Why do we care about Schur Positivity

Question

Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?

Answer 1

In algebraic combinatorics, conjectures that certain numbers are integers or are positive or are unimodal are really implicit challenges to find the hidden structure. And finding hidden structure is a large part of what mathematics is all about.

As lambda said in the comments, in the specific case of combinatorially defined symmetric functions whose Schur-coefficients are integers but not obviously positive, a positivity conjecture is an implicit challenge to find the representation of $\mathfrak{S}_n$ or $GL(n)$.

Let me give an example from my own research. For the sake of simplicity, I will use slightly nonstandard notation. If $P$ is a finite partially ordered set on the set $\{1,2,\ldots,n\}$ then define a coloring of $P$ to be a map $\kappa:P\to \mathbb{N}$ such that every pre-image $\kappa^{-1}(i)$ is a totally ordered subset of $P$. Then, given variables $x_1, x_2, \ldots$ we can define $$X_P := \sum_\kappa x_{\kappa(1)} x_{\kappa(2)}\cdots x_{\kappa(n)}$$ where the sum is over all colorings $\kappa$. It is easy to check that $X_P$ is a symmetric function. What is less obvious is:

Theorem (Haiman, Gasharov). If $P$ is a unit interval order then $X_P$ is Schur-positive.

When a combinatorialist sees a statement like this, it is not the bare fact of positivity that is so important. Rather, long experience has shown that statements like this indicate the presence of some structure lurking beneath the surface. Gasharov's proof gave a combinatorial interpretation of the Schur-coefficients in terms of something called $P$-tableaux. I won't define $P$-tableaux here but the point is that the Schur-positivity led to the search for this combinatorial structure that was not apparent from the original definition.

This is not the end of the story, however, because one can ask if there is a natural representation of the symmetric group whose character is given by $X_P$. (Haiman's proof has a representation-theoretic flavor but does not construct such a representation directly.) For the answer to this question, we had to wait about twenty years, when Shareshian and Wachs conjectured (and Brosnan and myself—and independently, Guay-Paquet—proved) that a certain action called the dot action of the symmetric group on the cohomology of a regular semisimple Hessenberg variety associated to $P$ has character equal to $X_P$ (or actually $\omega X_P$ but never mind this technicality for now). The exact definition of all these terms is not important here; again, the point is that Schur positivity was the "canary in the coal mine" that signaled the presence of this hidden connection between colorings of partially ordered sets and the cohomology of an algebraic variety.

The biggest unanswered question in this subject is:

Conjecture (Stanley–Stembridge). if $P$ is a unit interval order then $X_P$ is $e$-positive; i.e., it is a nonnegative combination of elementary symmetric functions.

Once again, the conjecture challenges us to find the hidden structure that would explain this empirically observed fact.