Type-A Hecke Specht modules are in the same block iff their labels have the same e-core

Central idempotents of $H$ from blocks of $S$. For a block $C$ of $S$ with primitive central idempotent ${ϵ}_C$ put ${\widetilde{ϵ}}_C:={ϵ}_{C}e=e{ϵ}_{C}e\in H$. This element is a central idempotent of $H$: it is idempotent because ${ϵ}_C$ is, and for $x=ese\in H$ centrality of ${ϵ}_C$ gives $x{\widetilde{ϵ}}_C=es{ϵ}_{C}e=e{ϵ}_{C}se={\widetilde{ϵ}}_{C}x$; it is nonzero because it acts as the identity on the nonzero module $W_F^{\lambda }e=S_{\mathrm{M}}^{\lambda }$ for any $\lambda$ with $W_F^{\lambda }\in C$, namely $S_{\mathrm{M}}^{\lambda }{\widetilde{ϵ}}_C=W_F^{\lambda }e{ϵ}_{C}e=W_F^{\lambda }{ϵ}_{C}e=S_{\mathrm{M}}^{\lambda }$. Moreover ${\widetilde{ϵ}}_C{\widetilde{ϵ}}_{C^{\mathrm{\prime }}}=e{ϵ}_C{ϵ}_{C^{\mathrm{\prime }}}e=0$ for $C\mathrm{\ne }{C}^{\mathrm{\prime }}$ and ${\sum }_C{\widetilde{ϵ}}_C=e{\sum }_C{ϵ}_{C}e=e^2=1_H$.

Number of q-Schur blocks is at least the number of Hecke blocks. Let $H={⨁}_i{B}_i$ be the block decomposition of $H$ with block identities $b_i$. Since $M={⨁}_{\nu }{M}^{\nu }$ and $H=M^{(1^n)}$ is a direct summand, $M={⨁}_{i}M{b}_i$ with each $M{b}_i\mathrm{\ne }0$ (it contains $H{b}_i=B_i$), and ${\mathrm{H}\mathrm{o}\mathrm{m}}_H(M{b}_i,M{b}_j)=0$ for $i\mathrm{\ne }j$, because a map $f$ satisfies $f(x)=f(x{b}_i)=f(x)b_i\in M{b}_j{b}_i=0$. Hence $S={\mathrm{E}\mathrm{n}\mathrm{d}}_H(M)={⨁}_{i,j}{\mathrm{H}\mathrm{o}\mathrm{m}}_H(M{b}_i,M{b}_j)={\prod }_i{\mathrm{E}\mathrm{n}\mathrm{d}}_H(M{b}_i)$ is a product of the nonzero algebras ${\mathrm{E}\mathrm{n}\mathrm{d}}_H(M{b}_i)$, and a product of nonzero algebras has at least one block per factor; so $\mathrm{\#}\{\text{blocks of }S\}\ge \mathrm{\#}\{\text{blocks of }H\}$.

Number of Hecke blocks is at least the number of q-Schur blocks. Decompose each nonzero central idempotent ${\widetilde{ϵ}}_C$ of step 1.1 into primitive central idempotents of $H$ using the unique decomposition of $1_H$: every primitive central idempotent ${\delta }_D$ of $H$ with ${\delta }_D{\widetilde{ϵ}}_C={\delta }_D$ belongs to exactly one ${\widetilde{ϵ}}_C$, because ${\delta }_D={\delta }_D\cdot 1_H={\sum }_C{\delta }_D{\widetilde{ϵ}}_C$ and pairwise orthogonality of the ${\delta }_D$ forces ${\delta }_D{\widetilde{ϵ}}_C\in \{0,{\delta }_D\}$. Since each ${\widetilde{ϵ}}_C\mathrm{\ne }0$ has at least one such ${\delta }_D$, the blocks $D$ of $H$ are partitioned into nonempty supports over the blocks $C$ of $S$, and therefore $\mathrm{\#}\{\text{blocks of }H\}\ge \mathrm{\#}\{\text{blocks of }S\}$.

A bijection of block sets, and the core criterion. By steps 2.1 and 1.2 the two block counts are equal, so each ${\widetilde{ϵ}}_C$ of step 1.1 has exactly one primitive summand and the map $C\mapsto {\widetilde{ϵ}}_C$ is a bijection from the blocks of $S$ onto the blocks of $H$. Since $S_{\mathrm{M}}^{\lambda }=W_F^{\lambda }e$ and $S_{\mathrm{M}}^{\mu }=W_F^{\mu }e$, the modules $S_{\mathrm{M}}^{\lambda }$ and $S_{\mathrm{M}}^{\mu }$ lie in the same Hecke block if and only if $W_F^{\lambda }$ and $W_F^{\mu }$ lie in the same q-Schur block: indeed the block of $S_{\mathrm{M}}^{\lambda }$ is the image under the bijection of the block of $W_F^{\lambda }$. By the block theorem [F4] the latter holds if and only if $C_e(\lambda )=C_e(\mu )$, which proves the q-Schur-normalised statement.

Ordinary normalisation and semisimplicity. By [F5], $S^{\lambda }$ and $S^{\mu }$ lie in the same block exactly when $S_{\mathrm{M}}^{{\lambda }^{\mathrm{\prime }}}$ and $S_{\mathrm{M}}^{{\mu }^{\mathrm{\prime }}}$ do. Step 3.1 makes this equivalent to $C_e({\lambda }^{\mathrm{\prime }})=C_e({\mu }^{\mathrm{\prime }})$, and the identity $C_e({\nu }^{\mathrm{\prime }})=C_e(\nu {)}^{\mathrm{\prime }}$ makes it equivalent to $C_e(\lambda )=C_e(\mu )$. This identity follows by conjugating every removable border strip in a core-removal sequence; the terminal core also conjugates, by independence of removal order. For $e=\mathrm{\infty }$, $S$ is semisimple by [F4]. Its corner $eSe$ is semisimple: by Wedderburn–Artin theorem for semisimple rings, in the Wedderburn decomposition of $S$, each idempotent selects a subspace in each matrix factor, and its corner is a product of matrix algebras over the same division rings. Thus $H\cong eSe$ is semisimple; the block bijection separates the partition labels. This proves all assertions.