The Brundan–Kleshchev blockwise Hecke–KLR isomorphism

The mutually inverse early maps. By [F3] the local KLR generators of the nonzero sector satisfy the KLR relations, so there is a unique algebra homomorphism $\rho \text{\hspace{0.17em}⁣}:R_{\alpha }^{\mathrm{\Lambda }}\to H_{\alpha }^{\mathrm{\Lambda }}$ sending $e(i)\mapsto e(i)$, $y_r\mapsto y_r$ and ${\psi }_r\mapsto {\psi }_r$; by [F4] the Hecke generators $X_r,T_r$ (respectively $x_r,s_r$) of the sector satisfy the affine Hecke and cyclotomic relations, so there is a unique homomorphism $\sigma \text{\hspace{0.17em}⁣}:H_{\alpha }^{\mathrm{\Lambda }}\to R_{\alpha }^{\mathrm{\Lambda }}$ with $\sigma (e(i))=e(i)$ for $i\in I_{\alpha }$. The early central sector isomorphism [F2] proves, by explicit generator composites that are literally the identity, that $\rho$ and $\sigma$ are mutually inverse over $F$ for every content, including the zero sectors, with $H_{\alpha }^{\mathrm{\Lambda }}=0$ if and only if $R_{\alpha }^{\mathrm{\Lambda }}=0$. No algebraic closure, root extraction, dimension argument or primitive-block premise was used to construct these maps.

Primitivity of the sector. By [F1] every nonzero $e_{\alpha }$ is primitive and every nonzero $H_{\alpha }^{\mathrm{\Lambda }}$ is an actual block. Hence for the content $\alpha$ of the Statement the sector $H_{\alpha }^{\mathrm{\Lambda }}$ is a block of $H_d^{\mathrm{\Lambda }}$, and the block terminology in the Statement is justified rather than assumed.

The blockwise isomorphism and its normalization. Restricting the mutually inverse maps of step 1.1 to the nonzero sector $H_{\alpha }^{\mathrm{\Lambda }}$ gives mutually inverse unital $F$-algebra isomorphisms $\rho \text{\hspace{0.17em}⁣}:R_{\alpha }^{\mathrm{\Lambda }}\to H_{\alpha }^{\mathrm{\Lambda }}$ and $\sigma \text{\hspace{0.17em}⁣}:H_{\alpha }^{\mathrm{\Lambda }}\to R_{\alpha }^{\mathrm{\Lambda }}$. The defining formulas of $\rho$ give $\rho (e(i))=e(i)$ and the stated dot normalization $\rho (y_{r}e(i))=(1-q^{-i_r}{X}_r)e(i)$ in the nondegenerate case and $\rho (y_{r}e(i))=(x_r-i_r)e(i)$ in the degenerate case, matching the dot images of [F2]. Because $R_{\alpha }^{\mathrm{\Lambda }}$ is the algebra generated by $\{e(i)\}\cup \{y_r\}\cup \{{\psi }_r\}$ subject only to the defining KLR relations and $\rho$ is an isomorphism, $H_{\alpha }^{\mathrm{\Lambda }}$ is generated by the images of those generators subject only to those relations. The degenerate and nondegenerate cases were proved separately by [F2]–[F4] and both hold in finite quantum characteristic $e$, with no characteristic-zero or level-one restriction.