### Abstract

Let F = F_{q} denote the finite field of order q, and let GL_{n} = GL(n, q) denote the general linear group of all nonsingular n × n matrices over F. A function f : GL_{n} → GL_{n} is called a scalar polynomial function on GL_{n} if there exists a polynomial f(x) ε{lunate} F[x] which, when considered as a matrix function under substitution, represents the function tf. In this paper we characterize those polynomials f(x) ε{lunate} F[x] which define scalar polynomial functions on GL_{n} and we characterize those polynomials f(x) which determine permutations of GL_{n}. Also we enumerate both the functions and the permutations of GL_{n} which are scalar polynomial functions.

Original language | English (US) |
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Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Linear Algebra and Its Applications |

Volume | 174 |

Issue number | C |

DOIs | |

State | Published - Sep 1992 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics