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Background: given a finite-dimensional real vector space V of dimension d, I can define a pointed cone in two ways: either as a set of the form $\{r_1v_1 + \cdots + r_nv_n \mid r_1, \dots, r_n \in R_{\ge 0}\}$ (here R is the real numbers, I can't seem to use mathbf?) where $v_1, \dots, v_n$ are some choice of generators, or as the intersection of finitely many half-spaces which doesn't contain any lines.

In particular, if I start with the generator description, one can define supporting hyperplanes as those with defining equation $a_1x_1 + \cdots + a_dx_d = b$ such that the cone lives in one of the two spaces $a_1x_1 + \cdots + a_dx_d \le b$ or $a_1x_1 + \cdots + a_dx_d \ge b$. A facet can be defined as the positive real span of a collection of d-1 vectors for which there exists a supporting hyperplane that vanishes on it. The relevant theorem here is that if I take the collection of supporting hyperplanes of facets, then the cone is the intersection of the appropriate half-spaces. All proofs of this that I have seen use the fact somewhere that R is an ordered field.

My question is whether one can generalize this situation to partially-ordered fields.

In particular, I am only interested in the following field. Take the ring of symmetric functions in n variables, and let K be its fraction field. Say that an element is "Schur positive" if it can be written in the form f/g where both f and g are Schur positive symmetric functions. (Note: it could be the case that f/g = f'/g' where both f and g are Schur positive and neither f' nor g' are Schur positive nor Schur negative, but whatever). And then I'll say that $x \ge y$ if $x-y$ is "Schur positive." This gives me a partial ordering on K compatible with addition and multiplication.

I'll now define a cone defined by generators in the same way, but instead of using coefficients from nonnegative reals, I'll use "Schur positive" coefficients. Now, is there any way of getting a dual "hyperplane" description for these kinds of cones, or any sensible definition of "facets"? There is a map from K to the rationals which sends a rational Schur polynomial to the evaluation at all 1's which sends "Schur positive" elements to positive numbers.

This all seems crazy, so let me explain my motivation. In the paper http://arxiv.org/abs/0712.1843 , Eisenbud and Schreyer show that the Betti tables of Cohen-Macaulay modules of a fixed codimension are positive rational combinations of pure Betti tables, and they use the generators / hyperplane duality mentioned in the beginning. In my paper http://arxiv.org/abs/0907.4505 with Weyman, we study equivariant resolutions, and wonder if their result generalizes in an equivariant way (see Section 4 specifically). It seems that the definition of K I give above is one sensible way to go to generalizing this. Going through Eisenbud and Schreyer's proof, everything can be done in an equivariant way, EXCEPT for knowing things about "Schur positive convex geometry."

I've thought about it for a while and looked for references, but have no idea where to begin to see if this kind of thing has been studied before. Hopefully someone else knows something about it.

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You can use mathbb rather than mathbf for black-board bold, if that's what you want. –  Theo Johnson-Freyd Jan 5 '10 at 16:23
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