Cyclotomic Hecke blocks inherit $\mathbb{Z}$-gradings

Transport of the grading. Fix a choice of admissible local normalising series, hence the isomorphism $\rho$ of [F1]. Declare each image of a homogeneous KLR generator to be homogeneous of the same degree; the images generate $H_{\alpha }^{\mathrm{\Lambda }}$ and the KLR relations of [F2] are homogeneous, so the resulting degree assignment is well defined. Products retain degree addition because $\rho$ is an algebra map, so the degree summands are exactly the transported homogeneous summands. Since the generator images span the algebra, their assigned degrees determine every degree summand and give uniqueness of the grading for that fixed choice, with $\mathrm{deg}e(i)=0$, $\mathrm{deg}{y}_r=2$ and $\mathrm{deg}{\psi }_{r}e(i)=-a_{i_r,i_{r+1}}$.

Independence of the series choice. If ${\rho }^{\mathrm{\prime }}$ is the isomorphism attached to another admissible series, then ${\rho }^{\mathrm{\prime }}\circ {\rho }^{-1}$ sends each homogeneous native generator image of the first choice to the corresponding new image and hence is graded by construction; the two transported algebras are graded-isomorphic, while the underlying identity map on $H_{\alpha }^{\mathrm{\Lambda }}$ need not be graded because the two series differ by units that need not be homogeneous of degree zero.

Triviality is not excluded. Assigning degree zero to every ordinary Hecke generator makes all defining relations homogeneous, so the degree-zero assignment is a legitimate grading of the ordinary Hecke presentation and no impossibility statement is asserted. A one-dimensional sector has only degree zero, because its unit has degree zero and it equals $F$; in particular the one-dimensional level-one rank-two $e=3$ blocks are trivially graded.

A concrete nontrivial witness. Consider level one, $e=2$, rank two. Of the four colour words only $(0,1)$ survives: the first-strand exponent zero kills $(1,0)$, and on $(0,0)$ the equal-colour mixed identities give ${\psi }_1{y}_2=1$ together with ${\psi }_1^2=0$, hence ${\psi }_1=0$ after left multiplication by ${\psi }_1$, giving $1=0$; so the equal-colour sector is zero, with $y_1=0$ and the crossing zero. The quadratic double-edge relation gives $y_2^2=0$. Conversely $e(0,1)=1$, $y_2=t$, $y_1={\psi }_1=0$ in $F[t]\mathrm{/}(t^2)$ satisfy all rank-two relations, so that sector is $F[t]\mathrm{/}(t^2)$ with $\mathrm{deg}t=2$, and its Brundan–Kleshchev image supplies an actual nontrivial transported grading, including in the characteristic-two degeneration. This is a witness that the construction can be nontrivial, not a claim about all sectors.

Conclusion. Each fixed admissible series choice transports the KLR homogeneous decomposition to a $\mathbb{Z}$-grading of $H_{\alpha }^{\mathrm{\Lambda }}$, unique for the induced generator degrees; different choices give graded-isomorphic transported algebras; and the grading may be nontrivial on some blocks while trivial on others, with the ordinary presentation retaining its degree-zero assignment.