No abstract

‫ﺍﻟﺨﻼﺼﺔ‬ : ‫ﺍﻟﺒﺤﺙ‬ ‫ﻫﺫﺍ‬ ‫ﻫﻭ‬ ‫ﹰ‬ ‫ﺃﺴﺎﺴﺎ‬ ‫ﻤﻭﻀﻭﻉ‬ ‫ﻓﻲ‬ ‫ﻨﻅﺭﻴﺔ‬ ‫ﺍ‬ ‫ﻟﺘﻤﺜﻴل‬ Representation theory ‫ﹰ‬ ‫ﻭﺘﺤﺩﻴﺩﺍ‬ ‫ﺠﺒﺭ‬ ‫ﻓﻲ‬ ‫ﺍﻟﻨﻭﻉ‬ ‫ﻤﻥ‬ ‫ﺸﻭﺭ‬ q (q-Schur algebra) ‫ﻫﻴﻜ‬ ‫ﻭﺠﺒﺭ‬ ‫ﺎ‬ (Hecke algebra) ‫ﺘﻠﻌﺏ‬ ‫ﺤﻴﺙ‬ ‫ﺃﻋﺩﺍﺩ‬ β ‫ﻷﻱ‬ ‫ﺘﺠﺯﺌﺔ‬ µ ‫ﻤﻥ‬ ‫ﺍﻟﻌﺩﺩ‬ ‫ﺍﻟﻤﻭﺠﺏ‬ ‫ﺍﻟﺼﺤﻴﺢ‬ r ‫ﻫﺫﺍ‬ ‫ﻤﻥ‬ ‫ﻜل‬ ‫ﻓﻲ‬ ‫ﹰ‬ ‫ﻤﻬﻤﺎ‬ ‫ﹰ‬ ‫ﺩﻭﺭﺍ‬ ‫ﻥ‬ ‫ﺍﻟﻨﻭﻋـﺎﻥ‬ ‫ﺍﻟﺠﺒﻭﺭ‬ ‫ﻤﻥ‬ . ‫ﻨﺘﺎﺌﺞ‬ ‫ﻭﺴﻌﻨﺎ‬ ‫ﺍﻟﻌﻤل‬ ‫ﻫﺫﺍ‬ ‫ﻓﻲ‬ Fayers ، ‫ﻗﺎﻡ‬ ‫ﺍﻟﺫﻱ‬ ‫ﺒﺈﻀﺎﻓﺔ‬ ‫ﻋﻤـﻭ‬ ‫ﺩ‬ (runner) ‫ﺠﺩﻴـﺩ‬ ‫ﻹﻋﺩﺍﺩ‬ β ، ‫ﻭﻗﻤﻨﺎ‬ ‫ﺒﺈﻀﺎﻓﺔ‬ ‫ﻋﺩﺓ‬ ‫ﺃﻋﻤﺩﺓ‬ ‫ﻭﺃﻁﻠﻘﻨﺎ‬ ‫ﺠﺎﻨﺏ‬ ‫ﻤﻥ‬ ‫ﻫﺫﺍ‬ ‫ﺒﺎﻟﺸﺠﺭﺓ‬ ‫ﻋﻠﻴﻬﺎ‬ ، ‫ﺠﺎﻨـﺏ‬ ‫ﻤﻥ‬ ‫ﺃﺨـﺭ‬ ‫ﺍﺨﺘﺯﺍل‬ ‫ﺴﻨﻘﺩﻡ‬ ‫ﺍﻷ‬ ‫ﻋﻤﺩﺓ‬ ‫ﹰ‬ ‫ﻭﺼﻭﻻ‬ ‫ﺇﻟﻰ‬ ‫ﺤﺎﻟﺔ‬ ‫ﺃﻁﻠﻘﻨﺎ‬ ‫ﺍﻟﺠـﺫﺭ‬ ‫ﺍﺴﻡ‬ ‫ﻋﻠﻴﻬﺎ‬ (radical) ‫ﺍﻟﺤـل‬ ‫ﻤﻘـﺩﻤﻴﻥ‬ ‫ﹰ‬ ‫ﻭﺒﺭﻤﺠﻴﺎ‬ ‫ﹰ‬ ‫ﺭﻴﺎﻀﻴﺎ‬ . Abstract:This is research is basically deals with the representation theory that is specifically on Hecke algebra and q-Schur algebra, where βnumbers of a partition µ 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.