Distance-transitive representations of the sporadic groups

On the Action of the Sporadic Simple Baby Monster Group on its Conjugacy Class 2B

“…For the Baction on the cosets of 2 1+22 .Co 2 , the sizes of the (2 1+22 .Co 2 )-orbits are already given in [14], up to a typo we are going to correct first. Moreover, the intersection Baby Monster action matrix of the shortest non-trivial (2 1+22 .Co 2 )-orbit, given in Table 3, has been determined independently in [26,27], using a 'by hand' strategy exploiting geometric arguments which yield a wealth of combinatorial data about the associated orbital graph.…”

“…Its character table in particular encodes information about the spectral properties of the orbital graphs associated with the permutation action, such as distance-transitivity or distance-regularity, see [14,5], or the Ramanujan property, see [7]. Here multiplicity-free actions, that is those whose associated endomorphism ring is commutative, have been of particular interest; for example a distance-transitive graph necessarily is an orbital graph associated with a multiplicity-free action.…”

Computing with graphs and groups

“…provides an overview of the state of the classification of distancetransitive graphs. Here we discuss the application of computing, which has been used in the discovery, analysis, and classification of certain distanceregular and distance-transitive graphs (see, for example, [37,31,23,18]). We also remark that computing is used in the determination of feasible intersection arrays for possible distance-regular graphs (see, for example, [3]).…”

“…Computation has been applied extensively in the classification of the graphs on which a sporadic simple group or its automorphism group acts primitively and distance-transitively (see [23,18]; the results of this classification are given in Chapter ??). In the process of this classification, the primitive multiplicity-free permutation characters for these sporadic groups were also determined, and for most of the primitive permutation representations of sporadic groups, a collapsed adjacency matrix was computed for the orbital graph of least degree.…”

“…Suppose that G acts distance-transitively on a graph G. Then G must be an orbital graph for G of the smallest or second-smallest vertex-degree (see [18]). Furthermore, each orbital for G must be self-paired, which is equivalent to the property that the permutation character of G on V is the sum of distinct complex irreducible characters, each with Frobenius-Schur indicator +1 (see [2, p. 64]).…”