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How to determine all the faithful irreducible representations of $\mathbb Z_n$ and $D_{2n}$ over $GF(p)$, where $p$ is a prime not dividing $n$?

Binzhou Xia
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  • See also http://mathoverflow.net/questions/102573/structure-of-f-pg-for-finite-group-g http://math.stackexchange.com/questions/172534/for-which-values-of-n-is-the-polynomial-px-1xx2-cdotsxn-irreducible/172540#comment396429_172540

    http://math.stackexchange.com/questions/172468/for-what-n-k-there-exists-a-polynomial-px-in-f-2x-s-t-degp-k-and

     – Alexander Chervov Jul 22 '12 at 11:23
  • http://math.stackexchange.com/questions/153429/irreducible-representations-of-a-cyclic-group-over-a-field-of-prime-order – Binzhou Xia Jul 22 '12 at 13:34
  • http://math.stackexchange.com/questions/167979/representation-of-cyclic-group-over-finite-field?lq=1 – Binzhou Xia Jul 24 '12 at 15:23

1 Answers1

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The faithful irreducible representations of $\mathbb{Z}_{n}$ over such a field $F$ have dimension $e$, where $e$ is the smallest positive integer such that $n$ divides $p^{e}-1.$ To find an explicit representation of that dimension is straightforward in theory, but may not be so easy in practice. It is necessary to find an irreducible factor, say $p(x)$ of degree $e$ of $x^{n}-1$ in $F[x].$ Given such a factor $p(x)$, there is an irreducible representation of $\mathbb{Z}_{n}$ which sends a generator of the cyclic group to the companion matrix of $p(x).$

Representing the dihedral group with $2n$ elements is a little more subtle. Let $z$ be a generator of the cyclic subgroup of index $2$. The issue is whether an irreducible representation of degree $e$ of $\langle z \rangle$ extends to the whole dihedral group or not. If it does not extend, then it induces to an irreducible representation of dimension $2e.$ So when does it extend? This depends on whether $z^{-1} = z^{p^{d}}$ for some $d$ with $1 \leq d \leq e.$ If yes, then the representation extends. If no, it does not.But since$e$ is the smallest positive integer such that $n$ divides $p^{e}-1,$ and $z$ has order $n,$ it reduces to checking whether $p^{\frac{e}{2}} +1$ is divisible by $n$ or not (except in the slightly unusual case $n = 2,$ in which case $e = 1$ and the representation does extend). So ( for $n \neq 2$), if $e$ is odd, the representation does not extend. If $e$ is even then $n \neq 2 $ and the representation extends if and only if $n$ divides $p^{\frac{e}{2}}+1.$

Geoff Robinson
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  • what is "companion matrix of p(x)" ? – Alexander Chervov Jul 22 '12 at 11:25
  • It is an $e \times e$ matrix which has $1$'s on the superdiagonal in rows $1$ through $e-1$ (zeroes elsewhere on those rows)and its entries in row $e$ apart from the diagonal are the coefficients of $p(x)$ each multiplied by signs to make its characteristic polynomial $p(x).$ Eg, the companion matrix of $x^{2} +1$ is $\left( \begin{array}{clcr} 0 & 1\-1 & 0 \end{array} \right)$. – Geoff Robinson Jul 22 '12 at 12:28
  • Can the number of inequivalent faithful irreducible representations be determined? – Binzhou Xia Jul 22 '12 at 13:22
  • For the cyclic group, it is the number of irreducible factors of degree $e$ of $x^{n}-1.$ For the dihedral group (apart from $n =2$), I think it is half this number if $n$ does not divide $p^{\frac{e}{2}}+1$ and twice that number if $n$ does divide $p^{\frac{e}{2}}+1,$ but I suggest you check that yourself: it is a question of Clifford theory. – Geoff Robinson Jul 22 '12 at 13:42
  • Thanks! How does a representation of $\langle z\rangle$ extend to the whole dihedral group? Can we construct that based on the representation of $\langle z\rangle$? – Binzhou Xia Jul 25 '12 at 15:00
  • An irreducible representation of $\langle z \rangle $ only extends in the second case I mention. As I said, it is a case of Clifford Theory. When it does not extend, you have to induce it. It would take long to explain in a comment. – Geoff Robinson Jul 25 '12 at 15:26
  • Are there some references about the modular representation of dihedral groups? – Binzhou Xia Jul 26 '12 at 00:09
  • Try reading up about Clifford's Theorem, eg in the book of Curtis and Reiner – Geoff Robinson Jul 26 '12 at 07:03
  • Hi Geoff (and others). If you have the time, I've asked a few clarifications to your answer over here: http://math.stackexchange.com/questions/412772/clarifications-on-the-faithful-irreducible-representations-of-the-dihedral-group Thank you! – john Jun 06 '13 at 07:13
  • maybe you have interests to answer this: http://math.stackexchange.com/questions/759668/which-non-abelian-finite-groups-contain-the-two-specific-centralizers-part-ii – annie marie cœur Apr 18 '14 at 21:37