Overcounting Verma module dimensions? I am having trouble following the discussion 7.1.2 in Conformal Field Theory by David Sénéchal, Philippe Di Francesco, and Pierre Mathieu. I summarize their statements below, hopefully accurately:
We associate to a Verma Module $V(c,h)$ generated by the Virasoro algebra the character generating function (character) $\chi_{c,h}(\tau)$ defined as
$$\chi_{c,h}(\tau) = \mathrm{Tr}\,q^{L_0 -c/24}$$
$$=\sum_{n=0}^\infty\mathrm{dim}(h+n)q^{n+h - c/24}$$
where the nome $q = \mathrm{e}^{2 \pi \mathrm{i} \tau}$ and $\mathrm{dim}(n+h)$ is the number of linearly independent states at level $n$ in the Verma module.
They go on to state that the character of a generic Verma module is written in terms of the generating function of the integer partitions of $n$, $p(n)$, as
$$\chi_{c,h}(\tau) = \frac{q^{h-c/24}}{\varphi(q)} = q^{h-c/24}\sum_{n=0}^\infty p(n)q^n = \sum_{n=0}^\infty p(n) q^{n+h-c/24}$$
where the last two expressions are included by me for clarity.
But isn't this (possibly) overcounting? In an earlier paragraph they note that $p(n)$ is the maximum number of independent states at level $n$, and even use this to argue for the uniform convergence of the character $\chi_{c,h}(\tau).$ I was under the impression that there were circumstances in which $p(n)$ was strictly greater than the number of independent states at level $n$. 
 A: By definition, the Verma module is the representation that has $p(n)$ independent states at level $n$, in other words you assume that the $p(n)$ states that you build by acting on your primary state with creation operators are independent. You can build another representation by assuming that these states are not independent, in other words by taking a quotient of the Verma module. 
In order for the resulting quotient not to be zero, you need some relation between $c$ and $h$, so that a null vector exists. However, when a null vector exists, nobody forces you to set it to zero. In this case, the Verma module is reducible i.e. it has a nontrivial subrepresentation generated by the null vector. In this situation there are two lowest-weight representations with the same conformal dimension: the Verma module (with a nonzero null vector) and the quotient (where the null vector is set to zero). In the quotient, if $n_0$ is the level of the null vector, the number of states at level $n$ is $p(n)$ if $n<n_0$, and strictly less than $p(n)$ for $n\geq n_0$.
