On minimal tilting complexes in highest weight categories

TL;DR

This paper constructs minimal tilting complexes in highest weight categories, linking their structure to Kazhdan-Lusztig polynomial coefficients for various algebraic representation categories.

Contribution

It provides a detailed construction of minimal tilting complexes and relates their multiplicities to Kazhdan-Lusztig polynomials in specific algebraic contexts.

Findings

Minimal tilting complexes are explicitly constructed for objects in highest weight categories.

Multiplicities of tilting objects in these complexes relate to Kazhdan-Lusztig polynomial coefficients.

Results apply to representations of Lie algebras, affine Kac-Moody algebras, and quantum groups at roots of unity.

Abstract

We explain the construction of minimal tilting complexes for objects of highest weight categories and we study in detail the minimal tilting complexes for standard objects and simple objects. For certain categories of representations of complex simple Lie algebras, affine Kac-Moody algebras and quantum groups at roots of unity, we relate the multiplicities of indecomposable tilting objects appearing in the terms of these complexes to the coefficients of Kazhdan-Lusztig polynomials.