Category O, KLR, and Hecke Categorification Route

This page is a cross-category route through existing theorem homes. It adds one route-convention definition and makes no duplicate theorem claim. The companion contains finite calculations that check the parameter boundaries: category-o-klr-and-hecke-categorification-route-examples.

Reading prerequisites

The plan lists algebraic-representation-and-tensor-symmetry-map and ariki-categorification-and-graded-decomposition-numbers as page-level prerequisites. The Ariki page is published. The algebraic gateway page is currently a draft; its actual authored interface and the remaining reconciliation are recorded in the batch evidence. This page uses the published theorem and definition carriers below directly and does not quote an unreviewed excerpt from that draft page.

Route map

Category O and the degenerate affine Hecke algebra

For ${\mathfrak{g}\mathfrak{l}}_N(\mathbb{C})$ and $M\otimes ({\mathbb{C}}^N{)}^{\otimes d}$, the Casimir and place-permutation operators give the degenerate affine Hecke action in Arakawa-Suzuki Casimir operators on tensor functors and The Casimir and place-permutation operators satisfy the degenerate affine Hecke relations. This route keeps the two normalizations visible: the relation theorem uses $x_j,s_i$, while the Arakawa–Suzuki source uses $y_j=x_j-z_M\mathrm{/}N+(N-1)\mathrm{/}2$.

The higher-level Schur–Weyl results distinguish the raw double centralizer, the Morita equivalence at an admissible origin, and the stable-range equality of the Hecke image with the full cyclotomic algebra. Outside the stable range the admissible-origin image is a central corner. These are separate clauses of The higher-level Schur-Weyl double centralizer and Schur quotient and The higher-level Schur-Weyl endomorphism isomorphism; a centralizer statement alone does not supply Morita equivalence.

For an admissible origin, the Category-O residue-translation comparison is Category-O translation intertwines residue induction and restriction. Its image residue functors agree with the full cyclotomic functors only when the selected algebras are faithfully realized, or with the corresponding central cuts when the selected contents are fully supported. The result is ungraded and asserts no Koszul or parity lift.

For an admissible origin, the theorem's natural isomorphisms give the functor square for the image-algebra residue functors. Here $B_d$ denotes the tensor-image algebra ${\mathrm{End}}_{\mathfrak{g}}(T_d{)}^{\mathrm{o}\mathrm{p}}$. Full support identifies the selected central cuts; it does not identify $B_d$ with the whole cyclotomic algebra. That whole-algebra equality is the separate stable-range conclusion above.

\mathcal O_\alpha \arrow[r,"\pi_\alpha"] \arrow[d,"f_i^{\mathcal O}"'] &
B_\alpha\text{-mod} \arrow[d,"f_i^B"] \\
\mathcal O_{\alpha+\alpha_i} \arrow[r,"\pi_{\alpha+\alpha_i}"'] &
B_{\alpha+\alpha_i}\text{-mod}

The horizontal arrows are the Schur quotient functors, and the vertical arrows are the O-side and image-algebra residue functors. The commutation means the natural isomorphisms $f_i^B\pi \cong \pi f_i^{\mathcal{O}}$, not literal equality of functor expressions. When full cyclotomic functors are used, the support conditions just stated are part of the diagram's hypotheses.

Finite cyclic type-A Hecke and KLR blocks

The blockwise bridge is The Brundan–Kleshchev blockwise Hecke–KLR isomorphism under finite quantum characteristic $e\ge 2$. Write ${\xi }_{\mathrm{B}\mathrm{K}}$ for its Hecke parameter (denoted $q$ in that theorem), independent of the grading variable $q$. Its nondegenerate case has ${\xi }_{\mathrm{B}\mathrm{K}}\mathrm{\ne }1$; its degenerate case has ${\xi }_{\mathrm{B}\mathrm{K}}=1$ and requires $\mathrm{char}F=e$. The Category-O route over $\mathbb{C}$ is degenerate affine Hecke theory and must not be confused with this positive-characteristic ${\xi }_{\mathrm{B}\mathrm{K}}=1$ cyclotomic branch. In the nondegenerate level-one route, write the Hecke parameter $\xi$ and keep the grading variable $q$ independent, as fixed by The nondegenerate root-of-unity level-one Hecke regime.

Ariki's canonical-basis theorem and the characteristic-zero graded decomposition-number identity apply in branch (C0), with the level-one, root-of-unity and grading conventions of that regime: Ariki's canonical basis theorem in characteristic zero and Graded decomposition numbers equal canonical-basis coefficients in characteristic zero. In branch (CP), use the adjustment factorization in Positive-characteristic decomposition polynomials factor through adjustment; the adjustment matrix is not asserted to be the identity. The general negative-half categorification and cyclotomic highest-weight projective-$K_0$ identification are distinct results with their own hypotheses and Axiom of Choice dependencies: The KLR category categorifies the negative quantum half and The cyclotomic projective $K_0$ is the integral highest-weight module.

Specht, Soergel, and Category-O multiplicities

The type-A Murphy/Specht cellular basis is The Murphy tableau elements form a cellular basis, valid over its stated commutative coefficient rings and unit parameter without a semisimplicity assumption. The local type-A Soergel character theorem is Soergel's character conjecture for type A; it supplies no general Coxeter or positive-characteristic extension. The singular Category-O multiplicity formula is spherical and evaluated at $1$ in Singular category O multiplicities are parabolic Kazhdan–Lusztig values. The parabolic-category formula uses antispherical coefficients in a regular ambient block, by Parabolic category O multiplicities are parabolic Kazhdan–Lusztig values. These two formulas are not interchangeable.

Axiom and scope ledger

No new axiom is introduced by this route. The all-origin double-centralizer and admissible-origin Schur comparisons retain their stated Axiom of Choice assumptions; the Category-O residue comparison also retains its Axiom of Choice assumption. Ariki's characteristic-zero theorem and the graded decomposition theorem inherit choice only through their QG-8/global-basis suppliers. The negative-half KLR theorem uses choice through the stated nondegeneracy of the quantum form, and the cyclotomic projective-$K_0$ theorem states its Axiom of Choice hypothesis. The finite Brundan–Kleshchev block isomorphism and the finite examples in the companion add no choice principle. The positive-characteristic adjustment theorem also assumes choice through its crystal-label and global-basis suppliers; its finite coefficient substitution adds no choice. The singular and parabolic Category-O multiplicity theorems state choice as a hypothesis for their regular-block, translation, and highest-weight arguments. Each assumption is used only within its cited theorem's hypotheses.

Reuse boundaries

• The plan's source inventory and the full design section are in the Hecke dossier.
• The exact diagram and parameter contracts are in definition and diagram contracts.
• The shifted-Yangian PBW result Ordered monomial PBW basis for the shifted Yangian is outside this route and is not used as a supplier.
• Existing claims stay at their canonical pages; no regular/singular, characteristic-zero/positive-characteristic, or degenerate/nondegenerate hypothesis is enlarged here.