1) What is the action of the coproduct on X_A?

data file 1

1') How is this related to the induced representations?

2) What is the action of the antipode on any basis (the X, P or M basis)?

data file 2

data file 4 - compose the antipode and \omega to help

3) Define the involution Ω(P_A) = (-1)^{|A| - \ell(A)} P_A. We proved a formula for Ω(X_A) and we can conjecture a formula for Ω(M_A).

data file 4

4) h_A expands in the X-basis postively. Give a formula for this expansion and give a represenation theoretic interpretation for the h-basis.

data file 5

5) The map \chi: NCSym -> Sym is defined by \chi(P_A) = p_{\lambda(A)}. What is \chi(X_A)?

data file 6

The programs that I used to generate this data.

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