The symmetric functions catalog

An overview of symmetric functions and related topics

2023-09-20

Schur positivity

There are several different techniques to prove that a symmetric function is Schur positive.

Robinson–Schensted–Knuth (RSK)

Examples of results and papers that use Robinson–Schensted–Knuth 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.

Austin Roberts give 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:

  • Type $B$ version of dual equivalence for shifted tableaux, [Def. 4.5.1, Rob14].
  • Schur-P and Schur-Q dual equivalence.
  • Dual equivalence can be generalized to quasi-symmetric Schur functions, see A. Roberts [Rob16].
  • Permutation classes that give rise to Schur positive expressions, see [ER16a].

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 sum 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 (a 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 non-symmetric Macdonald polynomials, [AG18] and Stanley symmetric functions, [MS15c], and Dual stable Grothendieck polynomials, [Gal17].

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

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 [ER16a], 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 non-negative 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 $S = \{i_1,i_2,\dotsc,i_\ell\}.$

References