# Dominance order

Today in class I claimed that the transition matrix from the elementary symmetric functions $\{e_\lambda\}$ to the monomial symmetric functions $\{m_\lambda\}$ is upper triangular. However, it is not trivial to decide what order the rows and columns of this matrix should be written in to see this. Let’s discuss this here.

Two useful concepts:

– A useful poset on the partitions of n is the “dominance order”: http://en.wikipedia.org/wiki/Dominance_order

– A useful operation on the partitions of n is “conjugation”. Conjugating a partition corresponds to flipping its diagram: http://mathworld.wolfram.com/ConjugatePartition.html

Prove these two claims:

1. When you multiply out $e_{\lambda} = \prod_i e_{\lambda_i}$, the monomial symmetric function $m_{\lambda^t}$ appears exactly once, where $\lambda^t$ is the conjugate of $\lambda$

2. All other monomials $m_{\mu}$ that appear have the property that $\mu \leq \lambda^t$ in the dominance order.

How does this imply our claim that the transition matrix can be made upper triangular?