Category O, KLR, and Hecke route conventions

This page uses the following route convention. Category O means the BGG category for a fixed finite-dimensional complex semisimple Lie algebra and a chosen positive system, as defined in The classical BGG category O. The Arakawa–Suzuki route is the degenerate affine route over $\mathbb{C}$: on $M\otimes ({\mathbb{C}}^N{)}^{\otimes d}$ use the operators $x_j$ and $s_i$ of Arakawa-Suzuki Casimir operators on tensor functors, with the verified right $H_d$-action of The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations. The source-normalized operators are $y_j=x_j-\frac{1}{N}{z}_M+\frac{N-1}{2};$ they are not new generators to be identified with a nondegenerate Hecke generator. Here $z_M$ is the central endomorphism specified in the cited definition. This convention asserts no extension of the cited hypotheses on $M$, origin, or tensor functor.

The finite cyclic type-A Hecke/KLR route uses a field $F$, a finite quantum characteristic $e\ge 2$, and its declared residue and cyclotomic conventions (Affine and cyclotomic type-A Hecke algebras). In the nondegenerate branch write the Hecke parameter as $\xi \mathrm{\ne }1$ and reserve $q$ for the grading and Laurent-polynomial variable. The blockwise identification is exactly The Brundan–Kleshchev blockwise Hecke–KLR isomorphism: write ${\xi }_{\mathrm{B}\mathrm{K}}$ for its Hecke parameter (denoted $q$ in that theorem), independent of the grading variable $q$. Its ${\xi }_{\mathrm{B}\mathrm{K}}=1$ branch requires $\mathrm{char}F=e$, while its ${\xi }_{\mathrm{B}\mathrm{K}}\mathrm{\ne }1$ branch is separate. For level-one Ariki statements, use only the regime in The nondegenerate root-of-unity level-one Hecke regime. In characteristic zero the canonical-basis and graded-decomposition conclusions are those of Ariki's canonical basis theorem in characteristic zero and Graded decomposition numbers equal canonical-basis coefficients in characteristic zero. In positive characteristic retain the adjustment factorization of Positive-characteristic decomposition polynomials factor through adjustment; no identity adjustment matrix is part of this route.

The negative-half KLR categorification and the cyclotomic highest-weight projective-$K_0$ theorem are distinct routes, respectively The KLR category categorifies the negative quantum half and The cyclotomic projective $K_0$ is the integral highest-weight module. Each keeps its own field, grading, weight and Axiom-of-Choice hypotheses (The Axiom of Choice). Likewise, the type-A Hecke cellular result The Murphy tableau elements form a cellular basis, the local type-A character theorem Soergel's character conjecture for type A, the singular spherical formula Singular category O multiplicities are parabolic Kazhdan–Lusztig values, and the regular-ambient parabolic antispherical formula Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values retain their distinct statements and ranges. This route convention introduces no new theorem, grading, equivalence, or cross-characteristic transfer.

The shifted-Yangian PBW result Ordered monomial PBW basis for the shifted Yangian is not a supplier for these Category-O, Hecke, or KLR conventions; the link records a scope boundary only.