Modular representations and invariants of elementary abelian p-groups

• Main Author: Parsons, Christopher M.
• Other Authors: Shank, R. James
• Published: University of Kent 2018
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.754859

The purpose of this thesis is to develop tools to more easily classify the modular representations of elementary abelian p-groups and better understand their invariant rings. Since these groups almost always have wild representation type complete classification of the indecomposables is considered impossible and as such an alternative perspective is required. We reformulate the representation classification in the perspective of classifying maximal abelian subgroups of unipotent groups. Thence we express the problem as determining finitely many 'covering' homomorphisms of the form σ : (F^d, +) → GL_n(F) whose images collectively contain the images of all such representations up to equivalence. To aid in this we attach a combinatorial equivalence invariant object to modular p-group representations thereby allowing us to segment the problem and more easily distinguish between inequivalent families. Using these tools we build upon the work of [11] and develop a full set of covering homomorphisms for all modular elementary abelian p-groups in GL_4(F), GL_5(F) and GL_6(F). In doing so we also provide covering homomorphisms for select families in arbitrary dimension with specific patterns in their combinatorial invariant. By way of example we use these to provide an explicit construction for the Sylow p-subgroups of the finite orthogonal groups. Thereafter our focus switches to invariant rings. Given a matrix group in the image of a homomorphism σ : (F^d, +) → GL_n(F) we explore methods of recovering the W ≤ (F^d, +) used to generate the group purely through its action on specific elements in the symmetric algebra of the dual, properties of which are indicative of properties of the invariant ring. Using this we provide an alternative explicit construction for the invariant rings implicitly generated in [8]. Further we generalise a long-exploited technique for inductively defining invariants from those of maximal subgroups. After classifying the invariant rings and fields of several hitherto classified families we focus on the four-dimensional modular elementary abelian p-groups with rank 2, providing their invariant rings if Cohen-Macaulay and algorithmic methods to procure it otherwise.