A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups

2008/10/21 by Carlos A. M. Andre, Carlos A. M. André, Andre, Carlos A. M. +2
Computer Science · Mathematics · #20C15 #20G40 #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #math.GR #math.RT #msc:20C15 #msc:20G40

paper · pdf · doi:10.48550/arxiv.0810.3761

References added

openalex publication_date 2008/10/21 · arxiv created 2008/10/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define the superclasses for a classical finite unipotent group U of type B_n(q), C_n(q), or D_n(q), and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of U, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).

Related

• Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups
• New supercharacter theory for Sylow subgroups in orthogonal and symplectic groups
• Supercharacters of unipotent and solvable groups
• Supercharacter theories for Sylow p-subgroups of Lie type G₂
• Construction of supercharacter theories of finite groups