nLab Schur positivity

Definition

A symmetric function $f$ with coefficients in the ground field is Schur positive if, when expanded in the linear basis of Schur polynomials $f = \sum_\mu a_\mu s_\mu$, all coefficients $a_\mu$ are positive numbers.

References

• Rebecca Patrias, What is Schur positivity and how common is it? (arXiv:1809.04448)

• Thomas Lam, Alexander Postnikov, Pavlo Pylyavskyy, Schur positivity and Schur log-concavity, math.CO/0502446

• James M. Borger, Witt vectors, semirings, and total positivity, arxiv/1310.3013

We extend the big and p-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the p-typical case, it uses positivity with respect to an apparently new basis of the p-typical symmetric functions.