A list of some open problems related to NCSym

For most of the following problems I have computer data and so if possible we should be able to answer.

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.