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edited Jun 22, 2022 at 7:16

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This is not a direct answer to your questions (which I still haven't understood completely from your formulation). But it seems important to place these questions within the extensive theoretical background, since I'm unconvinced that computing large examples (by brute force or otherwise) will provide much insight.

A convenient recent source is a short announcement by A. Khare here here. He and various collaborators have posted on arXiv a number of related papers on the faces of weight polytopes. For example, a paper with Ridenour was published in 2012 in Algebras and Representation Theory; the preprint is here here.

Some of this work refers back to work of Vinberg and others. Since the combinatorics of weight polytopes gets formidable even when the Weyl group is a symmetric group (presumably the best-behaved family of examples), it's a good idea to explore this literature and also to refine your own questions as far as possible.

This is not a direct answer to your questions (which I still haven't understood completely from your formulation). But it seems important to place these questions within the extensive theoretical background, since I'm unconvinced that computing large examples (by brute force or otherwise) will provide much insight.

A convenient recent source is a short announcement by A. Khare here. He and various collaborators have posted on arXiv a number of related papers on the faces of weight polytopes. For example, a paper with Ridenour was published in 2012 in Algebras and Representation Theory; the preprint is here.

Some of this work refers back to work of Vinberg and others. Since the combinatorics of weight polytopes gets formidable even when the Weyl group is a symmetric group (presumably the best-behaved family of examples), it's a good idea to explore this literature and also to refine your own questions as far as possible.

This is not a direct answer to your questions (which I still haven't understood completely from your formulation). But it seems important to place these questions within the extensive theoretical background, since I'm unconvinced that computing large examples (by brute force or otherwise) will provide much insight.

A convenient recent source is a short announcement by A. Khare here. He and various collaborators have posted on arXiv a number of related papers on the faces of weight polytopes. For example, a paper with Ridenour was published in 2012 in Algebras and Representation Theory; the preprint is here.

Some of this work refers back to work of Vinberg and others. Since the combinatorics of weight polytopes gets formidable even when the Weyl group is a symmetric group (presumably the best-behaved family of examples), it's a good idea to explore this literature and also to refine your own questions as far as possible.

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created Dec 22, 2013 at 18:30

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This is not a direct answer to your questions (which I still haven't understood completely from your formulation). But it seems important to place these questions within the extensive theoretical background, since I'm unconvinced that computing large examples (by brute force or otherwise) will provide much insight.

A convenient recent source is a short announcement by A. Khare here. He and various collaborators have posted on arXiv a number of related papers on the faces of weight polytopes. For example, a paper with Ridenour was published in 2012 in Algebras and Representation Theory; the preprint is here.

Some of this work refers back to work of Vinberg and others. Since the combinatorics of weight polytopes gets formidable even when the Weyl group is a symmetric group (presumably the best-behaved family of examples), it's a good idea to explore this literature and also to refine your own questions as far as possible.