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ON THE DIMENSION OF PERMUTATION VECTOR SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 100 / Issue 2 / October 2019
- Published online by Cambridge University Press:
- 03 April 2019, pp. 256-267
- Print publication:
- October 2019
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- Article
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Let
$K$ be a field that admits a cyclic Galois extension of degree 
$n\geq 2$. The symmetric group 
$S_{n}$ acts on 
$K^{n}$ by permutation of coordinates. Given a subgroup 
$G$ of 
$S_{n}$ and 
$u\in K^{n}$, let 
$V_{G}(u)$ be the 
$K$-vector space spanned by the orbit of 
$u$ under the action of 
$G$. In this paper we show that, for a special family of groups 
$G$ of affine type, the dimension of 
$V_{G}(u)$ can be computed via the greatest common divisor of certain polynomials in 
$K[x]$. We present some applications of our results to the cases 
$K=\mathbb{Q}$ and 
$K$ finite.
