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We provide examples for negativity of structure constants of the stably canonical basis of modified quantum $\mathfrak{gl}_n$ and an analogous basis of modified quantum coideal algebra of $\mathfrak{gl}_n$.
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H. Bao and W. Wang, A new approach to Kazhdan-Lusztig theory of type B B via quantum symmetric pairs
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K. Shigechi, Canonical bases of a coideal subalgebra in U q ( 𝔰 𝔩 2 ) U_{q}(\mathfrak{sl}_{2})
Cited in the paper.
B. Webster, Canonical bases and higher representation theory
Cited in the paper.