Direct Rotation b - Numbers

Abstract

For  any partition  u = (u₁, u₂, . . ., u_n)  of a non - negative integer number r there exist a diagram (A) of b- numbers for each e where e is a a positive integer number greater than or equal to two; which introduced by James in 1978.These diagrams (A) play a enormous role in Iwahori-Hecke algebras  and q-Schur algebras; as presented by Fayers in 2007. In this paper, we introduced some new  diagrams (A⁹⁰ ), (A¹⁸⁰) and (A²⁷⁰ ) by employing  the "direct rotation  application of three diffrent degrees namely 90^o , 180^o and 270^o " on the main diagram (A). We concluded that we can find the successive main diagrams (A⁹⁰), (A¹⁸⁰) and (A²⁷⁰) for the guides b₂, b₃,. . . and b_e depending on the main diagrams (A⁹⁰ ), (A¹⁸⁰) and (A²⁷⁰ ) for b1 and set these facts as rules named Rule (3.1.2), Rule (3.2.2) and Rule (3.3.2) respectively.We depended in our work on  the idea of the intersection of the main diagrams  (A)  given by Mahmood in 2011 and the "upside-down b-numbers" again given by Mahmood in 2013.