# Non-commutative association schemes and their fusion association schemes

## TL;DR

This paper establishes a criterion for creating fusion association schemes from non-commutative schemes and constructs examples using special matrices, advancing the understanding of algebraic combinatorial structures.

## Contribution

It provides a sufficient condition for non-commutative schemes to have fusion schemes and constructs new examples from specific matrices.

## Key findings

- Established a criterion for fusion schemes from non-commutative schemes
- Constructed non-commutative schemes using generalized matrices
- Obtained symmetric fusion schemes from these constructions

## Abstract

We give a sufficient condition for a non-commutative association scheme to have a fusion association scheme, and construct non-commutative association schemes from symmetric balanced generalized weighing matrices and generalized Hadamard matrices. We then apply the criterion to these non-commutative association schemes to obtain symmetric fusion association schemes.