Reductive Algebraic Groups Revisited: Structural Decompositions and Representation-Theoretic Proofs
Keywords:
Reductive groups; Algebraic groups; Root systems; Highest weight theory; Weyl groups; Representation theory; Bruhat decomposition; Cartan decomposition Mathematics Subject Classifica- tion (2020): 20G40; 22E46; 22E70Abstract
This paper presents an in-depth examination of reductive algebraic groups by providing detailed proofs of fundamental structural and representation-theoretic results. We prove the conjugacy of maximal tori, classify reductive groups via their root data, and establish the Cartan and Bruhat decompositions with complete proofs. We also prove the highest weight classification and the Weyl character formula for finite-dimensional representations. Finally, we introduce an original refinement of the Bruhat decomposition that elucidates the affine cell structure of double cosets in the flag variety. This work is intended for researchers and advanced graduate students interested in algebraic groups and their representations.
References
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