The nondegenerate root-of-unity level-one Hecke regime

Parameters. Let $F$ be a field and $\xi \in F^{\times }$ with finite quantum characteristic $e=e(F,\xi )\in \{2,3,\dots \}$ (The quantum characteristic of an invertible Hecke parameter). We work in the NONDEGENERATE branch $\xi \mathrm{\ne }1$; then $\xi$ is a primitive $e$-th root of unity in $F$, and all statements on this page live in one of the two sub-branches: (C0) $\mathrm{char}F=0$, or (CP) $\mathrm{char}F=p>0$ with $p\nmid e$. The Hecke parameter is written $\xi$ here; it is the letter $q$ of def-affine-and-cyclotomic-type-a-hecke-algebras.

Hecke side. Fix the level-one fundamental weight ${\mathrm{\Lambda }}_0$ of the cyclic affine type-$A$ datum with all $d_i=1$ and charge $0$ (The cyclic affine type A Cartan datum and its residues). With the conventions of def-affine-and-cyclotomic-type-a-hecke-algebras, $H_d^{\mathrm{\Lambda }}=H_{F,\xi }(S_d)$ is the finite type-$A$ Hecke algebra $T_r^2=(\xi -1)T_r+\xi$ (prop-cyclotomic-hecke-algebras-of-level-one-are-finite-hecke-algebras), and $H_{\alpha }^{\mathrm{\Lambda }}=e_{\alpha }{H}_d^{\mathrm{\Lambda }}$ are its residue-content blocks (def-residue-sequence-idempotents-and-cyclotomic-hecke-blocks).

KLR side. Use the type-$A$ specialization $Q^A$ of def-type-a-cyclotomic-specialization-of-the-qg-klr-algebras with the same $e$ and the STANDARD homogeneous first-strand cyclotomic quotient $R_{\alpha }^{\mathrm{\Lambda }}$ of The cyclotomic KLR quotient at a dominant integral weight: the cyclotomic relation is $y_1^{{\delta }_{i_1,0}}e(i)=0$, induction appends on the right, and the grading is $\mathrm{deg}e(i)=0$, $\mathrm{deg}{y}_r=2$, $\mathrm{deg}{\psi }_{r}e(i)=-a_{i_r{i}_{r+1}}$ (The KLR grading and the diagrammatic multiplication). The transported grading on $H_{\alpha }^{\mathrm{\Lambda }}$ uses a FIXED admissible local normalising series (cor-cyclotomic-hecke-blocks-inherit-a-nontrivial-z-grading); a different series gives a graded-isomorphic transported algebra via ${\rho }^{\mathrm{\prime }}{\rho }^{-1}$, and the graded multiplicities studied below do not depend on the choice.

Grading and Fock variables. All grading shifts and all global-basis polynomials are in the variable $q$, $A=\mathbb{Z}[q,q^{-1}]$. Thus $\xi \in F^{\times }$ (the Hecke parameter) and $q$ (the grading variable) are INDEPENDENT data; the phrase "Ariki's normalization $\xi =q^2$" is retained only as the source dictionary of [Ariki 1996], whose Hecke parameters are powers of a square, and no numerical relation or specialization between $\xi$ and $q$ is imposed.

Fock side. Use the level-one symmetric model of The symmetric-group level-one residue and partition-label convention: charge $0$, node residues $c-r$, standard vectors $M_{\mu }$ for ALL partitions $\mu$, basic/restricted labels the $e$-restricted partitions $R{P}_{\alpha }=\{\lambda :{\lambda }_a-{\lambda }_{a+1}<e,\mathrm{cont}(\lambda )=\alpha \}$, empty component $B({\mathrm{\Lambda }}_0)$. For $\lambda \in RP$ the conjugate ${\lambda }^{\mathrm{\prime }}$ is $e$-regular. Write $P(\nu )$ for the independently constructed ordinary Dipper--James Specht module and $D_{\mathrm{D}\mathrm{J}}^{\nu }$ for its form quotient in thm-dj-bridge-ordinary-specht-is-the-signed-conjugate-murphy-cell. Define $\theta (T_r)=(\xi -1)-T_r=-\xi T_r^{-1}$, the involutive sign automorphism of The Hecke sign automorphism, pullback and contragredient dual. After converting right modules to left modules through the star anti-involution, the intrinsic restricted Murphy simple with label $\lambda$ is the $\theta$-pullback of $D_{\mathrm{D}\mathrm{J}}^{{\lambda }^{\mathrm{\prime }}}$, not the untwisted ordinary module. This is an ungraded convention dictionary. The pullback can move blocks: on the multiplicative Jucys--Murphy elements $\theta (X_j)=X_j^{-1}$, so charge-zero residue content is sent from $\alpha$ to $-\alpha$. Thus the ordinary module before pullback has the conjugate content; no claim identifies it untwisted inside the same block. At $\xi =1$ the separate specialization theorem cor-dj-bridge-specialization-and-composition-multiplicity-dictionary identifies $\theta$ with sign tensoring over every field; $\xi =1$ remains outside the present nondegenerate regime.

No choice principle is used.