The type-A cyclotomic specialization of the QG-KLR algebras

Let $F,q,e,I,\mathrm{\Lambda },{\lambda }_i$ be as in Affine and cyclotomic type-A Hecke algebras (The quantum characteristic of an invertible Hecke parameter, The cyclic affine type A Cartan datum and its residues). On the cyclic affine type-A Cartan datum take the normalized symmetrizers $d_i=1$ and define the type-A KLR datum $Q^{\mathrm{A}}$ by $Q_{ii}^{\mathrm{A}}=0,Q_{ij}^{\mathrm{A}}=1\text{if }{a}_{ij}=0,$ $Q_{i,i+1}^{\mathrm{A}}(u,v)=v-u,Q_{i+1,i}^{\mathrm{A}}(u,v)=u-v(e\ge 3),$ $Q_{01}^{\mathrm{A}}(u,v)=Q_{10}^{\mathrm{A}}(u,v)=-(u-v{)}^2(e=2)\mathrm{.}$ Write $R_{\alpha }$ for the resulting generalized KLR algebra over $F$ (The quiver Hecke (KLR) algebra of a symmetrizable Cartan datum, The KLR grading and the diagrammatic multiplication) and $R_{\alpha }^{\mathrm{\Lambda }}$ for the standard cyclotomic quotient of The cyclotomic KLR quotient at a dominant integral weight (monic first-strand relations $x_1^{{\lambda }_{i_1}}e(\nu )=0$).

The specialization dictionary. Put $y_r:=x_r$ and ${\psi }_r:=-{\tau }_r$. With these normalizations, the relations (1.6)–(1.15) of the source hold in $R_{\alpha }^{\mathrm{\Lambda }}$, and conversely the quotient presented by those relations is the same algebra; in other words $R_{\alpha }^{\mathrm{\Lambda }}$ is the sign-modified cyclotomic Khovanov–Lauda algebra of the source for the cyclic quiver of quantum characteristic $e$. The verification of the dictionary is elementary and is recorded here so that no hidden convention is imported:

1. (Q1)–(Q4). Non-adjacent pairs have $a_{ij}=0$ and $Q_{ij}^{\mathrm{A}}=1$, homogeneous of degree $0$. For an adjacent pair $(i,i+1)$ with $e\ge 3$ one has $b_{ij}=a_{ij}=-1$, and $v-u$ is homogeneous of degree $2=-2b_{ij}$ with nonzero extremal coefficients of $u^{-a_{ij}}=u$ and $v^{-a_{ji}}=v$; the opposite orientation is the transpose, so (Q2) is built into the two formulae. For $e=2$ the pair $\{0,1\}$ has $a_{01}=a_{10}=-2$ and $b_{01}=-2$, and $-(u-v{)}^2$ is symmetric, homogeneous of degree $4=-2b_{01}$, with top coefficient $-1\mathrm{\ne }0$.
2. Dot sliding. Relation (1.8) comes from ${\tau }_{r}e(i)=e(s_{r}i){\tau }_r$; relations (1.12)–(1.13) reduce to the dot-sliding constants ${\tau }_a{x}_a-x_{a+1}{\tau }_a=e(\nu )$ and ${\tau }_a{x}_{a+1}-x_a{\tau }_a=-e(\nu )$ on equal colours, whose sign changes to the source's constants $+1$ under $\psi =-\tau$.
3. Quadratic and braid relations. Relation (1.14): equal colours give ${\tau }^2=Q_{ii}=0$; non-adjacent colours give ${\tau }^2=1=e(i)$; a forward arrow gives ${\tau }^2=v-u$, evaluated at $(y_r,y_{r+1})$ as $y_{r+1}-y_r$; a backward arrow gives $y_r-y_{r+1}$; the $e=2$ double edge gives $(y_{r+1}-y_r)(y_r-y_{r+1})$. Relation (1.15): the divided-difference numerator $[Q^{\mathrm{A}}(x_a,x_{a+1})-Q^{\mathrm{A}}(x_{a+2},x_{a+1})]\mathrm{/}(x_{a+2}-x_a)$ equals $\pm 1$ in the two single-edge cases with $i_{r+2}=i_r$, equals $y_r+y_{r+2}-2y_{r+1}$ in the $e=2$ double-edge case, and vanishes otherwise; the global sign $\psi =-\tau$ matches the source's three cases. The product order is explicit: the source difference is ${\psi }_r{\psi }_{r+1}{\psi }_r-{\psi }_{r+1}{\psi }_r{\psi }_{r+1}$, which under $\psi =-\tau$ equals the native reversed-word difference ${\tau }_{r+1}{\tau }_r{\tau }_{r+1}-{\tau }_r{\tau }_{r+1}{\tau }_r$; for $Q=-(u-v{)}^2$ the displayed quotient is $u+w-2v$. Thus no braid-order sign is suppressed, also in characteristic two.

Degrees of products are read on the successive permuted colour pairs as in The KLR grading and the diagrammatic multiplication. This definition introduces no new basis, trace or primitive-block assertion, and uses no choice principle.