The following question stems from page 10 of this dissertation. The Hilbert series is defined for $\mathbb{C}^n$ by

$$HS(t_1, t_2, \ldots, t_n; \mathbb{C}^n) = \sum_{i_1, \ldots, i_n = 0}^{\infty} t_1^{i_1} t_2^{i_2} \ldots t_{n}^{i_n} = \prod_{i=1}^{n}\frac{1}{1-t_i} = PE\left[\sum_{i=1}^{n}t_i\right]$$

Here $PE$ denotes the Plethystic exponential function, defined for a multivariable function $f(t_1, \ldots, t_n)$ satisfying $f(0, \ldots, 0) = 0$ by

$$PE[f(t_1, \ldots, t_n) := \exp\left(\sum_{r=1}^{\infty}\frac{f(t_1^r, \ldots, t_n^r)}{r}\right)$$

Now, in $HS(t_1, \ldots, t_n; \mathbb{C}^n)$ we do the following change of variables: $t_1, \ldots, t_n$ to $y_1, \ldots, y_{n-1}, t$ with map:

$$t_1 = t y_1, \qquad t_2 = t\frac{y_2}{y_1},\qquad \cdots \quad ,t_n = t \frac{1}{y_{n-1}}$$

so that

$$HS(t_1, t_2, \ldots, t_n; \mathbb{C}^n) = PE\left[\left(y_1 + \frac{y_2}{y_1} + \cdots + \frac{1}{y_{n-1}}\right)t\right]$$ $$ \qquad \qquad \qquad = PE\left[\chi\left([1,0, \ldots, 0]_{SU(n)}\right)t\right]$$ $$ \qquad \qquad = \sum_{k=0}^{\infty}\chi\left([k, 0, \ldots, 0]_{SU(n)}\right)$$

where $[n_1, \ldots, n_r]$ denotes the Dynkin label of the representation of the Lie algebra of $SU(n)$ and $\chi([n_1, \ldots, n_r])$ denotes the corresponding character.

The question is: how does one transition from the first line containing PE of a bunch of $y_i$'s to the next line containing the character? What happens to the $y_i$'s?

In terms of the original dissertation, how does one go from eqn (2.22) to (2.23), and then (2.23) to (2.24)?

I know that for the n-dimensional representation of $SU(2)$, there's a simple character formula

$$\chi(y) = \mbox{Tr }y^{2 J_3} = \frac{y^{n} - y^{-n}}{y - y^{-1}}$$

Presumably now that we're dealing with $SU(n)$, the single $y$ is replaced by $y_{1}, \ldots, y_{n-1}$.


\begin{align} y_1 + \frac{y_2}{y_1} + \cdots + \frac{1}{y_{n-1}}=\chi\left([1,0, \ldots, 0]_{SU(n)}\right) \end{align}

is what you expect if each of the $y_i \in U(1)$. If this is true, then each term in the sum is also in $U(1)$ and the product of all the terms in the sum in one. So the terms in the sum are the eigenvalues of some $U \in SU(n)$ and their sum is just the trace (ie. character) of the fundamental representation (there are exactly $n$ terms in the sum, as expected).

Now why it's true that each $y_i \in U(1)$ is a matter of digging around in the notation of that dissertation/subfield/physics. But it seems to be true, based on say Equation 2.2 of this paper, which that dissertation cites.


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