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I am interested in learning about standard monomial theory and Seshadri's program. I find the topic interesting, but I could not yet find a resource which kind of "dumbs it down" enough (a kind of introduction to a layman etc.). Could an expert please point me to some not so difficult to read introductions to SMT, if they exist? Eventually, I would like to understand Littelmann's path models, but I don't mind if the introduction doesn't get there. It will hopefully give me enough information to be able to understand Littelmann's work later on. Many thanks! (By the way, I am currently happy with working over $\mathbb{R}$ and $\mathbb{C}$ for the time being.)

Update: I found Seshadri's book really helpful to understand how SMT developed. Then it was easier to understand how Littelmann's path model came about. I also found some slides by Leonard Hardiman to be really helpful (http://math.univ-lyon1.fr/~hardiman/hardiman_slides_C.pdf), particularly because they contain some figures, which enables one to see what the path operators do, when applied to a particular path.

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Seshadri wrote a book, "Introduction to the Theory of Standard Monomials" (https://doi.org/10.1007/978-981-10-1813-8), which is very easy-going, especially in the beginning. But perhaps it does not cover exactly what you're interested in?

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    $\begingroup$ The paper link.springer.com/chapter/10.1007/978-93-86279-11-8_24 and the book Hodge Algebras by DeConcini, Eisenbud, and Procesi may be of interest. $\endgroup$
    – Richard Stanley
    Commented Oct 22, 2022 at 2:59
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    $\begingroup$ In case Springer changes its URLs in the future, the paper mentioned in Richard Stanley's comment is The Development of Standard Monomial Theory-I by C. Musili, in A Tribute to C. S. Seshadri, pages 385–420. $\endgroup$
    – Timothy Chow
    Commented Oct 22, 2022 at 12:38
  • $\begingroup$ Thank you @RichardStanley and TimothyChow for your additional help ! I now have enough reading material. Seshadri's book, by the way, at least at the beginning, is easy-to-read, which is great. Thank you all! $\endgroup$
    – Malkoun
    Commented Oct 22, 2022 at 22:27
  • $\begingroup$ Thank you Sam Hopkins for mentioning Seshadri's book. It was really helpful to me, especially the fact that for the special cases studied there (which include $G/P$ with $P$ being a maximal parabolic subgroup), they index the basis using admissible pairs in $W/W_P$, where $W$ is the Weyl group and $W_P$ is the Weyl group of $P$. I found that interesting. $\endgroup$
    – Malkoun
    Commented Nov 6, 2022 at 5:25
  • $\begingroup$ I should also mention that after reading parts of Seshadri's book, I found some nice slides by Leonard Hardiman to be helpful, because they had some figures (math.univ-lyon1.fr/~hardiman/hardiman_slides_C.pdf). I guess I need some figures to understand what the path operators are doing. I found this very helpful. $\endgroup$
    – Malkoun
    Commented Nov 6, 2022 at 5:28

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