# Functor from a group to a poset

Are there examples of functors from the category of a single group to the category a partially ordered set (some sort of representation of the group in a poset) ?

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Since you have only one object in a group and partially ordered sets are the small categories with at most one morphism between any two objects, there isn't much interesting to say... – t.b. Jul 26 '11 at 15:56
On the other hand, a functor from a group-as-a-category to the category of posets would indeed be a poset representation. Such a poset representation might be obtained by composing a $G$-set (thought of as a functor) with the covariant powerset functor. I'm not sure what you mean about viewing simplicial complexes as posets though... – Zhen Lin Jul 26 '11 at 16:14
I thought a simplicial complex could be viewed as a poset ordered by inclusion. Thus if a functor could be found between a group-as-category and a poset-as-category, one could do the same between the group and the complex. I will edit my question and remove that second part, since it raises more issues than expected. – AlexPof Jul 26 '11 at 17:24
(I guess one problem with too-broad, or too-amorphous questions, that verge on "can one make sense of X", is that they always have a (pointless) "yes" answer, insofar as a person sufficiently mathematically literate can contrive connections between any two things... upon demand, in effect. But that proves little. In general, telling the context in which a seemingly broad question arises is very helpful, enabling people to give useful, rather than pointless-formal responses.) – paul garrett Jul 26 '11 at 18:43
@paul garrett: I agree, but currently, if I start giving my motivations, it will be way too long and too boring for MathSE. To make it short : - I'm studying (as a hobby) mathematical music theory, in particular transformational music theory - I have intuitions about some new stuff. - However I hold a PhD in chemistry and my math education stopped at the undergraduate level, so that my reasoning is floppy. Hence the multiple "amorphous" questions on MathSE... – AlexPof Jul 27 '11 at 19:25

Maybe the question (and the elided second question) wants to be rewritten as asking about posets of subgroups of a finite group, and asking about construction of the "incidence geometry", a simplicial complex attached to that poset?

I suggest this revision because it has at least one worthwhile answer/discussion, namely, for Coxeter groups ... definable as groups $G$ generated by sets $S$ of reflections (elements $s\in S$ satisfy $s^2=e$), defined by relations $(st)^{m_{s,t}}=e$ for $s,t\in S$ for some integer $m_{s,t}$ (by convention $m_{s,t}=\infty$ if no power of $st$ is $e$). The simplest example is the symmetric group on $n$ things, which is generated by "reflections" consisting of "adjacent" transpositions.

The special subgroups of $G$ are those generated by subsets of the set of reflections. Two special subgroups are incident when their intersection is also special. Collections of mutually incident special subgroups are taken as simplices (maybe inclusion-reversing), and we obtain a simplicial complex (the associated "Coxeter complex").

It is a not-completely-trivial result that the resulting complex is a triangulation of a sphere for finite Coxeter groups. For certain infinite Coxeter groups ("affine" ones) the Coxeter complex is a triangulation of an affine space.

Another, related case of a meaningful simplicial complex attached to a group is the (spherical) building attached to parabolic subgroups $P$ of a reductive group $G$: two parabolics are incident when their intersection is another parabolic, and mutually incident collections are simplices (inclusion-reversing). These buildings are constructible by sticking together the Coxeter complexes ("apartments") made from conjugates of "Weyl groups". For example, for $G=GL_n(k)$ with a field $k$, the "Weyl group" $W$ is identifiable with permutation matrices in $G$, so is a Coxeter group, and the "apartments" are spheres.

More of this sort of discussion is possible, in many directions, ...

Edit: given the poser's (re-) framing of the question in the context of transformational (mathematical) music theory (an interesting idea, about which I read a few things some years ago...): very likely, the operationally useful idea would be that of "a group acting on [something]". Keyphrase "acting on", rather than "group qua group". Yes, relevant posets can easily arise, as "subsequent" states/ideas in a sequence of such, with the ordering simply being (essentially) "later", in a piece of music that is deterministically performed. (!) Even more literally, since we do not imagine music going backward in time, the truly relevant notion is semi-group acting on poset. [But now I must stop, because I have not thought about actual applications to music beyond this mild deconstruction of modern "theoretical music theory", and, also, I'm more sympathetic to the "tonal" tradition, and do not see music as essentially mathematical, any more than I see playing outfield as such... and, by this, I don't mean to disparage anyone...]

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This is a great answer, but I don't know if it has anything to do with the question. – Qiaochu Yuan Jul 28 '11 at 3:01
(Trying to guess what the questioner had in mind...) – paul garrett Jul 28 '11 at 11:59