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?

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: