Pseudo-Weyl Polytopes

Given a positive root (coroot?) \lambda there is a Weyl polytope W_\lambda which can be defined a few different ways. The most straightforward is W_\lambda=conv(W\cdot\lambda) the convex hull of the orbit of \lambda under the Weyl group. Another way to write this is as an intersection of cones:

W_\lambda=\bigcap_w C^{w\cdot\lambda}_w =\{\alpha \in\mathfrak{t}_\mathbb{R} : \langle \alpha, w\cdot\Lambda_i\rangle \ge\langle w_0\cdot\lambda,\Lambda_i\rangle \text{ for all } w \in W \text{ and fundamental weight } \Lambda_i \}

A pseudo Weyl polytope is a generalization where we replace (w\cdot\lambda)_{w\in W} with a collection (\mu_w)_{w\in W} satisfying \mu_v\ge_w\mu_w:

P(\mu_\cdot) := \bigcap_wC^{\mu_w}_w

(The inequality says that the vertex of each cone is inside every other cone and therefor a vertex of the polytope.)

An alternative way to describe any pseudo Weyl polytope (including Weyl polytopes) is by specifying an integer for every fundamental weight (M_\gamma)_\gamma\in\Gamma:

P(M_\cdot)=\{\alpha\in\mathfrak{t}_\mathbb{R} : \langle\alpha,\gamma\rangle\ge M_\gamma \text{ for all }\gamma\in\Gamma\}.

Converting from \mu_\cdot to M_\cdot can be done by the formula: M_{w\cdot\Lambda_i}=\langle\mu_w,w\cdot\Lambda_i\rangle.

Not all sets of integers will define a good polytope, the inequality imposed on weights imposes a set of inequalities on them called the edge inequalities. Furthermore if we have M that satisfy the edge inequalities there is a formula for obtaining the corrseponding \mu_\cdot.

2 Responses to “Pseudo-Weyl Polytopes”

  1. DN Says:
    December 9, 2007 at 8:10 am | Reply

    And MV polytopes are certain kinds of pseudo-Weyl polytopes, right?

  2. trdunlap2 Says:
    December 10, 2007 at 1:25 pm | Reply

    DN: Yes. Once I’ve sufficiently addressed the polytope-cycle relationship I think it’ll be clear why this is.

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