S(e_n) = (-1)^n h_n, and a mistake in lecture

1. In class I explained that S(e_{\lambda}) = (-1)^n h_{\lambda} for any partition \lambda of n, and that this follows from the equation

\sum_{k=0}^n (-1)^k h_k e_{n-k} = 0

for n \geq 1. Can someone provide a proof of this?

(Hint: there are elegant formulas for the generating functions \sum_n e_nt^n and \sum_n h_nt^n.)

2. I made a mistake in defining the inner product on Sym. I meant to decree that m_{\lambda} and h_{\lambda} are dual bases. (I wrote e instead of m). So we have:

\langle m_{\lambda}, h_{\mu} \rangle = 1 if \lambda = \mu, and \langle m_{\lambda}, h_{\mu} \rangle = 0 if \lambda \neq \mu.

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One comment

  1. Pingback: Two important comments on the scalar product « hopf algebras and combinatorics

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