# Questions about $p$-canonical basis" for $\widehat{\mathfrak{sl}_n}$ module (wedge power of natural representation)

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.

• It would help to explain your notation a bit further, starting with the notion of affine Lie algebra and the symbol $N$. – Jim Humphreys Apr 16 '18 at 14:11
• For the definition of affine Lie algebra, please see wikipedia: en.wikipedia.org/wiki/Affine_Lie_algebra – Vinoth Apr 16 '18 at 14:19
• P.S. When you refer to page 58, which version of the preprint are you pointing to? (The page numbering shifts a little.) – Jim Humphreys Apr 16 '18 at 14:21
• It's the beginning of Section 2; that should clarify any confusion. Thanks for the edits, Jim! – Vinoth Apr 16 '18 at 14:41
• I'm pretty sure you mean the beginning of Part 2 (currently on p. 58), not of Section 2 (currently on p. 17). Is the Chuang–Rouquier reference to Derived equivalences for symmetric groups and $\mathfrak{sl}_2$-categorification? – LSpice Dec 3 '18 at 15:43