[Paper Review] Linear independences of maps associated to partitions

Stefan Jung|arXiv (Cornell University)|Jun 25, 2019

Advanced Operator Algebra Research19 references4 citations

TL;DR

This paper establishes the linear independence of intertwiner maps associated with non-crossing partitions in the context of easy quantum groups, using a revised and corrected version of W. Tutte's matrix determinant framework. The key result proves that for $ N \geq 4 $, the maps $ T_p $ indexed by non-crossing partitions of type $ (0,n) $ are linearly independent, ensuring injectivity of the functor mapping categories of partitions to easy quantum groups.

ABSTRACT

Given a suitable collection of partitions of sets, there exists a connection to easy quantum groups via intertwiner maps. A sufficient condition for this correspondence to be one-to-one are particular linear independences on the level of those maps. In the case of non-crossing partitions, a proof of this linear independence can be traced back to a matrix determinant formula, developed by W. Tutte. We present a revised and adapted version of Tutte's work and the link to the problem above, believing that this self-contained article will assist others in the field of easy quantum groups. In particular, we fixed some errors in the original work and adapted notations, definitions, statements and proofs.

Motivation & Objective

• To establish the linear independence of intertwiner maps $ T_p $ associated with non-crossing partitions in the construction of easy quantum groups.
• To correct and re-interpret W. Tutte's 1993 work on the matrix of chromatic joints to fit the framework of easy quantum groups.
• To ensure the functor $ \Psi $ mapping categories of partitions to $ C^* $-categories is injective, thereby guaranteeing uniqueness of associated easy quantum groups.
• To provide a self-contained, notationally consistent, and corrected proof of linear independence for non-crossing partition maps, particularly for $ N \geq 4 $.

Proposed method

• Reformulates the injectivity problem of the functor $ \Psi $ as a linear independence problem on the maps $ T_p $ associated with partitions.
• Adapts Tutte’s matrix determinant approach for the matrix $ A(n,0) $, which encodes the linear dependence structure of the maps $ T_p $.
• Introduces corrected definitions for key combinatorial objects: $ r $-flaws, graphs $ G(p,q) $, $ H_r(p,q) $, and structures $[i]$, $[i,i+1]$, $[0]$, to fix errors in Tutte’s original work.
• Uses recursive column manipulation via the function $ F_r(p,q) $ to derive a determinant recursion formula for $ A(n,0) $.
• Establishes that $ \det(A(n,0)) \neq 0 $ by showing all factors in the determinant product are non-zero: quotients of reversed Beraha polynomials, powers of $ N $, and non-vanishing determinants of submatrices $ A(n,n-1) $.
• Applies the non-vanishing determinant of $ A(n,0) $ to conclude linear independence of the maps $ (T_p)_{p \in \mathcal{NC}(0,n)} $.

Experimental results

Research questions

• RQ1 Under what conditions is the functor $ \Psi $ mapping categories of partitions to $ C^* $-categories of intertwiner maps injective?
• RQ2 Can Tutte’s matrix determinant method be corrected and adapted to prove linear independence of maps $ T_p $ for non-crossing partitions?
• RQ3 Is the linear independence of $ (T_p)_{p \in \mathcal{NC}(0,n)} $ guaranteed for $ N \geq 4 $, ensuring uniqueness of associated easy quantum groups?
• RQ4 What are the precise combinatorial and algebraic conditions under which the determinant of the matrix $ A(n,0) $ is non-zero?

Key findings

• For $ N \geq 4 $, the determinant of the matrix $ A(n,0) $ is non-zero, as it is a product of non-vanishing terms including quotients of reversed Beraha polynomials and non-zero powers of $ N $.
• The determinant of the submatrix $ A(n,n-1) $ is $ N^{\lceil n/2 \rceil} \neq 0 $, contributing to the non-vanishing of $ \det(A(n,0)) $.
• The maps $ (T_p)_{p \in \mathcal{NC}(0,n)} $ are linearly independent for all $ n \in \mathbb{N} $ and $ N \geq 4 $, as a consequence of the non-vanishing determinant of $ A(n,0) $.
• The functor $ \Psi $ from categories of non-crossing partitions to $ C^* $-categories is injective for $ N \geq 4 $, ensuring that distinct categories yield distinct easy quantum groups.
• The corrected version of Tutte’s work provides a self-contained and rigorous foundation for the linear independence result, fixing errors in the original definitions and proofs.