Residue-sequence idempotents and cyclotomic Hecke blocks

Work in the charged finite-$e$ field conventions of Affine and cyclotomic type-A Hecke algebras. By Polynomial spectra and central content sectors of finite cyclotomic Hecke algebras, $H_d^{\mathrm{\Lambda }}$ is finite-dimensional and each minimal polynomial of the polynomial generator $z_r=X_r$ (nondegenerate normalization) respectively $x_r$ (degenerate normalization) splits over the residues $c_i=q_H^i$ respectively $i\cdot 1_F$.

Residue-sequence idempotents. Let ${\epsilon }_r(i)$ be the canonical CRT primary idempotent of that generator at the spectral value $c_i$ associated with residue label $i$ (zero when that value is absent), and put $e(\nu )={\prod }_{r=1}^d{\epsilon }_r({\nu }_r)$ for a residue sequence $\nu =({\nu }_1,\dots ,{\nu }_d)\in (\mathbb{Z}\mathrm{/}e\mathbb{Z}{)}^d$. On every finite $H$-module the element $e(\nu )$ projects onto the joint generalized residue space, and these mutually orthogonal idempotents sum to $1$. At content $\alpha$ put $e_{\alpha }={\sum }_{\nu \in I_{\alpha }}e(\nu ),$ the sum over residue sequences of content $\alpha$; it is central, and $H_d^{\mathrm{\Lambda }}$ is the direct sum of its central ideals $H_{\alpha }^{\mathrm{\Lambda }}=e_{\alpha }{H}_d^{\mathrm{\Lambda }}=e_{\alpha }{H}_d^{\mathrm{\Lambda }}{e}_{\alpha }\mathrm{.}$ By Central content sectors are zero or primitive cyclotomic Hecke blocks, every nonzero $e_{\alpha }$ is primitive, so every nonzero $H_{\alpha }^{\mathrm{\Lambda }}$ is an actual block (Blocks of a finite-dimensional algebra via primitive central idempotents); this block assertion is proved there, not retained as an unused citation. The algebra basis and dimension convention is that of Basis and dimension of the cyclotomic Hecke algebra.

Level one. At level one, $\mathrm{\Lambda }={\mathrm{\Lambda }}_j$, the polynomial generators correspond under the explicit finite-Hecke inverse of Cyclotomic Hecke algebras of level one are finite Hecke algebras to $X_r=q_H^j(1+(q_H-1)L_r)\text{(nondegenerate)},x_r=j+L_r\text{(degenerate)},$ where $L_r$ is the ordinary additive Jucys–Murphy element. Thus on a standard tableau node of content $m$ their eigenvalues are $q_H^{j+m}$ respectively $j+m$, while $L_r$ itself has additive $q$-content $[m{]}_{q_H}$. The nonzero joint projectors are precisely the residue words of standard tableaux (The Hecke Jucys-Murphy action is dominance triangular), and grouping those words by content gives the block labels.

Boundary conventions. At rank zero the empty projector is $1$ and the unique block is $F$; at level zero and positive rank all projectors and blocks are zero. The projection, centrality, and block-decomposition assertions recorded here are exactly the proved content-sector statements of the linked suppliers; this definition introduces no new primitive-block classification of its own. It uses no choice principle: all spectra, projectors, and sums are finite.