In question 5a), we’re given a definition for the Hilbert series as Hilb$(T(V):q)=\sum_{n=0}^\infty\dim(V^{\otimes k})q^k$. I’m unsure how to interpret this, since the index, $n$, doesn’t show up in the terms of the sum. Should I read this as $n=k$ Also, what is the domain of $q$? Real numbers?

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First question: Refer to the post by Lisa Clayton. (The answer is “yes” –there’s a typo.)

Second question: I strongly suspect that the answer is “it doesn’t matter.” This is combinatorics/algebra after all. More formally, we are talking about an object that lives in the ring of all formal power series, so where lives is immaterial to the question at hand.

By the way, if you type the word “latex” after the first dollar sign, the system will compile your LaTeX code.

I didn’t notice that at all! I just went through my HW and changed those n’s to k’s. Thanks!

Yes, is just a formal variable. (Although it wouldn’t hurt if you consider it to be a complex number and figure out when the series converges.)

I’m not sure about this, but I have heard before combinatorialists sometimes talking about what the poles are of a given power series. What do the singularities tell us about a generating function and what combinatorial information can we extract about the family under consideration?

It tells us roughly how fast the series grows, I think.