Positivity vs negativity of canonical bases

Abstract

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 ${\mathfrak{gl}}_n$. In contrast, we construct the canonical basis of the modified quantum coideal algebra of ${\mathfrak{sl}}_n$, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to the canonical bases are shown to be positive. Formulas for canonical basis of the iSchur algebra of rank one are obtained.

v4, 38 pages. Assertion on type B maximal parabolic KL polynomials replaced by "Formulas for canonical basis of the iSchur algebra of rank one are obtained". Final version, to appear in the Proceedings for Lusztig's 70th birthday conference, Bulletin of Institute of Mathematics Academia Sinica (N.S.)

References in corpus (4)

• A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs
• Geometric Schur duality of classical type
• Positivity of Canonical Bases under Comultiplication
• Canonical bases of a coideal subalgebra in $U_q({\mathfrak{sl}}_2)$

Cited by in corpus (3)

• Affine flag varieties and quantum symmetric pairs
• An elementary construction of monomial bases of modified quantum affine ${\mathfrak{gl}}_n$
• Canonical bases arising from $ı$quantum covering groups