Combinatorics Seminar 2018

Speaker: Alexander Ivanov, Imperial College London

Date: Mon, Apr 09, 2018

Time: 15:30 - 17:00

Venue: Middle Meeting Room

Title: The Monster Group: from Tilde Geometry to Majorana Theory


According to the Classification of Finite Simple Groups (CFSG), a finite non-abelian simple group is either an alternating group, a group of Lie type, or one of 26 sporadic simple groups. The Lie type groups enjoy a uniform theory of buildings and $BN$-pairs, while each of the sporadic simple groups has a story of its own. The fundamental question of what is the purpose for the sporadic groups to exist has both mathematical and philosophical flavor. For some groups the answer can be found in J. Tits' brilliant survey and in a later paper on the Monster group. The Witt's design and the Golay code are the canonical tools to study the Mathieu groups. These structures pave the way to Conway groups through the Leech lattice and while Fischer's groups are best viewed as 3-transposition groups. The largest and most remarkable sporadic simple group, the Monster group was discovered independently by B. Fischer and R. Griess around 1973 and constructed by R. Griess in 1980. Michael Atiyah famously said that `the discovery of the Monster alone is the most exciting output of the classification of finite simple groups.'

It is extremely desirable to locate the point where sporadic groups diverged from the groups of Lie type. A Borel subgroup $B$ related to the fixed prime $p$ is the normalizer of a Sylow $p$-subgroup so it exists in any group $G$ as long as $p$ divides its order. Next one considers parabolic subgroups $P_i$ where $i$ runs through some index set. These are subgroups containing $B$, and one considers the coset geometry ${\cal G}(G)$ of $G$ associated with the subamalgam in $G$ formed by the parabolic subgroups. If $G$ is of Lie type and $p$ is its natural characteristic then ${\cal G}(G)$ will be the corresponding building and its crucial feature is hidden in apartments stabilized by the Weyl subgroup, which in turn leads to the subgroup $N$ in the $BN$-pair definition. The apartments keep the whole structure together, in particular this forces the coset geometry to be simply connected. This can be considered as a story with a happy end. For sporadic groups the story is just beginning.The parabolic subgroups can be defined as 2-local subgroups containing a given Sylow 2-subgroup, although the absence of apartments allows the parabolic subgroups to grow along with $G$, which might or might not be the universal completion of the amalgam.

The construction outlined in the previous paragraph applied to the Monster group leads to the tilde geometry ${\cal M}$ discovered by M. Ronan and St. Smith and belonging to a rank five diagram, in which the leftmost edge stands for the famous triple cover (known as the Foster graph) of the generalized quadrangle of order 2. The automorphism group of the generalized quadrangle of order 2 is the symplectic group $Sp_4(2)$, which is isomorphic to the symmetric group $S_6$ of degree 6, while the diagram is just the double edge. The tilde above this edge in the diagram of ${\cal G}(M)$ stands for the triple cover, whose automorphism group is the non-split extension $3 \cdot S_6$. The diagram without the tilde belongs to a unique geometry ${\cal G}(Sp_{10}(2))$. The point stabilizers in the latter classical geometry and in the Monster are of the form $2^{10}.2^5.L_5(2)$ and $2.2^5.2^5.2^{10}.2^{10}.2^5.L_5(2),$ respectively.

The Monster group geometry ${\cal G}(M)$ was proved to be simply connected in and it is an ongoing research to turn the simple connectedness result into a new completely self-contained construction for the Monster as was done for the Fourth Janko's group $J_4$. The crucial step in the simple connectedness proof of ${\cal G}(M)$ is to prove that the fundamental group of the Monster graph is generated by the homology classes of paths along the triangles. The Monster graph is a graph on the class of $2A$-involutions in the Monster, where two vertex-involutions are adjacent whenever their product is again a 2A-involution. In order to turn this into a construction of the Monster one needs to show that $\Gamma(M)$ is uniquely recovered from its local structure. A major goal of my research is to accomplish this task. This lead the author to launching the Majorana Theory, which provides an axiomatic approach to studying the Monster group and its 196884-dimensional non-associative algebra.