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- Title
Shintani descent for standard supercharacters of algebra groups.
- Authors
André, Carlos A. M.; Branco Correia, Ana L.; Dias, João
- Abstract
Let (q) be a finite-dimensional nilpotent algebra over a finite field q with q elements, and let G (q) = 1 + (q). On the other hand, let denote the algebraic closure of q , and let = (q) ⊗ q . Then G = 1 + is an algebraic group over equipped with an q -rational structure given by the usual Frobenius map F : G → G , and G (q) can be regarded as the fixed point subgroup G F . For every n ∈ ℕ , the nth power F n : G → G is also a Frobenius map, and G F n identifies with G (q n) = 1 + (q n). The Frobenius map restricts to a group automorphism F : G (q n) → G (q n) , and hence it acts on the set of irreducible characters of G (q n). Shintani descent provides a method to compare F-invariant irreducible characters of G (q n) and irreducible characters of G (q). In this paper, we show that it also provides a uniform way of studying supercharacters of G (q n) for n ∈ ℕ. These groups form an inductive system with respect to the inclusion maps G (q m) → G (q n) whenever m | n , and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group G. Indeed, we show that Shintani descent permits the definition of a certain "superdual algebra" which encodes information about the supercharacters of G (q n) for n ∈ ℕ.
- Subjects
GROUP algebras; AUTOMORPHISM groups; FINITE fields; ALGEBRA; UNIFORM algebras
- Publication
International Journal of Algebra & Computation, 2024, Vol 34, Issue 2, p207
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196724500073