This is an optional argument for the schurRing and symmetricRing functions. It selects how the partition-indexed generators of the ring are interpreted as symmetric functions. The possible values are "Schur" (the default) and "Monomial".
When Basis => "Schur", the ring generator s_\lambda represents the Schur function indexed by \lambda. Multiplication uses the Littlewood-Richardson rule supplied by the Macaulay2 engine.
|
|
When Basis => "Monomial", the ring generator m_\lambda instead represents the monomial symmetric function indexed by \lambda. Multiplication is implemented by converting to the Schur basis, multiplying there via Littlewood-Richardson, and converting back to the monomial basis using Kostka numbers. The resulting ring is abstractly isomorphic to the Schur-basis ring but its elements are displayed and stored as linear combinations of monomial symmetric functions.
|
|
|
The consistency of the two bases can be verified against toM, which converts a symmetric function to the monomial basis:
|
|
|
A monomial-basis product agrees with the Schur-basis product after round-tripping through toS, verifying that the two rings are isomorphic with isomorphism toM/toS:
|
|
|
|
Converting back from the monomial basis to the Schur basis recovers the original Schur element:
|
The monomial-to-Schur transition is essentially indexed by kostkaNumber: the coefficient of m_\mu in s_\lambda equals K_{\lambda,\mu}. For example, the Kostka numbers with \lambda = (2,1) reproduce the monomial expansion of s_{(2,1)}:
|
|
|