{R}_n^\Lambda$ is homogeneous of degree 0, invertible and $\sigma(z) = \varepsilon z$.

*Proof.* First, recall from Section 3.1 that $e$ is finite thus $\mathcal{I}$ is finite and $z$ is well defined. We have $\deg z = 0$ since $\deg e(\mathbf{i}) = 0$ for all $\mathbf{i} \in \mathcal{I}^n$. Using the fact that $\{e(\mathbf{i}) : \mathbf{i} \in \mathcal{I}^n\}$ is a complete set of orthogonal idempotents we find that $z$ is invertible in $\mathcal{R}_n^\Lambda$ with inverse $\sum_{(i_1, \dots, i_n) \in \mathcal{I}^n} i_1 e(i_1, \dots, i_n)$. Finally, we have

$$ 
\begin{aligned}
\sigma(z) &= \sum_{(i_1, \dots, i_n) \in \mathcal{I}^n} i_1^{-1} e(\varepsilon i_1, \dots, \varepsilon i_n) \\
&= \sum_{(i_1, \dots, i_n) \in \mathcal{I}^n} (\varepsilon^{-1} i_1)^{-1} e(i_1, \dots, i_n) \\
&= \varepsilon z,
\end{aligned}
$$

which completes the proof.

□

**Corollary 5.12.** The automorphism $\sigma$ is $\varepsilon$-splittable in the sense of Definition 2.39.

Hence, all the results in Section 2.4 now apply to $\mathcal{R}_{p,n}^\Lambda$. Note that multiplication by $z$ induces an analogue of Lemma 5.11 for the blocks $\mathcal{R}_\alpha^\Lambda$ of $\mathcal{R}_n^\Lambda$, so the results in Section 2.4 also apply to the blocks $\mathcal{R}_{p,\alpha}^\Lambda$ of $\mathcal{R}_{p,n}^\Lambda$, for $[\alpha] \in \mathcal{I}_\sigma^n$.

By Theorem 3.14, all the results concerning the skew cellularity of $\mathcal{R}_{p,n}^\Lambda$, over a ring $R$ containing $\varepsilon$, can be deduced from $\mathcal{H}_{p,n}(q, \mathbf{Q}^{\vee\varepsilon})$ over the field $K$, which contains $\varepsilon$ by assumption. For example, the following holds:

**Corollary 5.13.** The algebra $\mathcal{H}_{p,n}(q, \mathbf{Q}^{\vee\varepsilon})$ is a graded skew cellular algebra. Moreover, if $p = 2$ then $\mathcal{H}_{2,n}(q, \mathbf{Q}^{\vee\varepsilon})$ is a graded cellular algebra.

Geck [11] has proved that the Iwahori–Hecke algebra of a finite Coxeter group is always an (ungraded) cellular algebra. By Corollary 5.13, the Iwahori–Hecke algebras of the Coxeter groups of types $A_{n-1}$, $B_n = C_n$, $D_n$ and $I_2(n)$ are graded skew cellular algebras. (These Coxeter groups are the complex reflection groups of types $G(1, 1, n)$, $G(2, 1, n)$, $G(2, 2, n)$ and $G(n, n, 2)$, respectively.)

The Iwahori-Hecke algebras of types $A_{n-1}$ and $B_n$ are graded cellular algebras by [18]. For the algebras of types $D_n$ and $I_2(n)$, Geck’s proof of the cellularity of the Iwahori-Hecke algebras $\mathcal{H}_q(D_n)$ assumes that $q^{1/2} \in K$. The following corollary generalises Geck’s result to the graded case and removes the assumption that $q^{1/2} \in K$.

**Corollary 5.14.** Suppose $\mathrm{char} K \neq 2$. The Iwahori-Hecke algebra $\mathcal{H}_q(D_n)$ of type $D_n$ is a $\mathbb{Z}$-graded cellular algebra.

*Proof.* The Coxeter group of type $D_n$ is the complex reflection group of type $G(2, 2, n)$ in the Shephard–Todd classification. The result thus directly follows from Corollary 5.13 for $d = 1$. □

It is an interesting open question whether the algebras $\mathcal{R}_{p,n}^\Lambda$ are graded cellular algebras when $p > 2$ (and $n > 2$ if $p = \ell$).

5.3. **Adjustment matrices.** As a final application we use the bases in this paper to describe “adjustment matrices” for the Hecke algebras of type $G(\ell, p, n)$, which relate decomposition matrices in different characteristics. For the Iwahori-Hecke algebras of finite Coxeter groups Geck and Rouquier used Lusztig’s asymptotic Hecke algebra to show that adjustment matrices exist whenever the Iwahori-Hecke algebra is defined over a field of “good characteristic”, which depends on the root system of the underlying Coxeter group; see [12, Table 1.4 and Theorem 3.6.3]. When $p = 1$ we recover the results of [8, §5.6] but, even in the ungraded case, these results appear to be new when $p > 2$.

Throughout this section fix $\Lambda$ as in Section 3.2. If $R$ is an integral domain let $\mathcal{R}_{p,n}^{\Lambda,R}$ be the quiver Hecke algebra over $R$ with weight $\Lambda$. Similarly, if $M$ is an $\mathcal{R}_{p,n}^{\Lambda,R}$-module then we write $M = M^R$ to emphasize that $M$ is an $R$-module. By Corollary 5.7, if $R$ contains a primitive $\ell$th root of unity and $p \in \mathbb{R}^\times$ then $\mathcal{R}_{p,n}^{\Lambda,R}$ is a skew cellular algebra with cell modules $C_\lambda^{(k),R}$, for $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^\ell$. By Theorem 2.17, if $R$ is a field, the graded simple $\mathcal{R}_{p,n}^{\Lambda,R}$-modules are shifts of the non-zero quotients $D_\lambda^{(k),R} = C_\lambda^{(k),R}/\mathrm{rad} C_\lambda^{(k),R}$, for $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^\ell$.
Fix a field $F$ of characteristic $c > 0$ that contains a primitive $p$th root of unity $\epsilon_F$. In particular, this implies that $c$ does not divide $p$. The subfield $F_{p,c} = \mathbb{Z}/c\mathbb{Z}(\epsilon_F)$ of $F$ is a splitting field for the $p$-th cyclotomic polynomial $\Phi_p$ over $\mathbb{Z}/c\mathbb{Z}$. By [25, Theorem 2.47], the field $F_{p,c}$ is the finite field with $c^r$ elements, where $r > 0$ is minimal such that $c^r \equiv 1 \pmod p$.

Let $\pi: \mathbb{Z}[\epsilon] \to F_{p,c}$ be the unique ring homomorphism determined by $\pi(\epsilon) = \epsilon_F$. Then $\mathfrak{p} = \ker \pi$ is a prime ideal of $\mathbb{Z}[\epsilon]$ and the localisation $\mathcal{O} = \mathbb{Z}[\epsilon]_{\mathfrak{p}}$ of $\mathbb{Z}[\epsilon]$ at the prime ideal $\mathfrak{p}$ is a discrete valuation ring with maximal ideal $\mathfrak{p}\mathcal{O}$. The next result follows from elementary properties of localisation.

**Lemma 5.15.** The residue field $\mathcal{O}/\mathfrak{p}\mathcal{O}$ of $\mathcal{O}$ is isomorphic to $F_{p,c}$.

Consequently, if $M^{\mathcal{O}}$ is an $\mathcal{R}_{p,n}^{\mathcal{A},\mathcal{O}}$-module then, by base change, $M^{F_{p,c}} = F_{p,c} \otimes_{\mathcal{O}} M^{\mathcal{O}}$ is an $\mathcal{R}_{p,n}^{\mathcal{A},F_{p,c}}$-module and $M^F = F \otimes_{F_{p,c}} M^{F_{p,c}} \cong F \otimes_{\mathcal{O}} M^{\mathcal{O}}$ is an $\mathcal{R}_{p,n}^{\mathcal{A},F}$-module.

**Remark 5.16.** By Theorem 2.17 every field is a splitting field for the algebra $\mathcal{R}_{p,n}^{\mathcal{A}}$, so there would be no loss of generality in assuming that $F = F_{p,c}$.

Let $\mathcal{Q} = \text{Frac}(\mathcal{O}) = \mathbb{Q}(\epsilon)$ be the field of fractions of $\mathcal{O}$. By Definition 2.6, $C_{\lambda}^{(k),\mathcal{Q}} \cong \mathcal{Q} \otimes_{\mathcal{O}} C_{\lambda}^{(k),\mathcal{O}}$ and $C_{\lambda}^{(k),F} \cong F \otimes_{\mathcal{O}} C_{\lambda}^{(k),\mathcal{O}}$, for $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$. Now recall from (2.13) that

$$ \text{rad } C_{\lambda}^{(k),\mathcal{O}} = \{y \in C_{\lambda}^{(k),\mathcal{O}} \mid \phi_{\lambda}^{(k)}(x, y) = 0 \text{ for all } x \in C_{\lambda}^{(k),\mathcal{O}}\} $$

and that $D_{\lambda}^{(k),\mathcal{O}} = C_{\lambda}^{(k),\mathcal{O}} / \text{rad} C_{\lambda}^{(k),\mathcal{O}}$ by Definition 2.16.

**Lemma 5.17.** Let $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$. Then $D_{\lambda}^{(k),\mathcal{O}}$ is an $\mathcal{O}$-free graded $\mathcal{R}_{p,n}^{\mathcal{A},\mathcal{O}}$-module. Moreover, $D_{\lambda}^{(k),\mathcal{Q}} \cong \mathcal{Q} \otimes_{\mathcal{O}} D_{\lambda}^{(k),\mathcal{O}}$ as graded $\mathcal{R}_{p,n}^{\mathcal{A},\mathcal{Q}}$-modules.

*Proof.* Let $\{D_t^{(k)} | t \in \text{Std}_{\sigma}(\lambda, k)\}$ be the standard basis of the cell module $C_{\lambda}^{(k),\mathcal{O}}$. Since $\text{rad} C_{\lambda}^{(k),\mathcal{O}}$ is a pure $\mathcal{O}$-submodule of $C_{\lambda}^{(k),\mathcal{O}}$, $D_{\lambda}^{(k),\mathcal{O}}$ is free as an $\mathcal{O}$-module. The final claim that $D_{\lambda}^{(k),\mathcal{Q}} \cong \mathcal{O} \otimes_{\mathcal{O}} D_{\lambda}^{(k),\mathcal{O}}$ is immediate because $\text{rad} C_{\lambda}^{(k),\mathcal{Q}} \cong \mathcal{Q} \otimes_{\mathcal{O}} \text{rad} C_{\lambda}^{(k),\mathcal{O}}$ since $\mathcal{Q}$ is a field of fractions of $\mathcal{O}$. $\square$

Let $\mathcal{P}_{\sigma,p,n,0}^{\ell} = \{(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell} | D_{\lambda}^{(k),\mathcal{Q}} \neq 0\}$. For convenience, set

$$ E_{\lambda}^{(k),F} = F \otimes_{\mathcal{O}} D_{\lambda}^{(k),\mathcal{O}}, \quad \text{for } (\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}. $$

In general, if $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$ then the $\mathcal{R}_{p,n}^{\mathcal{A},F}$-modules $D_{\lambda}^{(k),F}$ and $E_{\lambda}^{(k),F}$, and $\text{rad} C_{\lambda}^{(k),F}$ and $F \otimes_{\mathcal{O}} \text{rad} C_{\lambda}^{(k),\mathcal{O}}$, are not isomorphic. In particular, the $\mathcal{R}_{p,n}^{\mathcal{A},F}$-module $E_{\lambda}^{(k),F}$ is not necessarily irreducible.

Suppose that $R$ is a field and let $M^R$ be an $\mathcal{R}_{p,n}^{\mathcal{A},R}$-module. Recall from (2.20) that if $(\mu, l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$ and then the graded decomposition multiplicity of $D_{\mu}^{(l),R}$ in $M^R$ is

$$ [M^R : D_{\mu}^{(l),R}]_t = \sum_{s \in \mathbb{Z}} [M^R : D_{\mu}^{(l),R}(\langle s \rangle)] t^s \quad \in \mathbb{N}[t, t^{-1}]. $$

For $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$ and $(\mu, l), (\nu, m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$ define graded decomposition numbers

$$ d_{(\lambda,k)(\mu,l)}^{\mathcal{Q}}(t) = [C_{\lambda}^{(k),\mathcal{Q}} : D_{\mu}^{(l),\mathcal{Q}}]_t, $$

$$ d_{(\lambda,k)(\mu,l)}^{F}(t) = [C_{\lambda}^{(k),F} : D_{\mu}^{(l),F}]_t, $$

$$ \alpha_{(\mu,l)(\nu,m)}^{F}(t) = [E_{\mu}^{(l),F} : D_{\nu}^{(m),F}]_t. $$

Let $D_{\mathcal{Q}}(t) = (d_{(\lambda,k)(\mu,l)}^{\mathcal{Q}}(t))$, $D_{F}(t) = (d_{(\lambda,k)(\mu,l)}^{F}(t))$ and $A_{F}(t) = (\alpha_{(\mu,l)(\nu,m)}^{F}(t))$ be the corresponding matrices, where $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$ and $(\mu, l), (\nu, m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$. Then $D_{\mathcal{Q}}(t)$ and $D_{F}(t)$ are the graded decomposition matrices of $\mathcal{R}_{p,n}^{\mathcal{A},\mathcal{Q}}$ and $\mathcal{R}_{p,n}^{\mathcal{A},F}$, respectively. The matrix $A_{F}(t)$ is the **graded adjustment matrix** for these two algebras.

Let $\text{Rep}_{\mathcal{R}_{p,n}^{\mathcal{A},\mathcal{Q}}}$ and $\text{Rep}_{\mathcal{R}_{p,n}^{\mathcal{A},F}}$ be the Grothendieck groups of finitely generated graded $\mathcal{R}_{p,n}^{\mathcal{A},\mathcal{Q}}$- and $\mathcal{R}_{p,n}^{\mathcal{A},F}$-modules, respectively. Both Grothendieck groups are free $\mathbb{Z}[t, t^{-1}]$-modules where $t$ acts by grading shift. If $M$ is a module for one of these algebras let $[M]$ be the image of $M$ in the corresponding Grothendieck group.
**Lemma 5.18.** There is a unique abelian group homomorphism $d_F^Q: \text{Rep} \mathcal{R}_{p,n}^{\Lambda,Q} \to \text{Rep} \mathcal{R}_{p,n}^{\Lambda,F}$ such that
$d_F^Q[C_{\lambda}^{(k),Q}] = [C_{\lambda}^{(k),F}]$ and $d_F^Q[D_{\lambda}^{(k),Q}] = [E_{\lambda}^{(k),F}]$, for $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$.

*Proof.* Since $\mathcal{O}$ is a discrete valuation ring, for any $\mathcal{R}_{p,n}^{\Lambda,Q}$-module $M^Q$ there exists an $\mathcal{R}_{p,n}^{\Lambda,\mathcal{O}}$-module $M^\mathcal{O}$, a full $\mathcal{O}$-lattice, such that $M^Q \cong \mathcal{Q} \otimes_{\mathcal{O}} M^\mathcal{O}$. Define $M^F = F \otimes_{\mathcal{O}} M^\mathcal{O}$. The choice of $\mathcal{O}$-lattice is not unique but $[M^F]$ is independent of the choice of $\mathcal{O}$-lattice; compare with [10, Proposition 16.16]. Hence, we define $d_F^Q[M^Q] = [M^F]$. By Definition 2.6, if $(\nu, m) \in \mathcal{P}_{\sigma,p,n}^\ell$ then $d_F^Q[C_\lambda^{(k),Q}] = [C_\lambda^{(k),F}]$. Moreover, we have $d_F^Q[D_\lambda^{(k),Q}] = [E_\lambda^{(k),F}]$ because $E_\lambda^{(k),Q} \cong D_\lambda^{(k),Q}$ by Lemma 5.17. Finally, this establishes the uniqueness of $d_F^Q$ since $\{D_\lambda^{(k),Q} | (\lambda, k) \in \mathcal{P}_{\sigma,p,n,0}^\ell\}$ is a basis of $\text{Rep} \mathcal{R}_{p,n}^{\Lambda,Q}$.
$\square$

Let $\overline{\mathbb{Z}}: \mathbb{Z}[t, t^{-1}] \to \mathbb{Z}[t, t^{-1}]$ be the unique $\mathbb{Z}$-linear map such that $\overline{t^k} = t^{-k}$, for $k \in \mathbb{Z}$. Observe that $\dim_t M^* = \dim_t M$ if $M$ is a graded module. Hence, we can extend $\overline{\mathbb{Z}}$ to a map of the Grothendieck $\text{Rep} \mathcal{R}_{p,n}^{\Lambda,Q}$ and $\text{Rep} \mathcal{R}_{p,n}^{\Lambda,F}$ by setting $[M] = [M^*]$.

**Proposition 5.19.** Let $(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}$ and $(\mu, l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$. Then

a) $\alpha_{(\mu,l)(\mu,l)}^F(t) = 1$,

b) $\alpha_{(\lambda,k)(\mu,l)}^F(t) \neq 0$ only if $(\lambda, k) \geq \rho(\mu, l)$,

c) $\alpha_{(\lambda,-k)(\mu,-l)}^F(t) = \overline{\alpha_{(\lambda,k)(\mu,l)}^F(t)}$.

Moreover, $D_{\mu}^{(l),F} \neq 0$ if and only if $(\mu, l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$.

*Proof.* By construction, $F \otimes_{\mathcal{O}} \text{rad} C_{\lambda}^{(k),\mathcal{O}}$ is an $\mathcal{R}_{p,n}^{\Lambda,F}$-submodule of $\text{rad} C_{\lambda}^{(k),F}$, so parts (a) and (b) are immediate from Proposition 2.21. For part (c), in $\text{Rep} \mathcal{R}_{p,n}^{\Lambda,F}$ we have

$$[E_{\lambda}^{(k),F}] = \sum_{(\mu,l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} \alpha_{(\lambda,k)(\mu,l)}^{F}(t) [D_{\mu}^{(l),F}].$$

Taking duals and using Proposition 2.19 this becomes

$$[\overline{E_{\lambda}^{(k),F}}] = \sum_{(\mu,l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} \overline{\alpha_{(\lambda,k)(\mu,l)}^{F}(t)} [D_{\mu}^{(-l),F}].$$

On the other hand, $[\overline{E_{\lambda}^{(k),F}}] = \overline{d_F^Q[D_{\lambda}^{(k),Q}]} = \overline{d_F^Q[D_{\lambda}^{(k),Q}]} = \overline{d_F^Q[D_{\lambda}^{(-k),Q}]} = [\overline{E_{\lambda}^{(-k),F}}]$, where the second last equality comes from the same argument used in the proof of [10, 16.16] and Proposition 2.19, and the last equality from Lemma 5.18. This proves (c).

Finally, we prove that $D_{\lambda}^{(k),F} \neq 0$ if and only if $(\lambda, k) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$. First, observe that if $D_{\lambda}^{(k),F} \neq 0$ then $\text{rad} C_{\lambda}^{(k),F} \neq C_{\lambda}^{(k),F}$, implying that $\text{rad} C_{\lambda}^{(k),Q} \neq C_{\lambda}^{(k),Q}$. Hence, $D_{\lambda}^{(k),Q} \neq 0$ and $(\lambda, k) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$. Conversely, if $(\lambda, k) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}$ then $D_{\lambda}^{(k),Q} \neq 0$ so that $E_{\lambda}^{(k),F} \neq 0$. Hence, $E_{\lambda}^{(k),F} \neq 0$ by part (a). $\square$

We can now prove the main result of this section.

**Theorem 5.20.** Suppose that $F$ is a field of characteristic $c$ that $F$ contains a primitive $p$th root of unity $\epsilon_F$. Let $(\lambda, k) \in \mathcal{P}_n^\ell$ and $(\mu, l) \in \mathcal{P}_{\sigma,p,n,0}^\ell$. Then

$$d_{(\lambda,k)(\mu,l)}^F(t) = \sum_{(\nu,m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} d_{(\lambda,k)(\nu,m)}^Q(t) \alpha_{(\nu,m)(\mu,l)}^F(t).$$

That is, $D_F(t) = D_Q(t) A_F(t)$.
Proof. Using Lemma 5.18, and Proposition 2.21 twice,

$$
\begin{align*}
\sum_{(\boldsymbol{\mu},l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} d_{(\boldsymbol{\lambda},k)(\boldsymbol{\mu},l)}^{F}(t) [D_{\boldsymbol{\mu}}^{(l),F}] &= [C_{\boldsymbol{\lambda}}^{(k),F}] = d_{F}^{Q}[C_{\boldsymbol{\lambda}}^{(k),Q}] \\
&= d_{F}^{Q} \left( \sum_{(\boldsymbol{\nu},m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} d_{(\boldsymbol{\lambda},k)(\boldsymbol{\nu},m)}^{Q}(t) [E_{\boldsymbol{\nu}}^{(m),Q}] \right) \\
&= \sum_{(\boldsymbol{\nu},m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} d_{(\boldsymbol{\lambda},k)(\boldsymbol{\nu},m)}^{Q}(t) [E_{\boldsymbol{\nu}}^{(m),F}] \\
&= \sum_{(\boldsymbol{\nu},m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} d_{(\boldsymbol{\lambda},k)(\boldsymbol{\nu},m)}^{Q}(t) \sum_{(\boldsymbol{\mu},l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} \alpha_{(\boldsymbol{\nu},m)(\boldsymbol{\mu},l)}^{F}(t) [D_{\boldsymbol{\mu}}^{(l),F}] \\
&= \sum_{(\boldsymbol{\mu},l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} \left( \sum_{(\boldsymbol{\nu},m) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}} d_{(\boldsymbol{\lambda},k)(\boldsymbol{\nu},m)}^{Q}(t) \alpha_{(\boldsymbol{\nu},m)(\boldsymbol{\mu},l)}^{F}(t) \right) [D_{\boldsymbol{\mu}}^{(l),F}].
\end{align*}
$$

Since $\{[D_{\boldsymbol{\mu}}^{(l),F}] | (\boldsymbol{\mu},l) \in \mathcal{P}_{\sigma,p,n,0}^{\ell}\}$ is a basis of Rep$\mathcal{R}_{p,n}^{\Lambda,F}$, the result follows by comparing the coefficient of $[D_{\boldsymbol{\mu}}^{(l),F}]$ on both sides of this equation. $\square$

Let $\mathcal{F}$ be a field of characteristic zero that contains a primitive $p$th root of unity $\varepsilon$ and let $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon})$ be a Hecke algebra of type $G(\ell, p, n)$ over $\mathcal{F}$. Similarly, let $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon})$ be the corresponding Hecke algebra over the field $\mathcal{F}$, which contains a primitive $p$th root of unity $\varepsilon_{\mathcal{F}}$. By Theorem 3.14, $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon}) \cong \mathcal{R}_{p,n}^{\Lambda,F}$, so $D_{\mathcal{F}} = D_{\mathcal{F}}(1)$ is the decomposition matrix of $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon})$, where $D_{\mathcal{F}}(1)$ is the graded decomposition matrix of $\mathcal{R}_{p,n}^{\Lambda,F}$ evaluated at $t = 1$. By Theorem 2.17(b), every field is a splitting field for $\mathcal{R}_{p,n}^{\Lambda,Q}$, and $\mathcal{R}_{p,n}^{\Lambda,F} \cong \mathcal{F} \otimes_{\mathbb{Q}} \mathcal{R}_{p,n}^{\Lambda,Q}$, so $D_{\mathbb{Q}}(t)$ is the graded decomposition matrix of $\mathcal{R}_{p,n}^{\Lambda,F}$. On the other hand, $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon}) \cong \mathcal{R}_{p,n}^{\Lambda,F}$ by Theorem 3.14, so $D_{\mathcal{F}} = D_{\mathbb{Q}}(1)$ is the decomposition matrix of $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon})$. Finally, set $A_{\mathcal{F}} = A_{\mathcal{F}}(1)$. Hence, by Theorem 5.20 we obtain the following.

**Corollary 5.21.** Suppose that $\mathcal{F}$ is a field of characteristic 0 containing a primitive $p$th root of unity $\varepsilon$ and that $\mathcal{F}$ is a field of characteristic $c > 0$ that contains a primitive $p$th root of unity $\varepsilon_{\mathcal{F}}$. Then the decomposition matrix $D_{\mathcal{F}}$ of $\mathcal{H}_{p,n}^{\mathcal{F}}(q, \mathbf{Q}^{\vee\varepsilon})$ factorises as $D_{\mathcal{F}} = D_{\mathcal{F}}A_{\mathcal{F}}$.

**Remark 5.22.** This section does not really use the machinery of skew cellular algebras. Rather the results in this section follow from the fact that the skew cellular basis of Corollary 5.7 is defined over the ring $\mathbb{Z}[\varepsilon]$, which makes it easy to apply standard modular reduction arguments. The existence of adjustment matrices usually requires a delicate choice of modular system. The beauty of using KLR algebras is that we can work over $\mathbb{Z}[\varepsilon]$, which makes this result almost trivial.

**5.4. Graded simple modules.** Let $\mathcal{F}$ be a field. The algebra $\mathcal{R}_{p,n}^{\Lambda,F}$ is a skew cellular algebra by Corollary 5.7, so $\{D_{\boldsymbol{\lambda}}^{(k),F} \langle s \rangle | (\boldsymbol{\lambda}, k) \in \mathcal{P}_{\sigma,p,n,0}^{\ell} \text{ and } s \in \mathbb{Z}\}$ is a complete set of pairwise non-isomorphic graded simple modules by Theorem 2.17 (and Proposition 5.19). The aim of this section is to explicitly describe the set $\mathcal{P}_{\sigma,p,n,0}^{\ell}$.

By definition, $\mathcal{P}_{\sigma,p,n,0}^{\ell} = \{(\boldsymbol{\lambda}, k) \in \mathcal{P}_{\sigma,p,n}^{\ell} | D_{\boldsymbol{\lambda}}^{(k),F} \neq 0\}$, where $\mathcal{P}_{\sigma,p,n}^{\ell} = \mathcal{P}_{\sigma,n}^{\ell} \times \mathbb{Z}/\mathcal{O}_{\boldsymbol{\lambda}}\mathbb{Z}$ and $\mathcal{P}_{\sigma,n}^{\ell}$ is a fixed set of representatives in $\mathcal{P}_{n}^{\ell}$ under the action of $\sigma_{\mathcal{P}}$.

For $\boldsymbol{\lambda} \in \mathcal{P}_{n}^{\ell}$ let $C_{\boldsymbol{\lambda}}^{F}$ be the corresponding cell module and $D_{\boldsymbol{\lambda}}^{F} = C_{\boldsymbol{\lambda}}^{F}/\mathrm{rad} C_{\boldsymbol{\lambda}}^{F}$ be a graded simple $\mathcal{R}_{n}^{\Lambda,F}$-module (or zero). By Corollary 5.12, $\sigma_{n}^{\Lambda}$ is $\varepsilon$-splittable, so we can apply Theorem 2.48 to deduce the following:

**Lemma 5.23.** Suppose that $\mathcal{F}$ is a field. Then $\mathcal{P}_{\sigma,p,n,0}^{\ell} = \{(\boldsymbol{\lambda}, k) \in \mathcal{P}_{\sigma,p,n}^{\ell} | D_{\boldsymbol{\lambda}}^{F} \neq 0\}$.

The simple $\mathcal{R}_{n}^{\Lambda,F}$-modules have been independently classified by Bowman [3] and Kerschul [21]. To state their result we first need some new notation.

Let $\langle \cdot, \cdot \rangle$ be the total order on the set of nodes $\mathcal{N}_{n}^{\ell}$ where $A \langle \cdot, \cdot \rangle B$ if $\chi_{\rho}(A) < \chi_{\rho}(B)$. Extending notation from Section 3.4, for $i \in \mathcal{I}$ let

$$ \mathcal{A}_i(\boldsymbol{\lambda}) = \{A \in \mathcal{A}(\boldsymbol{\lambda}) | \mathrm{res}(A) = i\} \quad \text{and} \quad \mathcal{R}_i(\boldsymbol{\lambda}) = \{A \in \mathcal{R}(\boldsymbol{\lambda}) | \mathrm{res}(A) = i\} $$
be the sets of addable and removable $i$-nodes of $\lambda \in \mathcal{P}_n^\ell$. If $A \in \mathcal{A}_i(\lambda) \cup \mathcal{R}_i(\lambda)$ define

$$d_A(\lambda) = \#\{B \in \mathcal{A}_i(\lambda) | A <_x B\} - \#\{B \in \mathcal{R}_i(\lambda) | A <_x B\}.$$

Given a second node $C$ with $A <_x C$ set

$$d_A^C(\lambda) = \#\{B \in \mathcal{A}_i(\lambda) | A <_x B <_x C\} - \#\{B \in \mathcal{R}_i(\lambda) | A <_x B <_x C\}.$$

A good $i$-node of $\lambda$ is a removable $i$-node $A \in \mathcal{R}_i(\lambda)$ that is minimal node with respect to $<_x$ such that $d_A(\lambda) \le 0$ and $d_A^C(\lambda) < 0$ whenever $A <_x C$ and $C \in \mathcal{R}_i(\lambda)$.

**Definition 5.24.** If $n > 0$ then the set of $\rho$-Uglov $\ell$-partitions of $n$ is

$$\mathcal{U}_{\rho,n}^{\Lambda} = \{\lambda \in \mathcal{P}_n^{\ell} | \lambda = \mu \cup \{A\} \text{ where } A \text{ is a good } i\text{-node of } \lambda \text{ for some } i \in \mathcal{I}\},$$

where $\mathcal{U}_{\rho,0}^{\Lambda} = \mathcal{P}_0^{\ell}$.

For our particular choice of $\chi_\rho$-coordinate function, a special case of the results of Bowman [3] and Kerschl [21] is the following:

**Theorem 5.25** (Bowman [3, Theorem B] and Kerschl [21, Main Theorem]). Suppose that $F$ is a field. Then $\{D_\lambda(s) | \lambda \in \mathcal{U}_{\rho,n}^\Lambda \text{ and } s \in \mathbb{Z}\}$ is a complete set of pairwise non-isomorphic graded simple $\mathcal{R}_n^{\Lambda,F}$-modules.

Hence, by Lemma 5.23 we obtain:

**Corollary 5.26.** Suppose that $F$ is a field. Then $\mathcal{P}_{\sigma,p,n,0}^{\ell} = \{(\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell} | \lambda \in \mathcal{U}_{\rho,n}^{\Lambda}\}$. That is, $\{D_{\lambda}^{(k)}(s) | (\lambda, k) \in \mathcal{P}_{\sigma,p,n}^{\ell}, \lambda \in \mathcal{U}_{\rho,n}^{\Lambda} \text{ and } s \in \mathbb{Z}\}$ is a complete set of pairwise non-isomorphic graded simple $\mathcal{R}_{p,n}^{\Lambda,F}$-modules.

INDEX OF NOTATION

<table><thead><tr><th>Symbol</th><th>Description</th><th>Page</th></tr></thead><tbody><tr><td>α</td><td>An <i>I</i>-composition of <i>n</i></td><td>19</td></tr><tr><td>[α]</td><td>The σ-orbit of α</td><td>33</td></tr><tr><td>A<sub>n</sub><sup>ρ</sup></td><td>A rational Cherednik algebra</td><td>26</td></tr><tr><td>A<sub>n</sub><sup>ρ</sup></td><td>A block of A<sub>n</sub><sup>ρ</sup></td><td>26</td></tr><tr><td>A<sub>n</sub><sup>ρ</sup>(ω)</td><td>The ω-weight space of A<sub>n</sub><sup>ρ</sup></td><td>29</td></tr><tr><td>A<sub>n</sub><sup>ρ</sup></td><td>= ⊕<sub>β∈[α]</sub> A<sub>β</sub><sup>ρ</sup></td><td>33</td></tr><tr><td>B<sub>t</sub><sup>reg</sup></td><td>A regular diagram indexed by t</td><td>37</td></tr><tr><td>B<sub>t</sub><sup>sing</sup></td><td>A singular diagram indexed by t</td><td>37</td></tr><tr><td>B<sub>t</sub></td><td>The diagram B<sub>t</sub><sup>reg</sup> B<sub>t</sub><sup>sing</sup></td><td>38</td></tr><tr><td>B<sub>λ</sub><sup>[k]</sup></td><td>The diagram B<sub>λ</sub><sup>[k],sing</sup> B<sub>λ</sub><sup>[k],reg</sup></td><td>40</td></tr><tr><td>B<sub>λ,t</sub><sup>[k],reg</sup></td><td>The diagram B<sub>λ</sub><sup>[k],reg</sup> B<sub>t</sub><sup>reg</sup> ∈ A<sub>n</sub><sup>ρ</sup>(μ̂<sub>t</sub><sup>k</sup>, μ̂<sub>t</sub>)</td><td>42</td></tr><tr><td>C<sub>n</sub><sup>ℓ</sup></td><td>The set of <i>I</i>-compositions of <i>n</i></td><td>19</td></tr><tr><td>C<sub>st</sub></td><td>Cellular basis element of A<sub>n</sub><sup>ρ</sup></td><td>28</td></tr><tr><td>ε</td><td>A fixed <i>p</i>th root of unity</td><td>18</td></tr><tr><td>E<sub>ω</sub></td><td>An idempotent in A<sub>n</sub><sup>ρ</sup></td><td>29</td></tr><tr><td>H<sub>n</sub>(q, Q<sup>νε</sup>)</td><td>Hecke algebra of type G(ℓ, 1, <i>n</i>)</td><td>18</td></tr><tr><td>H<sub>p,n</sub>(q, Q<sup>νε</sup>)</td><td>Hecke algebra of type G(ℓ, <i>p</i>, <i>n</i>)</td><td>18</td></tr><tr><td>I</td><td>The vertex set {ε<sup>j</sup>q<sup>i</sup> | <i>i</i>, <i>j</i> ∈ <i>Z</i>} for the quiver Γ</td><td>19</td></tr><tr><td>κ</td><td>The multicharge κ = (ρ<sub>1</sub>, ..., ρ<sub>d</sub>)</td><td>21</td></tr><tr><td>λ</td><td>An ℓ-partition of <i>n</i></td><td>22</td></tr><tr><td>n, p, d = ℓ/p</td><td>The parameters for R<sub>p,n</sub><sup>Λ</sup></td><td>18</td></tr><tr><td>N<sub>n</sub><sup>ℓ</sup></td><td>The set of nodes</td><td>22</td></tr><tr><td>o<sub>λ</sub></td><td>The size of the orbit of λ ∈ P under the action of σ<sub>P</sub></td><td>8</td></tr><tr><td>P</td><td>A poset</td><td>3</td></tr><tr><td>P<sub>n</sub><sup>ℓ</sup></td><td>The set of ℓ-partitions of <i>n</i></td><td>22</td></tr><tr><td>P<sub>σ,n</sub><sup>ℓ</sup></td><td>A set of σ-orbit representatives in P<sub>n</sub><sup>ℓ</sup></td><td>38</td></tr></tbody></table>
<table><thead><tr><th>Symbol</th><th>Description</th><th>Page</th></tr></thead><tbody><tr><td><i>P</i><sup><i>l</i></sup><sub><i>σ,p,n</i></sub></td><td>The set <i>P</i><sup><i>l</i></sup><sub><i>σ,n</i></sub> × Z/<sub>σλ</sub>Z</td><td>45</td></tr><tr><td><i>P</i><sup><i>l</i></sup><sub><i>σ,p,n,0</i></sub></td><td>The set {⟨<i>λ</i>,<i>k</i>⟩ ∈ <i>P</i><sup><i>l</i></sup><sub><i>σ,p,n</i></sub> | <i>D</i><sub><i>λ</i></sub><sup>(<i>k</i>),<i>Q</i></sup> ≠ 0}</td><td>47</td></tr><tr><td><i>q</i></td><td>The Hecke parameter</td><td>18</td></tr><tr><td><i>Q</i><sup><i>√ε</i></sup></td><td>The cyclotomic parameters of <i>H</i><sub><i>n</i></sub>(<i>q</i>, <i>Q</i><sup><i>√ε</i></sup>)</td><td>18</td></tr><tr><td>res <i>γ</i></td><td>The residue of <i>γ</i> ∈ <i>N</i><sub><i>n</i></sub><sup><i>l</i></sup></td><td>24</td></tr><tr><td><i>R</i><sup>Λ</sup><sub><i>n</i></sub></td><td>The quiver Hecke algebra of type <i>G</i>(<i>l</i>, 1, <i>n</i>)</td><td>20</td></tr><tr><td><i>R</i><sup>Λ</sup><sub><i>α</i></sub></td><td>A block of <i>R</i><sup>Λ</sup><sub><i>n</i></sub></td><td>20</td></tr><tr><td><i>R</i><sup>Λ</sup><sub><i>p,n</i></sub></td><td>The quiver Hecke algebra of type <i>G</i>(<i>l</i>, <i>p</i>, <i>n</i>)</td><td>21</td></tr><tr><td><i>σ</i></td><td>An automorphism of <i>H</i><sub><i>n</i></sub>(<i>q</i>, <i>Q</i><sup><i>√ε</i></sup>) or <i>R</i><sup>Λ</sup><sub><i>n</i></sub></td><td>18</td></tr><tr><td><i>σ</i><sub><i>p</i></sub></td><td>A poset automorphism of <i>P</i></td><td>7</td></tr><tr><td><i>σ</i><sup>Λ</sup><sub><i>n</i></sub></td><td>An automorphism of <i>R</i><sup>Λ</sup><sub><i>n</i></sub></td><td>21</td></tr><tr><td><i>σ</i><sub><i>p</i></sub></td><td>An automorphism of <i>P</i><sup><i>l</i></sup><sub><i>n</i></sub></td><td>31</td></tr><tr><td><i>σ</i><sub>Std</sub></td><td>An automorphism of Std(<i>P</i><sup><i>l</i></sup><sub><i>n</i></sub>)</td><td>31</td></tr><tr><td>Std(<i>λ</i>)</td><td>The set of standard tableaux of shape <i>λ</i></td><td>27</td></tr><tr><td>Std<sub><i>σ</i></sub>(<i>λ</i>)</td><td>A set of <i>σ</i>-orbit representatives of standard tableaux of shape <i>λ</i></td><td>38</td></tr><tr><td><i>s</i>, <i>t</i></td><td>Standard tableaux</td><td>27</td></tr><tr><td><i>ν̄</i></td><td>A generalised partition</td><td>24</td></tr><tr><td><i>ν̄</i><sub><i>t</i></sub></td><td>A generalised partition constructed from <i>t</i></td><td>37</td></tr><tr><td><i>W</i><sub><i>ρ</i></sub></td><td>The set of (isotopy classes of) Webster diagrams</td><td>25</td></tr><tr><td><i>ω</i></td><td>The <i>l</i>-partition (1<sup><i>n</i></sup> | ... | 0) ∈ <i>P</i><sup><i>l</i></sup><sub><i>n</i></sub></td><td>27</td></tr><tr><td><i>x</i><sub><i>ρ</i></sub></td><td>The loading function <i>x</i><sub><i>ρ</i></sub>: <i>N</i><sub><i>n</i></sub><sup><i>l</i></sup> → Q</td><td>22</td></tr><tr><td><i>▷</i><sub><i>ρ</i></sub></td><td>The <i>ρ</i>-dominance order on <i>l</i>-partitions</td><td>24</td></tr></tbody></table>

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SCHOOL OF MATHEMATICAL AND STATISTICS, BEIJING INSTITUTE OF TECHNOLOGY, BEIJING, 100081, P.R. CHINA
*Email address: junhu404@bit.edu.cn*

SCHOOL OF MATHEMATICS AND STATISTICS, UNIVERSITY OF SYDNEY, NSW 2006, AUSTRALIA
*Email address: andrew.mathas@sydney.edu.au*

UNIV RENNES, CNRS, IRMAR - UMR 6625, F-35000 RENNES, FRANCE
*Email address: salim.rostam@ens-rennes.fr*