# Orbits of the action of $A_6$ on $\mathbb{P}_2$

By a paper of Scott Crass http://xxx.lanl.gov/pdf/math/9903111v1.pdf we know that $A_6$ (Permutation on 6 elements) is an automorphism group of $\mathbb{P}_2$ which fix a sextic. What is the geometry of this action i.e., what are the orbits of this action explicitly.

• Is $A_6$ the symmetric group? That type of notation is usually reserved for the alternating group, i.e., the subgroup of even symmetries. – Vidit Nanda Aug 7 '13 at 21:52
• I mean the simple group with 360 elements. Yes. Symmetric. – user13559 Aug 7 '13 at 22:25
• You mean "Yes. Alternating." ("Symmetric" would be $S_6$, not $A_6$.) This group has a nontrivial triple cover $3.A_6$, with a faithful $3$-dimensional representation that projectivizes to an action of $A_6$ on ${\bf P}^2$. There's some information and reference in the Wikipedia page on the "Valentiner group": en.wikipedia.org/wiki/Valentiner_group $\phantom\infty$ It's known that $A_6$ and $A_7$ are the only simple alternating groups whose Schur multiplier is cyclic of order $6$; the others have Schur multiplier ${\bf Z}/2{\bf Z}$. – Noam D. Elkies Aug 7 '13 at 23:09

As noted in the comments, this is a classical object, credited to [Valentiner 1889]; the paper by Robert Crass that you cite refers to that paper and also to work of Wiman  and Fricke  that should contain the answer to your question, and Crass himself collects those answers starting on page 13.

The generic point of course has trivial stabilizer. For each of the $45$ involutions $i \in A_6$ there's a fixed line $l_i$, and most points on each $l_i$ are stabilized by just $i$, and thus have an orbit of size $180$. [The normalizer $N_i$ is an $8$-element dihedral group, and $N_i/\{1,i\}$ is a Klein four-group acting on $l_i$; the orbit of the generic point on $l_i$ consists of its four images under that four-group, and four further images on each of the other $44$ lines, for a total of $4 \cdot 45 = 180$.] There are two special orbits of size $60$, and one each of size $45$ and $36$, with dihedral stabilizers of size $6$, $8$, $10$ [NB there are two conjugacy classes of $S_3$'s in $A_6$, related by an outer automorphism of $S_6$]; when the stabilizer is dihedral of size $2n$, the point is at the intersection of $n$ lines $l_i$ corresponding to the $n$ non-central involutions in the group. There is also a $90$-point orbit with cyclic stabilizer of order $4$.

You asked for "the orbits of this action explicitly"; if you want explicit coordinates, you must specify coordinates for ${\bf P}^2$ and matrices for (generators of) the projective action of $A_6$. Fortunately Crass gives these coordinates and generators on pages 4ff. $-$ actually two choices ("octahedral cordinates" and "icosahedral coordinates", citing and slightly modifying Fricke ).

Alternatively you could go to the ATLAS  entry for $A_6$, and go to the "$2 \times 3 . A_6$" paragraph of the Constructions section. This gives coordinates for the $45$-point orbit on the lift of the Valentiner group to a complex reflection group of $3 \times 3$ matrices (#27 in the list of Shephard and Todd ). Regarding them as dual coordinates for $45$ lines $l_i$, you should be able to compute all the intersections that give orbits shorter than $180$, and also to find explicit matrices for the involutions $i$ that generate the group (and thus in particular the $4$-cycles that stabilize the $90$-point orbit).

References (mostly from Crass):

[ATLAS 1985] John H[orton]. Conway et al.: ATLAS of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford: Clarendon Press, 1985.

[Fricke 1926] Robert Fricke: Lehrbuch der Algebra 2. Vieweg, 1926.

[Shephard and Todd 1954] G.C. Shephard and T.A. Todd: Finite Unitary Reflection Groups, Canadian Journal of Mathematics 6 (1954), 274$-$304.

[Valentiner 1889] H. Valentiner: De endelige Transformations-Gruppers Theori, Videnskabernes Selskabs Skrifter, Sjette Raekke, Naturvidenskabelig og Mathematisk Afdeling 2. Bianco Lunos, 1889.

[Wiman 1895] Anders Wiman: Ueber eine einfache Gruppe von 360 ebenen Collineationen, Mathematische Annalen 45 (1895), 531$-$556.