I’m going to go ahead and start a discussion on HW #1, mainly because I have questions of my own.
The rules, per Dr. Ardila: discussing general ideas is ok, even a slight hint or two, but no solutions should be posted.
Helping each other out, however, is highly encouraged.
hopf algebras and combinatorics 
About question #5:
1. Does anyone have any good resources for the Hilbert series? Mathworld was oddly unhelpful, Wikipedia only marginally so, and the blogs I read are a more advanced than I need.
2. About the generators of
and 
: am I correct in thinking that the ideals being generated won’t belong in every tensor product of 
?
Section 11.5 of Dummit and Foote talks about these algebras, including the dimension of each term. My understanding is that the Hilbert series is just the sum of these dimensions over the whole thing.
As Katrina said, the book is tremendously helpful in solving these problems. I have been picturing these tensor algebras as something similar to polynomial rings where instead of having multiplication given by
, it is given by 
. Thus a “polynomial” multiplied by another “polynomial” results in a “polynomial” of higher “degree”. Seeing it this way helped me see what elements of the ideals looked like (think of polynomials)
As I understand it, given that an algebra can be decomposed (graded) into a direct sum of different “degrees”, then the Hilbert series is a formal power series that simultaneously describes the dimensions of each of the parts (like a generating function). Keeping this in mind for parts b and c is helpful because it might not be clear at first what we’re finding the dimension of (at least it wasn’t too clear for me initially).