The level-one rank-two cyclotomic KLR algebra at e=3

The cyclotomic relation on $I^2$. Enumerate $I^2=\{0,1,2{\}}^2$. The cyclotomic exponent of the sequence $i$ is the pairing $({\mathrm{\Lambda }}_0,{\alpha }_{i_1})={\delta }_{i_1,0}$, so the defining cyclotomic relation reads $y_1^{{\delta }_{i_1,0}}e(i)=0$. For the three sequences with $i_1=0$ this is $y_{1}e(i)=0$; for the six sequences with $i_1\mathrm{\ne }0$ the exponent is zero and the relation is $y_1^{0}e(i)=e(i)=0$, so those idempotents vanish rather than merely a degree-one dot relation.

The quadratic relation. The KLR quadratic relation distinguishes the colour cases: $i_1=i_2$, $i_2=i_1+1$, $i_2=i_1-1$. For $e=3$ and $I=\{0,1,2\}$ every ordered pair of distinct colours is adjacent in the $3$-cycle, so these are all cases, and substituting gives ${\psi }_1^{2}e(i)=0$, $(y_2-y_1)e(i)$, and $(y_1-y_2)e(i)$ respectively.

The two nonzero blocks. By Cyclotomic Hecke algebras of level one are finite Hecke algebras the level-one algebra $H_2^{\mathrm{\Lambda }}$ is the finite Hecke algebra $H_{F,q}(S_2)$ with standard basis $\{1,T_1\}$, so it is two-dimensional, and $T_1^2-(q-1)T_1-q=0$ gives $H_2^{\mathrm{\Lambda }}\cong F[T]\mathrm{/}((T-q)(T+1))\cong F\times F$, the two factors being distinct because $q\mathrm{\ne }1$ and $q\mathrm{\ne }-1$ for a primitive cube root in characteristic $\mathrm{\ne }3$. Hence there are exactly two nonzero one-dimensional blocks. Their labels come from the polynomial generator: on the trivial module $X_1=q^0=1$ and $X_2=q^{-1}{T}_1{X}_1{T}_1=q^{-1}{q}^2=q$, giving residue sequence $(0,1)$; on the sign module $X_1=1$ and $X_2=q^{-1}(-1{)}^2=q^{-1}=q^2$, giving $(0,2)$; equivalently these are the $q$-contents of the standard tableaux of $(2)$ and $(1,1)$ (The Hecke Jucys-Murphy action is dominance triangular). Every nonzero block contains at least one of these surviving idempotents, so the remaining four blocks of height $2$ are zero.