Let $p$ be a prime number. Consider the natural representation of the affine Lie algebra $\widehat{\mathfrak{sl}_p}$, defined as follows.

$$A = \bigoplus_{i=1}^N \mathbb{C}a_i; \qquad \text{nat}_p = A \otimes \mathbb{C}[t, t^{-1}]$$

There's a natural action of $\widehat{\mathfrak{sl}_p}$ on $\bigwedge^n \text{nat}_p$ (here $\bigwedge$ denotes the wedge power). Let $\textbf{k}$ be an algebraically closed field with characteristic $p$, and $G = \text{SL}_n(\textbf{k})$. [Chuang-Rouquier] categorify this action as follows; see pg 58 (beginning of Section 2) in Williamson-Riche for a detailed exposition.

\begin{align*} K^0(\text{Rep}(G)) \simeq \bigoplus_{\lambda \in X^+} \mathbb{C}[\Delta(\lambda)] \end{align*}

In the above isomorphism, $\Delta(\lambda)$ denotes the Weyl module with highest weight $\lambda$, and $X^{+}$ denotes the set of all dominant weights. The Weyl modules give a basis in the Grothendieck group.

\begin{align*} \varphi: K^0(\text{Rep}(G)) &\rightarrow {\bigwedge}^{n} \text{nat}_p \\ \varphi([\Delta(\lambda)]) &= m_{\lambda_1} \wedge m_{\lambda_2 - 1} \wedge \cdots \wedge m_{\lambda_n-n+1} \end{align*}

**Definition:** Let the $p$-exotic canonical basis denote the classes of the simple modules in $\text{Rep}(G)$. As $p$ approaches infinity, then it's known that it stabilizes; we refer to that as the exotic canonical basis.

How can we express the ``exotic canonical basis'' in terms of Lusztig's canonical basis for $\bigwedge^n \text{nat}_p$, viewed as a module for $U_q(\widehat{\mathfrak{sl}_p})$ (specialized at a suitable parameter)?

**Remark:** Let $p$ be a sufficiently large prime. Lusztig's conjecture describes the classes of the simple objects in the Grothendieck group, and involves affine Kazhdan-Lusztig polynomials; see Section 4 of Jantzen's expository article ``Character formulae from Weyl to the present" here. This shows that the $p$-exotic canonical basis stabilizes as $p$ approaches infinity, and gives a method of computing the exotic canonical basis. However, the answer to the above question takes additional work.

Part2 (currently on p. 58), not ofSection2 (currently on p. 17). Is the Chuang–Rouquier reference to Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification? $\endgroup$ – LSpice Dec 3 '18 at 15:43