I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs.

Their basic setup is as follows:

They take a finite group $G$ acting faithfully, linearly and orthogonally on Euclidean space ($E=\mathbb{R}^n$) and a vector $v \in E$ such that $gv \neq v$ holds for all $g \neq 1$ in $G$. Then they consider the convex hull $P(G,v)$ of $Gv$ which is obviously a polytope (since $G$ is finite) and therefore a natural CW-Complex. The group acts on $P(G,v)$ by permuting faces in each dimension and we can conclude that the cellular chain complex $C_*(P)$ is in fact a complex of $\mathbb{Z}G$-modules (I am actually somewhat unclear on the definition of the corresponding differential but luckily there are algorithms that can compute it if necessary).

My problem starts when they describe the action on $C_k(P)$: The module $C_k(P)$ is $\mathbb{Z}$-free (not necessarily $\mathbb{Z}G$-free) with free generators which can be identified with the $k$-faces of $P(G)$ and $G$-action $ge=\pm f$ if $g$ maps the $k$-face $e$ to $f$ with sign depending on the orientation with which $g$ maps $e$ to $f$. I am quite unsure about the meaning of this.

  1. If I have a $1$- or a $2$-face I can give it an orientation in a somewhat natural way but what is the orientation of some $5$-face of a polytope in dimension $8$ (e.g.).

  2. Even if there was a way to give each $k$-face an orientation wouldn`t the choice be kind of arbitrary?

  3. What confuses me most is the implementation: In the HAP-package for the computer algebra system Gap there is an algorithm which computes $C_*(P)$ for given $G$ and $v$. Among other things a function "action($k,j,g$)" is computed with output $\pm1$ depending on how the element $g$ acts on the permutation on the cell "j" in dimension $k$. However if you have a look at the code you can see that the output does not depend on $k$ and $j$ at all but merely on the property of $g$ to belong to the subgroup $G_{ev}$ which is generated by all products of two generators of $G$ (the generating set for $G$ being computed via a Gap-command). This is not even invariant under the choice of the generating system. Why should the action on the orientation only depend on this choice of a generating system?

Thanks in advance for any help.

[1] available at http://hamilton.nuigalway.ie/ (preprints)

  • 1
    $\begingroup$ Avoiding the verb to irritate might be a good idea. $\endgroup$ – Mariano Suárez-Álvarez Apr 26 '13 at 8:09
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    $\begingroup$ The choice of orientations is arbitrary. See a textbook treating the cellullar complex of a CW-complex; here the CW-complex is regular so the differential is of the form described (and not more complicated) $\endgroup$ – Mariano Suárez-Álvarez Apr 26 '13 at 8:17
  • $\begingroup$ It may be the case that only the HAP author will be able to comment on (3). Please try to contact him directly or ask your question in the GAP Forum $\endgroup$ – Alexander Konovalov Apr 26 '13 at 9:00

1) As my understanding, the orientation of g.e depends on both k-cell e and the element g. It could be defined as below: Suppose that B the chosen basis for the k-dimensional space containing k-cell e and B' the chosen basis for the k-dimensional space containing f. The orientation is defined by the sign of the determinant of the matrix which exactly is the matrix of basis transformation from g.B to B'.

2) As I mentioned above the orientation depends on your choice of bases.

3) The function Action(k,j,g) of course depends on k and j. First step, we try to assign the element g with an element h in the stabilizer of the jth k-cell (which actually the representative of the jth k-orbit). The question "how to assign?" well, it's your choice. In the function it was the author's choice. So now, we just care about the sign of the stabilizer elements which exactly is the orientation defined above. In the function you can see the RotSubgroup, it is actually the subgroup of all "sign=1" elements of the stabilizer mentioned above.



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