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If $H_1, ..., H_n$ are hyperplanes in $\mathbb{R}^m$ such that the complement of the union $\cup_i H_i$ is the interior of a complete polyhedral fan, then how does one determine ray generators for each face of the fan?

This question came about by trying to figure this out when the hyperplanes come from the Weyl chamber decomposition associated to a semisimple group. In dimension 1 or 2 one can just draw it and see what the answer is. But even in dimension three this seems difficult and seems to offer little help in higher dimensions.

Its relatively simple to describe the 1-dimensional cones (Mariano's answer) but it seems like more work needs to be done to be able to describe the higher dimensional cones.

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It is sufficient to find generators for the minimal cones in the fan, for every face is generated by the generators of the minimal cones it contains. In the case of your Weyl groups, the minimal cones are half-lines, so you really need just a non-zero vector in those minimal cones. Now, a minimal cones in this situation is half of an intersection of $n-1$ independent of your hyperplanes.

So: make a list of all sets of $n-1$ independent hyperplanes, find the intersection of each, and pick one vector in each direction in the resulting line. Put them all in a set $G$. To find generators for a cone $\sigma$ in your fan, then, just compute $\sigma\cap G$.

In the Weyl case, you'll end up with a simplicial complex with vertex set $G$.

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Yes I agree with this, but I guess my problem is that I don't have a good description or enumeration of non minimal cones. For example which two elements subsets of $G$ generate a two dimensional cone? So perhaps my question should have been, if you are given the $H_i$ as just the vanishing of linear equations, how do you obtain from this data a description of all cones $\sigma$ such that your answer above can actually be applied? – solbap Jun 18 '11 at 23:09

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