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.

  • 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.

  • Implicit in many of the above, and perhaps too obvious to note: when the function can be expressed as a sum of symmetric functions that are already known to be Schur positive. – Zachary Hamaker Jun 19 '17 at 16:28
  • Sometimes you can write a symmetric function as a sum of products of Schur functions, which are then Schur positive (via LR-rule or representation theory, if you want). – Sam Hopkins Jun 19 '17 at 22:08
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    There is an identity of Kirillov that I like a lot: $(s_{c^r})^2 = s_{c^{r-1}}s_{c^{r+1}} + s_{(c-1)^r}s_{(c+1)^r}$. See: A consequence is that $(s_{c^r})^2-s_{c^{r-1}}s_{c^{r+1}}$ is Schur positive, which I think is not so obvious otherwise. See: (Conjecture 1) for an interesting conjecture about Schur positivity of expressions of this form. – Sam Hopkins Jun 19 '17 at 22:13

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.

  • Ah, right! Forgot about Gasharov's proof. His proof somehow produces the 'highest-weights' in a hypothetical crystal graph, but no-one has published a full crystal graph structure yet. – Per Alexandersson Jun 16 '17 at 6:05
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    @PerAlexandersson : If you will forgive a shameless plug, Brosnan and I have a preprint on the ArXiv proving a conjecture of Shareshian and Wachs about a representation-theoretic interpretation of those Schur-function coefficients. So this gives a representation-theoretic viewpoint, though I have to say that I haven't thought about this problem in terms of crystals. – Timothy Chow Jun 16 '17 at 14:03
  • Ah, right, the Hessenberg variety connection. I have worked a bit (on the combinatorial side), of similar functions. Perhaps there is some 'incomplete flag' Hessenberg variety explaining the various positivity conjectures/results in the paper w. G. Panova we recently put on arxiv.. – Per Alexandersson Jun 16 '17 at 15:27

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.

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