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

• Published in: Journal of algebra Vol. 322; no. 4; pp. 1273 - 1294
• Main Authors: André, Carlos A.M., Neto, Ana Margarida
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.08.2009
• Subjects: Basic set of positive roots; Finite unipotent group; Orthogonal group; Positive root; Supercharacter; Supercharacter theory; Superclass; Symplectic group; Superclass; Supercharacter; Orthogonal group; Positive root; Finite unipotent group; Symplectic group; Supercharacter theory; Basic set of positive roots
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2009.05.025

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 [C.A.M. André, A.M. Neto, Supercharacters of the Sylow p-subgroups of the finite symplectic and orthogonal groups, Pacific J. Math. 239 (2) (2009) 201–230], they form a supercharacter theory in the sense of [P. Diaconis, I.M. Isaacs, Supercharacters and superclasses for algebra groups, Trans. Amer. Math. Soc. 360 (5) (2008) 2359–2392]. 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 a 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).