Abstract
Abstract: This is research is basically deals with the representation theory that is specifically on Hecke algebra and q-Schur algebra, where β-numbers of a partition has sufficient effect in both types of algebra. The objective of this work is to expand the results of Fayers that is by adding new runners to β-numbers, to represent a "tree", and by other hand, we decide to reduce the runners reaching another new definition "radical" by using both mathematically and computer programming ways.
