What techniques are there to prove Schur positivity? As the title says, what methods exists for proving that a symmetric polynomial (or function) is Schur positive, perhaps involving extra parameters, in which case coefficients should be polynomials in the parameters with non-negative coefficients.
This is what I have seen in the literature:


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*Representation-theoretical proof. Construct a (graded) $S_n$-module, and show that the Frobenius map of the decomposition into irreducibles gives the polynomial. This often involves finding recursions, and does not give formulas for the coefficients. Examples: Modified Macdonald polynomials, and things in diagonal harmonics. This MO-post.

*RSK-type proof. This essentially gives a way to convert words to semi-standard tableaux, and is therefore a type of bijective proof. Examples: skew-Schur functions (and thus the Littlewoord-Richardson rule).

*Via Gessel fundamental quasisymmetric expansion. It is usually easy to find the Gessel expansion of a combinatorially defined polynomial. Once this is known,
the goal is to gather these pieces into Schur-polynomials.
The 'Schur-expansion' is technically known from the Gessel expansion, but it is given in the form of functions $S_{\alpha}$, where one needs to modify the compositions $\alpha$ according to the 'slinky' rule to obtain partitions. This can introduce signs, that needs to be taken care of via a sign-reversing involution or similar.

*Type-A crystal proof. If the polynomial is given as a sum over combinatorial objects, one can try to define a certain graph structure on these objects, fulfilling some combinatorial (fairly local) axioms (Stembridge did this characterization, if I recall). Each connected component of this graph will each correspond to a Schur polynomial in the Schur expansion. This is related to my old question, before I knew about crystals, and it turns out it is enough to consider three variables 
at a time (in the Stembridge axioms). Basically, if you can give a crystal graph in three variables, it should generalize to $n$ variables without any issue. The crystal structure is closely related to RSK, and also provides a representation-theoretical connection, as well as a (crystal) bijection to SSYTs. Examples: Stanley symmetric functions. Dual Grothendieck.

*Dual equivalence graph, introduced by S. Assaf. Similar idea as crystals/RSK/Gessel. From the fundamental quasisymmetric expansion, define a graph structure that gathers these pieces into Schur-positive parts.
Example: This article, which has a non-symmetric counterpart as well.
 A: Gasharov's proof that Stanley's chromatic symmetric function is Schur-positive for incomparability graphs of (3+1)-free posets doesn't seem to fit neatly into any of your categories (Discrete Math. 157  (1996), 193–197).  He expresses the coefficients of the Schur-function expansion as a signed sum of inner products with complete homogeneous symmetric functions.  The summands have a combinatorial interpretation and he defines a sign-reversing involution on them to cancel out everything except certain positive terms.  As a byproduct he obtains a combinatorial interpretation of the Schur-function coefficients.
More generally, if you have a combinatorial interpretation of the coefficients when the function is expanded in terms of some other symmetric function basis, then you can try applying the change-of-basis matrix (which in most cases has some kind of combinatorial interpretation, although sometimes it's very complicated), thereby expressing the Schur-function coefficients as some kind of signed sum of combinatorial objects.  Then you can look for a sign-reversing involution or some other combinatorial argument to cancel stuff out.
A: Some further examples of Schur positivity: Exercises 7.38, 7.46, and 7.91 in EC2, and Problems 116(b,d) and 137(a,b) in http://www-math.mit.edu/~rstan/ec/ch7supp.pdf (version of 14 March 2021).
A: There are also geometric arguments of Schur positivity. The outline is very similar to the representation-theoretic strategy. Construct a variety that deforms into a union of Grassmannians, and show that the cohomology class of the variety gives the polynomial. This often involves finding recursions, and does not give formulas for the coefficients. Examples: Stanley symmetric functions via the Lascoux-Schutzenberger tree, toric Schur functions and $k$-Schur functions.
Note, all I have done is replace "$Sn$-module" with "variety that deforms into a union of Grassmannians" and "Frobenius map of the decomposition into irreducibles" with "cohomology class of the variety" from your representation theoretic outline.
