Arbitrary-level cyclotomic Hecke, Murphy cell and multipartition Schur modules

Let $R$ be a commutative ring, let $n\ge 0$ and $\mathrm{\ell }\ge 1$, and let $z,U_1,\dots ,U_{\mathrm{\ell }}$ be units of $R$ (Algebras over a commutative ring, central structure maps, and algebra homomorphisms). The arbitrary-level cyclotomic Hecke algebra $H_n=H_n(R;z;U_1,\dots ,U_{\mathrm{\ell }})$ is the $R$-algebra presented by invertible commuting elements $X_1,\dots ,X_n$ and generators $T_1,\dots ,T_{n-1}$ subject to the relations $(T_i-z)(T_i+1)=0,T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},T_i{T}_j=T_j{T}_i(\mathrm{\mid }i-j\mathrm{\mid }>1),$ $T_i{X}_i{T}_i=z{X}_{i+1},X_j{T}_i=T_i{X}_j(j\mathrm{\notin }\{i,i+1\}),{\prod }_{a=1}^{\mathrm{\ell }}(X_1-U_a)=0.$ For $n=0$ set $H_0=R$. The grading variable of this page is $v$, not the unit $z$: the two are identified only where a statement says so explicitly.

Multipartitions and Murphy data. An $\mathrm{\ell }$-multipartition of $n$ is an ordered tuple $\lambda =({\lambda }^{(1)},\dots ,{\lambda }^{(\mathrm{\ell })})$ of partitions with ${\sum }_b\mathrm{\mid }{\lambda }^{(b)}\mathrm{\mid }=n$ (Tableaux and standard tableaux). Multipartitions are partially ordered by dominance, $\lambda ⊵\mu$ meaning that ${\sum }_{b<k}\mathrm{\mid }{\lambda }^{(b)}\mathrm{\mid }+{\sum }_{j\le m}{\lambda }_j^{(k)}\ge {\sum }_{b<k}\mathrm{\mid }{\mu }^{(b)}\mathrm{\mid }+{\sum }_{j\le m}{\mu }_j^{(k)}$ for all $k,m$. Fill the diagram $t^{\lambda }$ with $1,\dots ,n$ in component order, then row order, then column order. Put $a_k:={\sum }_{b<k}\mathrm{\mid }{\lambda }^{(b)}\mathrm{\mid }$, $u_{\lambda }:={\prod }_{k=1}^{\mathrm{\ell }}{\prod }_{j\le a_k}(X_j-U_k)$, $x_{\lambda }:={\sum }_{w\in S_{\lambda }}{T}_w$, and $m_{\lambda }:=x_{\lambda }{u}_{\lambda }$, where $S_{\lambda }$ is the row subgroup: the subgroup of $S_n$ stabilizing the rows of $t^{\lambda }$ (The finite symmetric group $S_n$, one-line notation, and cycle notation, The generic type-A Hecke algebra). Write $M_{\lambda }:=m_{\lambda }{H}_n$ for the right $H_n$-submodule generated by $m_{\lambda }$ (Unital left and right modules over a ring; unqualified module means left module). The presentation involution $\ast$ of $H_n$ fixes every $X_j$ and every $T_i$ and reverses products. For standard tableaux $s,t$ of shape $\lambda$ put $m_{st}:=T_{d(s)}^{\ast }{m}_{\lambda }{T}_{d(t)}$, where the right tableau action is $t\cdot w:=w^{-1}\cdot t$ (the left action relabels entries), and $d(s)$ is determined by $t^{\lambda }\cdot d(s)=s$. Denote by $H^{>\lambda }$ the span of the higher Murphy shapes once their basis and cell law is established by The arbitrary-level cyclotomic Hecke Murphy basis is cellular ↗, and by $S_{\lambda }$ the right cell (Specht) module with basis $m_t$ and the action given by that cell law; its cellular form defines $D_{\lambda }:=S_{\lambda }\mathrm{/}\mathrm{rad}{S}_{\lambda }$, which may vanish.

Types. The types used below are exactly all $\mathrm{\ell }$-multipartitions of $n$, including tuples with empty components — not all multicompositions. A semistandard tableau $S$ has entries the labels $(i,k)$ (row $i$ in component $k$) ordered by component first and then by row, weakly increasing along rows, strictly increasing down columns, and a label of component $a$ may occur only in physical component $b\le a$ (Semistandard tableaux and Kostka numbers); the type of $S$ is the multipartition counting each label with the multiplicity of the corresponding row. For standard $s$ let $\mu (s)$ be the tableau obtained from $s$ by replacing each entry by its row-and-component in $t^{\mu }$. Put $m_{(S,t)}:={\sum }_{s:\mu (s)=S}{m}_{st},m_{(S,T)}:={\sum }_{s,t:\mu (s)=S,\nu (t)=T}{m}_{st}\mathrm{.}$ Let $M:={⨁}_{\lambda \text{ multipartition of }n}{M}_{\lambda },C_n:={\mathrm{End}}_{H_n}(M),$ with ordinary composition (The direct sum of an indexed family of modules). For $S,T$ of shape $\lambda$ and types $\mu ,\nu$, the map ${\phi }_{ST}$ — once justified by Cyclotomic semistandard sums give the actual Schur endomorphism basis ↗ — sends $m_{\alpha }h$ to ${\delta }_{\alpha ,\nu }{m}_{(S,T)}h$. Its right cell module ${\mathrm{\Delta }}_{\lambda }$ has basis $v_T$ over the semistandard $T$ of multipartition types, with contravariant cell form and quotient $L_{\lambda }$. Let $T^{\lambda }$ be the unique tableau of shape and type $\lambda$, and let $e_{\lambda }$ be the identity projection onto $M_{\lambda }$. Put $\omega :=(\mathrm{\varnothing },\dots ,\mathrm{\varnothing },(1^n))$ and $e:=e_{\omega }$; then $m_{\omega }=1$, so $M_{\omega }=H_n$, and $H_n$ is an actual direct summand of $M$. The weight projections used by the Jantzen theory below are the $e_{\nu }$ with $\nu$ a multipartition. Write $K_{(\lambda ,\nu )}:=\mathrm{\#}{\mathrm{SStd}}_{\nu }(\lambda )$ for the number of semistandard tableaux of shape $\lambda$ and type $\nu$.

Boundary of the definition. All asserted module, form and map laws are discharged by the linked local suppliers The arbitrary-level cyclotomic Hecke Murphy basis is cellular ↗, Cyclotomic semistandard sums give the actual Schur endomorphism basis ↗ and Cyclotomic Schur standards and corner functor control block linkage ↗; the definition itself claims no proof of them and no primitive residue-content classification. No choice principle is used: only finite bases, finite subgroup and coset decompositions, finite filtrations and explicit polynomial identities occur.