Positivity properties of canonical bases

TL;DR

This work proves strong positivity properties for the canonical basis of the modified quantum group $\dot{\mathbf{U}}$ and for the canonical basis of tensor products involving mixed lowest/highest weight modules, under symmetric Cartan data. The central method, the thickening construction, embeds suitable approximate tensor products into the negative part of a larger quantum group $\tilde{\mathbf{U}}^{-}$, allowing positivity to be inherited from the known positivity of $\tilde{\mathbf{B}}^{-}$. The main results show that structure constants for both multiplication and the action on tensor products lie in $\mathbb{N}[v,v^{-1}]$, and in finite type every canonical basis element of $\dot{\mathbf{U}}$ is spherical-parabolic, giving broad positivity. The paper extends positivity beyond finite type to arbitrary tensor products using Demazure modules and thickening realizations, providing a unified framework for positivity in quantum groups and their representations. This advances connections between canonical bases, total positivity principles, and geometric/categorical perspectives in quantum group theory.

Abstract

We prove that the canonical basis of a modified quantum group $\dot{\mathbf{U}}$ exhibits strong positivity properties for the canonical basis elements arising from spherical parabolic subalgebras. Our main result establishes that the structure constants for both the multiplication with arbitrary canonical basis elements in $\dot{\mathbf{U}}$ and the action on the canonical basis elements of arbitrary tensor products of simple lowest and highest weight modules by these elements belong to $\mathbb{N}[v,v^{-1}]$. This implies, in particular, for quantum groups of finite type, the structure constants for multiplication and for action on tensor product with respect to canonical basis are governed by positive coefficients. A key ingredient is the thickening construction, an algebraic technique that embeds a suitable approximation of the tensor of a lowest weight module and a highest weight module of $\dot{\mathbf{U}}$ into the negative part $\tilde{\mathbf{U}}^-$ of a larger quantum group. This allows us to inherit the desired positivity for the tensor product from the well-established positivity of the canonical basis of $\tilde{\mathbf{U}}^-$.

Theorems & Definitions (58)

• Theorem 1: = Theorem \ref{['Positivity in general case']}
• Corollary 2
• Proposition 3: = Proposition \ref{['positivity of dot B on simple module']}
• Theorem 4: = Theorem \ref{['positivity of dot B']}
• Corollary 5
• Theorem 1.1: Lusztig-1993
• Theorem 1.2: Lusztig-1993
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5