Symmetric Functions (New) [HB 107]

New Features:

• Algebras of symmetric functions can be created with any one of five possible bases: a basis consisting of Schur functions, Elementary, Monomial, Homogeneous or Power Sum functions.

• Elements of algebras of symmetric functions can be created as linear combinations of basis elements (indexed by partitions) or from coercing a polynomial or a scalar.

• Algebras of symmetric functions are rings so there are a number of ring predicates which are available for these algebras also.

• The print style of symmetric functions can be altered using an attribute on the algebra.

• Symmetric functions can be added, subtracted, multiplied and composed (plethysm). They can be tested for homogeneity and equality.

• Symmetric functions can be decomposed into a sequence of basis elements and coefficients thereof. A coefficient corresponding to the basis element of a given partition can be directly accessed. The number of basis elements with non zero coefficient in an element and the degree of an element can also be determined.

• Symmetric functions can be coerced into polynomial rings.

• A Frobenius homomorphism may be applied to symmetric functions. Inner products of symmetric functions can be taken. A set of tableaux for which a Schur function is the generating function can be retrieved and the character of the symmetric group corresponding to a symmetric function can be created.

• Matrices converting from any of the five bases to any other of the five bases can be calculated.