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Canonical bases of the oriented skein category

Yaolong Shen

TL;DR

This work develops a diagrammatic framework for the oriented skein category $\mathcal{OS}(z,t)$, introducing a bar involution and Kazhdan-Lusztig type canonical bases on all morphism spaces. The canonical bases arise from a positive-crossing, reduced-ribbon foundation and are compatible with Lusztig's KL theory, with transition matrices independent of the parameter $t$. By specializing $z$ to $q-q^{-1}$, the quantized walled Brauer algebra $\mathfrak B_{m|n}(q,t)$ is realized as a morphism space in $\mathcal{OS}(q-q^{-1},t)$ and inherits a bar involution and a canonical basis indexed by matchings. The results show positivity in low ranks and suggest deeper connections with quantum group representations via mixed Schur-Weyl duality and HOMFLY-PT skein theory, providing a diagrammatic route to KL-type structures in these algebras.

Abstract

We develop a bar involution and canonical basis for every morphism space of the oriented skein category through a diagrammatic approach. In particular, our construction gives rise to Kazhdan-Lusztig type bases on quantized walled Brauer algebras.

Canonical bases of the oriented skein category

TL;DR

This work develops a diagrammatic framework for the oriented skein category , introducing a bar involution and Kazhdan-Lusztig type canonical bases on all morphism spaces. The canonical bases arise from a positive-crossing, reduced-ribbon foundation and are compatible with Lusztig's KL theory, with transition matrices independent of the parameter . By specializing to , the quantized walled Brauer algebra is realized as a morphism space in and inherits a bar involution and a canonical basis indexed by matchings. The results show positivity in low ranks and suggest deeper connections with quantum group representations via mixed Schur-Weyl duality and HOMFLY-PT skein theory, providing a diagrammatic route to KL-type structures in these algebras.

Abstract

We develop a bar involution and canonical basis for every morphism space of the oriented skein category through a diagrammatic approach. In particular, our construction gives rise to Kazhdan-Lusztig type bases on quantized walled Brauer algebras.
Paper Structure (10 sections, 10 theorems, 40 equations)

This paper contains 10 sections, 10 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Oriented skein category
  3. Definition
  4. A standard basis
  5. A bar involution
  6. Canonical bases
  7. Quantized walled Brauer algebra
  8. Basic properties
  9. A bar involution
  10. Canonical bases

Key Result

Proposition 2.2

Tur90 The morphism space $Hom_{\mathcal{OS}(z,t)}(\bf a,\bf b)$ is a free $\mathbb K$-module with a basis given by any set consisting of a reduced lift for each of the $(\bf a,\bf b)$-matchings.

Theorems & Definitions (27)

  • Example 2.1
  • Proposition 2.2
  • Example 2.3
  • Corollary 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • Lemma 2.7
  • proof
  • Theorem 2.8
  • ...and 17 more