toM -- Monomial (m-) basis representation

Given a symmetric function f, the function toM returns a representation of f as a linear combination of monomial symmetric functions. The output is a RingElement in a SchurRing with Basis => "Monomial", so that the basis element m_\mu represents the monomial symmetric function indexed by the partition \mu.

Internally, the conversion uses Kostka numbers: the Schur-to-monomial transition is s_\lambda = \sum_\mu K_{\lambda,\mu} m_\mu.

If the target monomial ring M is not specified, the first call to toM on an element of S creates and caches an associated monomial-basis ring with the same number of generators and the same coefficient ring as S; subsequent calls reuse this cached ring.

Alternatively one can supply the target monomial ring explicitly.

The function also accepts elements of a Symmetric ring, in which case the input is first converted to the Schur basis via toS.

The identity h_n = \sum_\mu m_\mu, summed over all partitions of n, reflects the fact that the Kostka numbers K_{(n),\mu} are all 1:

Power-sum functions admit a particularly simple monomial expansion: p_n = m_{(n)} identically, since p_n = \sum_i x_i^n. This can be read off for any n:

Longer partitions can still be handled; the coefficients are the corresponding Kostka numbers:

The map toM is invertible via toS, so a Schur element roundtrips through the monomial basis and back to the same Schur ring: