Cyclotomic Schur standards and corner functor control block linkage

Centers and blocks of an endomorphism algebra containing the regular module. Let $M$ be a finite right $H$-module containing $H$ as a direct summand with projection $e$, and $C={\mathrm{End}}_H(M)$. Then $eCe={\mathrm{End}}_H(H)\cong H$ by left multiplication, and $Ce\cong {\mathrm{Hom}}_H(H,M)\cong M$ by evaluation at $1$. If $c\in Z(C)$, then $c$ restricts on the $H$-summand to left multiplication by some $h$; for each $m\in M$ the $H$-map $f_m:H\to M$, $a\mapsto ma$, is the off-diagonal block $j{f}_m\pi \in C$ through $H$, and commuting it with $c$ and evaluating at $1$ gives $c(m)=mh$. Since $c$ is $H$-linear, $c(ab)=c(a)b$ for $a,b\in H$ gives $abh=ahb$, so with $a=1$ we get $hb=bh$ and $h\in Z(H)$. Conversely every $h\in Z(H)$ defines the $H$-linear endomorphism $m\mapsto mh$, which commutes with every $H$-endomorphism of $M$, hence is central in $C$. These constructions are inverse unital algebra maps, so $Z(C)\cong Z(H)$ and primitive central idempotents correspond bijectively ([F4]). For $M={⨁}_{\lambda }{M}_{\lambda }$ we have $M_{\omega }=H_n$, so this applies with $e=e_{\omega }$.

Schur standards and nonzero heads. By [F2] the right cell module ${\mathrm{\Delta }}_{\lambda }$ of $C_n$ has basis the $v_T$ over semistandard $T$ of shape $\lambda$. The unique tableau $T^{\lambda }$ of shape and type $\lambda$ gives ${\phi }_{(T^{\lambda },T^{\lambda })}=e_{\lambda }$, the identity projection onto $M_{\lambda }$, an actual idempotent; evaluating the cell form at $v_{T^{\lambda }}$ gives $⟨v_{T^{\lambda }},v_{T^{\lambda }}⟩=1$, so every standard is nonzero with nonzero form. Applying the finite cellular-head theory [F3] to $C_n$ (finite-dimensional and cellular over $F$ by [F2]) shows every head $L_{\lambda }={\mathrm{\Delta }}_{\lambda }\mathrm{/}\mathrm{r}\mathrm{a}\mathrm{d}{\mathrm{\Delta }}_{\lambda }$ is nonzero, absolutely simple, the $L_{\lambda }$ are pairwise distinct and exhaust the simple $C_n$-modules, and $[{\mathrm{\Delta }}_{\lambda }:L_{\mu }]\mathrm{\ne }0$ implies $\lambda ⊵\mu$ with $[{\mathrm{\Delta }}_{\lambda }:L_{\lambda }]=1$.

Split quasiheredity by an explicit heredity chain. Order the multipartitions in a linear extension of dominance and remove the maximal remaining cell layers successively; in the relevant quotient algebra $B$ the next layer $J$ has basis $c_{st}$ with a distinguished index $u$ whose Gram value is $G(u,u)=1$. Then $e=c_{uu}$ is idempotent, $c_{su}{c}_{ut}=c_{st}$, and the multiplication map $Be{\otimes }_{F}eB\to J$ is an isomorphism on these basis elements; also $eBe=Fe$, since the cell law and maximality of $J$ show every $ebe$ is a scalar multiple of $e$. Hence $Be$ is projective, $J$ is a direct sum of copies of $Be$, and $e\mathrm{r}\mathrm{a}\mathrm{d}(B)e=0$ because the corner is a field. Thus $J$ is a split heredity ideal, and iterating gives a split heredity chain whose standards are the cell modules ${\mathrm{\Delta }}_{\lambda }$; so $C_n$ is split quasihereditary over $F$.

Weight dimensions and the unitriangular Kostka matrix. The weight projections $e_{\nu }={\phi }_{(T^{\nu },T^{\nu })}$ are self-adjoint because ${\phi }_{ST}^{\ast }={\phi }_{TS}$ ([F2]), and the cell action gives $v_T{e}_{\nu }={\delta }_{\mathrm{type}(T),\nu }{v}_T$, so $\dim {\mathrm{\Delta }}_{\lambda }{e}_{\nu }$ is the number $K_{(\lambda ,\nu )}$ of semistandard tableaux of shape $\lambda$ and type $\nu$; this already shows the standards and their forms are defined over the universal ring, free on the $v_T$. A semistandard tableau of shape $\lambda$ and type $\nu$ forces $\lambda ⊵\nu$: for a component $k$ and row cutoff $r$, labels of type components below $k$ cannot occupy physical components at least $k$, labels of type component $k$ with row at most $r$ cannot occupy physical components greater than $k$ and occupy at most $r$ nodes per column of physical component $k$ by strict column increase, so the count ${\sum }_{a<k}\mathrm{\mid }{\nu }^{(a)}\mathrm{\mid }+{\sum }_{i\le r}{\nu }_i^{(k)}$ is at most ${\sum }_{a<k}\mathrm{\mid }{\lambda }^{(a)}\mathrm{\mid }+{\sum }_{i\le r}{\lambda }_i^{(k)}$, which is exactly the multipartition dominance inequality; and $T^{\lambda }$ is the unique tableau of shape and type $\lambda$, so the Kostka matrix is unitriangular.

The standard basis of $G_0$ and block purity. The matrix of standard multiplicities is unitriangular with diagonal $1$ by step 1.2, so inverting it by backward substitution shows that the classes $[{\mathrm{\Delta }}_{\lambda }]$ form a $\mathbb{Z}$-basis of $G_0(C_n\text{-mod})$ ([F5]). Projecting to blocks is exact. Every ${\mathrm{\Delta }}_{\lambda }$ is nonzero and block-pure by [F3], so if a class supported in one block equals ${\sum }_{\lambda }{a}_{\lambda }[{\mathrm{\Delta }}_{\lambda }]$, then every coefficient of a standard lying in another block vanishes by this basis independence; this is the block-separation input used by the Jantzen equivalence.

The corner functor and the radical comparison. Put $e=e_{\omega }$; then $eCe\cong H_n$ by step 1.1, and $V\mapsto Ve$ is exact. Semistandard tableaux of type $\omega$ are canonically standard multitableaux: replacing the label $(r,\mathrm{\ell })$ by $r$ is a bijection onto the standard tableaux, and the surviving standard vectors $v_t$ of ${\mathrm{\Delta }}_{\lambda }e$ carry the same right-$H_n$ action as the Murphy vectors $m_t$ of $S_{\lambda }$ by the cell multiplication [F2]; this is an explicit isomorphism ${\mathrm{\Delta }}_{\lambda }e\cong S_{\lambda }$. The forms also agree: in $C_n$, $⟨v_s,v_t⟩e_{\lambda }\equiv {\phi }_{(T^{\lambda },s)}{\phi }_{(t,T^{\lambda })}$ modulo higher $C_n$-shapes, and evaluating the two factors at $m_{\lambda }$ gives the product $m_{(t^{\lambda },s)}{m}_{(t,t^{\lambda })}$, whose coefficient of $m_{\lambda }$ modulo higher $H_n$-shapes is exactly the Specht form entry; a higher Schur map evaluates to a sum of higher Murphy shapes by its explicit definition, so no shape error is lost. Since $e^{\ast }=e$, the $e$ and $1-e$ parts of ${\mathrm{\Delta }}_{\lambda }$ are orthogonal: if $x$ pairs to zero against ${\mathrm{\Delta }}_{\lambda }e$, then $⟨x,y(1-e)⟩=⟨x(1-e),y⟩=0$, so $\mathrm{r}\mathrm{a}\mathrm{d}({\mathrm{\Delta }}_{\lambda })e=\mathrm{r}\mathrm{a}\mathrm{d}({\mathrm{\Delta }}_{\lambda })\cap {\mathrm{\Delta }}_{\lambda }e=\mathrm{r}\mathrm{a}\mathrm{d}({\mathrm{\Delta }}_{\lambda }e)$. Hence $L_{\lambda }e\cong D_{\lambda }$ for every $\lambda$, with $D_{\lambda }$ allowed to be zero. Applying the exact functor to a composition series of ${\mathrm{\Delta }}_{\lambda }$, each simple factor $L_{\mu }$ becomes $D_{\mu }$ or $0$, and the nonzero $D_{\mu }$ are pairwise distinct and exhaustive by [F3]; hence $[{\mathrm{\Delta }}_{\lambda }:L_{\mu }]=[S_{\lambda }:D_{\mu }]$ for every $\mu$ with $D_{\mu }\mathrm{\ne }0$, with no simple-label bijection through $e$ assumed.

Typed Schur/Specht block linkage. The center isomorphism of step 1.1 sends a central block idempotent $b$ of $H_n$ to the same right-action projection on $M$, hence to the corresponding central idempotent $b_C$ of $C_n$, and on the corner $V{b}_{C}e=(Ve)b$. Every ${\mathrm{\Delta }}_{\lambda }$ is nonzero and block-pure by [F3], and ${\mathrm{\Delta }}_{\lambda }e\cong S_{\lambda }$ is nonzero because $S_{\lambda }$ has its Murphy standard-tableau basis; consequently the block supporting ${\mathrm{\Delta }}_{\lambda }$ corresponds to exactly the block supporting $S_{\lambda }$. Therefore ${\mathrm{\Delta }}_{\lambda },{\mathrm{\Delta }}_{\mu }$ share a $C_n$-block if and only if $S_{\lambda },S_{\mu }$ share an $H_n$-block, and [F3] describes both sides as linkage by chains of cell modules with neighbouring terms sharing a simple composition factor; step 2.2 supplies the multiplicity transfers when a nonzero simple is needed. No equality-of-residue-contents assertion is used.