# Modular Irreducible Representations of the FpW4-Submodules of the Modules as Linear Codes, where W4 is the Weyl Group of Type B4

• Jinan F. N. Al-Jobory Department of Mathematics, College of Education, Mustansiriyah University, Baghdad, Iraq
• Emad B. Al-Zangana Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq
• Faez Hassan Ali Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq
Keywords: Field of characteristic 0 (infinite field), Finite field Fp  GF(p), Weyl group Wn of type Bn, Group ring FpWn, FpWn-module, FpWn-submodule, Pair of partitions of a positive integer n, Specht polynomial, Specht module, -tableau, Standard -tableau, Vector space, Generating matrix, Linear code

### Abstract

The modular representations of the FpWn-Specht modules  as linear codes is given in our paper [6], and the modular irreducible representations of the FpW4-submodules  of the Specht modules  as linear codes where W4 is the Weyl group of type B4 is given in our paper [5]. In this paper we are concerning of finding the linear codes of the representations of the irreducible FpW4-submodules  of the FpW4-modules  for each pair of partitions  of a positive integer n = 4, where Fp = GF(p) is the Galois field (finite field) of order p, and p is a prime number greater than or equal to 3. We will find in this paper a generator matrix of a subspace  representing the irreducible FpW4-submodulesof the FpW4-modulesand give the linear code of for each prime number p greater than or equal to 3. Then we will give the linear codes of all the subspaces  for all pair of partitions  of a positive integer n = 4, and for each prime number p greater than or equal to 3.

We mention that some of the ideas of this work in this paper have been influenced by that of Adalbert Kerber and Axel Kohnert [13], even though that their paper is about the symmetric group and this paper is about the Weyl groups of type Bn.

Published
2021-06-27
How to Cite
F. N. Al-Jobory, J., B. Al-Zangana, E., & Hassan Ali, F. (2021). Modular Irreducible Representations of the FpW4-Submodules of the Modules as Linear Codes, where W4 is the Weyl Group of Type B4. Al-Nahrain Journal of Science, 24(2), 48-63. Retrieved from https://anjs.edu.iq/index.php/anjs/article/view/2394
Section
Articles