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:


* 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.](https://mathoverflow.net/questions/254782/a-schur-positivity-conjecture-related-to-row-and-column-permutations)
* 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](https://mathoverflow.net/questions/240213/schur-positivity-on-2-letter-alphabets-implies-schur-positivity-on-n-letters), 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](https://arxiv.org/pdf/1703.02466.pdf), which has a non-symmetric counterpart as well.



What techniques are there to prove Schur positivity?
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.


