Positivity properties for canonical bases of modified quantum affine ${\widehat{\mathfrak{sl}}}_n$

The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in $\mathbb{N}[v, v^{-1}]$. Let $\mathbf{U}$ be the quantum group over $\mathbb{Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part $\mathbf{U}^{+}$ of $\mathbf{U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{\mathbf{U}}$ of $\mathbf{U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{\mathbf{U}} ({\widehat{\mathfrak{sl}}}_n)$ coincide with certain structure constants for the canonical basis of $\mathbf{U} {({\widehat{\mathfrak{sl}}}_N)}^{+}$ for $n \lt N$. In particular, the positivity property for $\dot{\mathbf{U}}({\widehat{\mathfrak{sl}}}_n)$ follows from the positivity property for $\mathbf{U} {({\widehat{\mathfrak{sl}}}_N)}^{+}$.

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The first author was supported by the National Natural Science Foundation of China (11671297), Fok Ying Tung Education Foundation (131004), Shanghai Education Development Foundation and Shanghai Municipal Education Commission (Shuguang Program 16SG16).

Received 12 June 2016

Accepted 28 January 2017

Published 5 July 2018