I have made a study of polytopes with the books of Ziegler and "Integer Programming" of Conforti, my main goal is to study matroid polytopes; to study matroids I have thought about the book "MATROID THEORY" of Welsh. But how do I connect the two topics, i.e. polytopes with matroids? I am not clear if in the same text of Welsh this is specified, or if there is any additional reference to do this work? Any suggestions are welcome!
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3$\begingroup$ I don't think there is a book on matroid polytopes. And learning about matroids in general, like from the textbook of Welsh, is a good idea, but I'm not sure it will help you understand contemporary research in matroid polytopes. I might suggest you learn about generalized permutohedra (alias polymatroids), a slightly larger class of polytopes that includes matroid polytopes. The classic papers here are "Permutohedra, associahedra, and beyond" by Postnikov arxiv.org/abs/math/0507163 and "Faces of Generalized Permutohedra" by Postinkov-Reiner-Williams arxiv.org/abs/math/0609184. $\endgroup$– Sam HopkinsCommented Dec 12, 2023 at 16:18
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$\begingroup$ Sounds nice! Please write down the stuff you learn and share! $\endgroup$– Per AlexanderssonCommented Dec 12, 2023 at 19:08
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$\begingroup$ @SamHopkins Do you think that studying the type of matroids that you indicate can be used to write a monograph, which is the final work for undergraduate studies in mathematics? I say this because it can be a topic that, in depth, can be something that exhibits a section interesting of the polytopes, or on the contrary, do you think that related to this there is some result/theorem/example that is more interesting?. Your recommendation is very useful! I appreciate it; $\endgroup$– WrloordCommented Dec 13, 2023 at 2:38
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$\begingroup$ I don't quite understand your last question, but if you're asking if matroid polytopes would be a reasonable topic for an undergraduate thesis, then I would say yes. And in fact, a quick search on Google lead me to this bachelor thesis on matroid polytopes by Christoph Pegel, which you might be interested in: iazd.uni-hannover.de/fileadmin/iazd/Pegel/… $\endgroup$– Sam HopkinsCommented Dec 13, 2023 at 2:40
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$\begingroup$ @SamHopkins Exactly, my question is what you mention; Furthermore, I was wondering if I could lead the thesis towards Postnikov's article with the article "Faces of Generalized Permutohedra", studying this article in depth would be a good approach? I appreciate your comments from your experience. pd:The last article you attached looks really interesting, thank you so much for it! $\endgroup$– WrloordCommented Dec 13, 2023 at 2:44
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2 Answers
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You might like the book Coxeter Matroids by Borovik, Gelfand, and White.
In some sense, this book is actually about generalizations of matroids. One generalization is from matroids to flag matroids (which are sequences of matroids related by quotient/concordance). The other direction of generalization is to other Coxeter types. Usual matroids are type A (and related to the permutohedron stuff mentioned in the comment by Sam Hopkins).
While many books will be completely dedicated to matroid via various axioms, this book already introduces the polytope perspective in the first chapter.
answered Dec 12, 2023 at 17:01
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Matroid polytopes are a standard topic in the field of combinatorial optimization, and as such I would recommend the beautiful text Combinatorial Optimization - Polyhedra and Efficiency by Lex Schrijver. This is an encyclopedic treatment of the subject covering far more than matroid polytopes. Matroid polytopes (and many other topics related to matroids) are covered in Part IV. For example, Edmonds' classic inequality description of the independent set polytope of a matroid is presented in Part IV.
answered Dec 12, 2023 at 20:50