Seminar in Representation Theory
Sporadic groups and Automorphic forms
2026/11/10 (Tue.) 15:30~16:30
- Date : 2026/11/10 (Tue.) 15:30~16:30
- Location : Seminar Room 638, Institute of Mathematics (NTU Campus)
-
Speaker : Gerald Höhn (Kansas State University)
- Organizer : Ching Hung Lam (AS)
- Abstract :
Twenty of the 26 sporadic simple groups belong to families organized
around the Golay code, the Leech lattice, and the Moonshine module;
the remaining six are called $pariahs$. The Leech lattice and the
Moonshine module have deep connections with automorphic forms for
orthogonal groups.
In this talk, I will present an analogous picture for the Rudvalis group,
one of the pariahs, and the exceptional Lie group of type $E_6$.
The exceptional theta correspondence for
${E_6\times\mathrm{SU}(2,1)\subset E_8}$ leads to the mass identity
\begin{aligned}
&\frac{1}{|{}^3D_4(2):3|}
+\frac{1}{|2.W(E_8)^+.S_4|}
+\frac{1}{|4.2^8.S_{10}|}\\[4pt]
&\quad+\frac{1}{|2^5.(2^6:\mathrm{GL}_3(2)).S_4|}
+\frac{1}{|4^6.W(E_6)|}
+\frac{1}{|2^6.W(E_6)'|}
+\frac{2}{|{}^2F_4(2)|}\\[4pt]
&\qquad=\frac1{64}\Bigl|
\zeta(-1)\zeta(-5)\zeta(-7)\zeta(-11)
L(-4,\chi_{-4})L(-8,\chi_{-4})
\Bigr|,
\end{aligned}
for a genus of eight definite integral $E_6$-structures.
The resulting geometry yields the intrinsic order formula
$ |\mathrm{Ru}|
=\dim_{\mathbb C}\operatorname{Sym}^3(\mathbb C^{28})
\cdot |{}^2F_4(2)|
=2^{14}\cdot3^3\cdot5^3\cdot7\cdot13\cdot29$.
