Low-dimensional representations of quasi-simple groups (Q2709398)

The authors present a list of all the absolutely irreducible representations of degree at most 250 of all quasi-simple finite groups. The representations may be defined over a field of arbitrary characteristic, but the list excludes the representations of groups of Lie type over fields of the same characteristic as the defining characteristic of the group. The Tits simple group is, however, included in the list for all characteristics, as it is not considered to be a group whose defining characteristic is 2. (The authors remark that, for a given group \(G\) of Lie type of Lie rank \(l\), F. Lübeck has determined all absolutely irreducible representations of \(G\) of degree at most \(l^3/8\) in the defining characteristic of \(G\), as well as those of degree at most 250.)NEWLINENEWLINENEWLINEThe main body of the paper is presented in a 29 page list of increasing degrees of representations, with the corresponding group being described in Atlas notation. The authors include the Frobenius-Schur indicator of the representation. This is defined to be \(0\) if the representation is not self-dual. If the characteristic of the field is not 2, the indicator takes the value \(1\) if the representation preserves a non-zero symmetric bilinear form, and the value \(-1\) if the representation preserves a non-zero alternating bilinear form. If the characteristic of the field is 2, the indicator takes the value \(1\) if the representation preserves a non-zero quadratic form, and the value \(-1\) if the representation preserves a non-zero alternating bilinear form but no non-zero quadratic form. In characteristic 2, calculation of the value of the indicator (when it is not \(0\)) can present considerable calculational problems.NEWLINENEWLINENEWLINEUsing the classification of finite simple groups, the authors restrict their attention to simple groups of Lie type, alternating groups and sporadic simple groups, as well as their covering groups. For groups of Lie type, the Landazuri-Seitz-Zalesskii bound rapidly reduces the size of the list of groups to be considered. Excluding the groups of type \(L_2(q)\), which are easily handled, there are about 70 remaining groups of Lie type that may occur. Results available in the literature reduce this to about nine groups. These are handled in various ad hoc ways, including algorithmic computer algebra systems, developed in part by the first author. The alternating groups \(A_n\) are investigated using tables of decomposition numbers, which exist for \(n\leq 14\). Once the value of \(n\) is 14 or more, the only irreducible representations that occur correspond to partitions of \(n\) into a small number of parts, such as \((n-1,1)\), \((n-2,2)\) or \((n-2,1^2)\). The covering groups of the alternating groups are handled using results of Wales (1979). The sporadic groups are investigated using the Modular Atlas, as well as research papers, published or unpublished, of various authors. The last entry of the list is the Thompson sporadic group, which arises from its integral representation of degree 248, which translates into an irreducible representation in all characteristics.NEWLINENEWLINENEWLINEThe paper is a useful addition to the research literature on group representations, providing a valuable supplement to the Atlas and Modular Atlas. The list of 61 references is particularly gratifying, as it helps to direct the interested reader to sources that may not be well known, such as diploma theses or relevant web sites. (Also submitted to MR).