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I'm making this post to ask for a reference about combinatorics: I'm a PhD student in representation theory/algebraic geometry. My background is mostly in algebra and geometry (and also mostly theoretical unfortunately).

Right now I'm interested in the cohomology of quiver and character varieties and their links with cohomological Hall algebras, quantum groups and the character ring of $\operatorname{GL}(n,\mathbb{F}_q)$. There's a lot of interesting combinatorics going on, but I really know very little about it. Especially regarding MacDonald polynomials etc.

Outside of the classic book by Macdonald "Symmetric functions and Hall polynomials" what could be a good reference to get into this area of combinatorics? Ideally I would like a book or notes with strong link to representation theory/cohomology theories etc

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    $\begingroup$ On the very combinatorial side (so not really satisfying your request to have strong links to representation theory, etc.) is Jim Haglund's book on $q$,$t$-Catalan numbers: www2.math.upenn.edu/~jhaglund/books/qtcat.pdf. $\endgroup$
    – Sam Hopkins
    Commented Nov 12, 2021 at 14:46
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    $\begingroup$ Why 'unfortunately' in "mostly theoretical unfortunately"? $\endgroup$
    – LSpice
    Commented Nov 12, 2021 at 16:27
  • $\begingroup$ Combinatorics of Coxeter groups google.com/books/edition/Combinatorics_of_Coxeter_Groups/… $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 12, 2021 at 18:16
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    $\begingroup$ Possibly of interest is F. Bergeron, Algebraic Combinatorics and Coinvariant Spaces. See bergeron.math.uqam.ca/wp-content/uploads/2013/12/book_debut.pdf. $\endgroup$
    – Richard Stanley
    Commented Nov 12, 2021 at 18:39
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    $\begingroup$ Crystal Bases: Representations and Combinatorics by Bump and Schilling may be relevant to your interests. They don't focus on cohomology per se, but they do give a very nice combinatorial development of (some of) the representation theory that you allude to. $\endgroup$
    – Timothy Chow
    Commented Nov 13, 2021 at 0:33

2 Answers 2

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M. Haiman "Notes on Macdonald polynomials and the geometry of the Hilbert scheme of points on $\mathbb{P}^2$". By one of the greatest specialists of interactions between combinatorics and algebraic geometry.

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I am not that strong on the representation-theory side, but know more about the combinatorics side. If you want to get an overview of the symmetric functions and the associated combinatorics (crystal bases, RSK etc), then one starting point (with references!) is www.symmetricfunctions.com.

I am the admin for this site, so all errors and issues are completely my fault :)

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