A Note on the Classical Dickson Invariants
Shin-Yao Jow · Algebra Colloquium · 2010
Let 𝔽q be the finite field with q elements, where q is a power of some prime p. In the classical paper [2], Dickson defined the polynomial [Formula: see text], where e1,…, en are non-negative integers. He observed that any [e1,…, en] is divisible by [0,1,…,n-1], and the quotient is GL n(𝔽q)-invariant. He then went on to show that one can pick n of these invariants to generate the entire invariant subring of GL n(𝔽q). In this paper, we answer the following two natural questions: How to express any given [e1,…, en]/[0,1,…,n-1] in terms of the fundamental generators? In general, when does [e1,…, en] divide [f1,…, fn]? The answers are Theorems 1.2 and 1.3, respectively.