Novel construction of Specht modules for Monomial groups

In this thesis, I introduce a new method for constructing Specht modules using Specht matrices for families of symmetric groups, monomial groups, and hyperoctahedral groups. Specht modules are central to the representation theory of symmetric groups, connected to the concepts of Young symmetrizers and the subsequent development of polytabloids. These polytabloids form a basis for the Specht module associated with a particular partition of a natural number. In this work, I introduce a new combinatorial approach to replace polytabloids with columns of Specht matrices. The columns corresponding to the standard tableaux of shape λ, where λ is a partition of n ∈ N, form the basis of the Specht module S λ . This new construction is straightforward and easy to grasp, as it relies on basic linear algebra techniques, such as solving systems of linear equations. A longstanding open question in the representation theory of finite groups is whether a base change matrix can be found for a finite group G that transforms permutation matrices corresponding to irreducible representations into block forms. My thesis addresses this question for specific families of finite groups, including symmetric groups, monomial groups, and hyperoctahedral groups. It shows that a submatrix of the Specht matrix serves as a base change matrix that accomplishes this transformation, enabling us to obtain all irreducible representations of these groups over C. The methodologies employed in this thesis are computational, involving the development of several algorithms and computer programs to support the theoretical work. These have been implemented in the computer algebra system GAP (version 4).

Funder

College of Science and Engineering, University of Galway

Publisher

University of Galway

Rights

Attribution-NonCommercial-NoDerivatives 4.0 International

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Collections

University of Galway Theses (PhD Theses)