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  • Are there good examples of algebraic structures whose Hilbert-Poincaré series are that of special functions? I'm particularly interested in cases where complex analytic reasoning about those series sheds light on the algebraic structures in question.

  • Is there a dictionary of algebraic structures whose Hilbert-Poincaré series correspond to special functions?

Edit: I've heard of Monstrous Moonshine -- and I'm aware that there is a Mathieu Moonshine, but neither of those gets us outside of the territory of modular forms.

graveolensa
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  • If not modular, what kind of special functions would you like to have then? – Start wearing purple Jun 26 '13 at 20:17
  • Well, I meant that besides modular functions and Moonshine, and say the polylogarithms (see http://mathoverflow.net/questions/25428/what-is-special-about-polylogarithms-that-leads-to-so-many-interesting-identitie but it's not clear if the variation of mixed Hodge structure arises from treating polylogarithms as Hilbert Poincare series). But there are a lot which don't have integral or rational coefficients (Airy, Bessel, hypergeometrics) for instance, and given how fecund both Moonshine and the polylogarithms are, I can't imagine that there isn't data about algebraic structures in them. – graveolensa Jun 26 '13 at 21:24
  • Of course there is a lot of algebraic structures in them, there is a huge amount of books on this (example). But you are asking about Hilbert series - this is much more specific requirement. – Start wearing purple Jun 26 '13 at 21:38

1 Answers1

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Example 1

Let $G$ be the Monster group and let $\rho_0,\rho_1,\rho_2\ldots$ be its irreducible representations ordered by dimension. Then $$\mathrm{dim}\left(V_{-1}\right)q^{-1}+\sum_{k=1}^{\infty}\mathrm{dim}\left(V_k\right)q^k=j(\tau)-744,\qquad q=e^{2\pi i \tau},$$ where $j(\tau)$ denotes the $j$-function and $V_{-1}=\rho_0$, $V_1=\rho_1\oplus\rho_0$, $V_2=\rho_2\oplus\rho_1\oplus\rho_0$, $V_3=\rho_3\oplus\rho_2\oplus\rho_1\oplus\rho_1\oplus\rho_0\oplus\rho_0$ etc.

There are many other examples of this type (e.g. McKay-Thompson series) and very far-reaching generalizations known under the general name of Monstrous Moonshine.

Example 2

Another example is the character of the generic Verma module of the Virasoro algebra $$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m+n,0}.$$ Such module $M(c,\Delta)$ is generated by the action of $L_{n<0}$ on the highest weight state $|\Delta\rangle$ annihilated by all $L_{n>0}$ and satisfying $L_0|\Delta\rangle=\Delta|\Delta\rangle$. The states $L_{-n_k}\ldots L_{-n_1}|\Delta\rangle$ of the module are naturally labeled by partitions. Now the character is \begin{align} \chi(c,\Delta|q)=\mathrm{Tr}\,q^{L_0-c/24}=\frac{q^{\Delta+(1-c)/24}}{\eta(\tau)},\qquad q=e^{2\pi i\tau}, \end{align} where $\eta(\tau)$ is the Dedekind eta function arising from the sum over these partitions.

Start wearing purple
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