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#Schur positivity

There are several different techniques to prove that a symmetric function is Schur-positive, meaning that it expands in the Schur basis with nonnegative coefficients.

For a brief survey on Schur polynomials and the probability that a symmetric function is Schur-positive, see [Pat19, PW18].

R. Orellana, F. Saliola, A. Schilling, and M. Zabrocki give a method for recovering the Schur expansion of a symmetric function from its fundamental quasisymmetric expansion using only the coefficients indexed by partitions [OSSZ24]. The method is phrased in terms of the quasi-Kostka matrix and has applications to symmetric chain decompositions and plethysm.

R. M. Adin, A. Berenstein, J. Greenstein, J.-R. Li, A. Marmor, and Y. Roichman study Gallai and transitive colorings, including analogues for Coxeter systems, matroids, and commutative algebras [ABGL+23]. For type \(A\) root systems, the maximal-color transitive colorings carry a descent-set map whose quasisymmetric generating function is symmetric and Schur-positive.

A. R. Mayorova and E. A. Vassilieva give a \(q\)-deformed type \(B\) Cauchy identity connected to C.-O. Chow’s quasisymmetric functions, and use domino-tableau methods for type \(B\) Schur-positivity [MV22]. This is a type \(B\) analogue of the usual Schur-positive expansions obtained from tableaux. Their \(q\)-domino functions are related to LLT polynomials, but the possible zero labels in the domino tableaux give the variable \(x_0\) a special role, so the resulting functions need not be symmetric.

#Robinson–Schensted–Knuth (RSK)

Examples of results and papers that use the Robinson–Schensted–Knuth (RSK) correspondence:

#Dual equivalence

The idea behind dual equivalence is to start with the Gessel fundamental expansion of a symmetric function, as a sum over some set of combinatorial objects, and define a graph structure on these objects such that connected components sum to Schur functions.

A. Roberts gives a modified list of axioms, allowing for a local characterization of dual equivalence graphs [Rob13]. It is then enough to verify all graphs with at most six vertices, which usually can be done on a computer.

The following are extensions and analogues of dual equivalence:

#Crystal graphs

The idea behind crystal graphs is to define a graph structure on the (combinatorial) objects that generate the monomial expansion. By showing that the graph satisfies a set of axioms, it follows that each connected component sums to a Schur function. For example, one can define a crystal graph on skew SSYT in order to prove that skew Schur functions are Schur-positive.

A crystal graph also comes with an \(\symS_n\)-action on the combinatorial objects, so that one obtains an \(\symS_n\)-module, or representation, whose Frobenius image is exactly the original symmetric function.

For example, one can define a crystal graph on words of length \(n,\) and the \(\symS_n\)-action is generated by the Lascoux–Schutzenberger involutions \(s_i,\) acting on the words. This proves that \((x_1+x_2+x_3+\dotsb)^n\) is Schur-positive.

Notable examples include nonsymmetric Macdonald polynomials [AG18], Stanley symmetric functions [MS15], and dual stable Grothendieck polynomials [Gal17].

A flagged version of skew dual stable Grothendieck polynomials is shown to be key-positive using crystals in [Kun23]. Kohnert polynomials are key-positive with crystal structure defined in [Ass21].

The group \(\GL_m \times \GL_n\) acts on \(\setC[z_{ij}] / I_k\) where \(I_k\) is the determinantal variety, generated by the vanishing of the \((k+1) \times (k+1)\) minors of the matrix \([z_{ij}]_{1 \leq i \leq m, 1 \leq j \leq n}.\) An example using crystals in this setting is [PSY25].

#Crystal skeletons

There is a notion that unifies dual equivalence graphs and crystal graphs, called crystal skeletons. These were introduced in [Maa23], and further developed in [BCDS25].

#Representation theory

For a background, see the page on representation theory.

Examples: modified Macdonald polynomials, LLT polynomials, Eulerian symmetric functions.

#Edelman–Greene

The Edelman–Greene bijection is used to show that the Stanley symmetric functions are Schur-positive, see [EG87].

#The slinky rule

The idea is to use the slinky rule to convert an expansion in the Gessel fundamental quasisymmetric basis into a signed Schur expansion, and then one must invent some sign-reversing involution. An example of a paper that uses this strategy is [Ser17].

#Positive classes of permutations

In [ER16], the authors characterize several classes of permutations \(A \subseteq \symS_n,\) such that \[\sum_{\sigma \in A} \gessel_{n,\DES(\sigma)}(\xvec)\] is Schur-positive. We say that \(A\) is Schur-positive if the above sum is Schur-positive.

In [AR15], it is proved that \(A \subseteq \symS_n\) is Schur-positive if and only if there are nonnegative integers \(a_\lambda\) such that \[\sum_{\sigma \in A} \yvec_{\DES(\sigma)} = \sum_{\lambda \vdash n} a_\lambda \sum_{T \in \SYT(\lambda)} \yvec_{\DES(T)}.\] Here, \(\yvec_{S} = y_{i_1} \dotsm y_{i_\ell}\) where \(S = \{i_1,i_2,\dotsc,i_\ell\}.\)

More recent results can be found in [BER20] where cyclic descent sets are used.

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