Question

Hi,

In the following, we consider the quantum group of type $G_2=(a_{ij})_{i,j\in I}$ where $I={1,2}$ I=\{1,2\}$ and $\alpha_2$ is the long root.

We choose a reduced expression for the longest element $w_0$ of the Weyl group $W(G_2)$ as $w_0=s_2s_1s_2s_1s_2s_1$ and thus identify $\mathbb{N}^{6}$ and the set Kashiwara's crystal $B(\infty)$ via Lusztig's parametrization, \begin{align*} \mathbb{N}^6 \stackrel{\sim}{\longrightarrow} B(\infty)=L(\infty)/q^{-1}L(\infty), (a,b,c,d,e,f)\longmapsto e_2^{(a)}e_{12}^{(b)}e_{11122}^{(c)}e_{112}^{(d)}e_{1112}^{(e)}e_1^{(f)}+{q^{-1}L(\infty)} \end{align*} Here $e_{*}$ are suitable elements defined by Lusztig's braid group symmetry which we don't repeat it here.

For each $b\in B(\infty)$, we denote by $G^{\ast}(b)$ the corresponding dual canonical basis element.

Let $\mathcal{H}_n$ be Khovanov-Lauda-Rouquier algebra of type $G_2$ over $\mathbb{Q}$. You can see that $G^{\ast}(0,0,1,0,0,3)$ corresponds to an irreducible representation of dimension 168 of $\mathcal{H}_8$.

Note that we have the negative occurrence \begin{align*} e_2G^{\ast}(0,0,1,0,0,3) =G^{\ast}(1,0,1,0,0,3) +q^{-3}G^{\ast}(0,3,0,0,0,3) -q^{-3}G^{\ast}(0,2,0,1,0,2) +q^{-6}G^{\ast}(0,0,1,0,1,0). \end{align*}

I checked that \begin{align*} G^{\ast}(1,0,1,0,0,3), G^{\ast}(0,3,0,0,0,3)-G^{\ast}(0,2,0,1,0,2), G^{\ast}(0,2,0,1,0,2), G^{\ast}(0,0,1,0,1,0) \end{align*} correspond to irreducible representations of dimension 168,1176,168,168 of $\mathcal{H}_9$ respectively.

Thus, the irreducible $\mathcal{H}_9$-module $V$ whose character in the quantum Shuffle is given by $G^{\ast}(0,3,0,0,0,3)-G^{\ast}(0,2,0,1,0,2)$ is an example you ask. $V$ is realizable over $\mathbb{Z}$ and its irreducibility is always preserved under the modulo-$p$ reduction for every prime $p\geq 2$.