Precomputed Data

The operation of FiSGO requires some data that has already been computed at some point.

In this folder, we store the files containing key information to be used in the different modules of FiSGO. The data currently stored includes:

• The first 10^5 prime numbers.

• Data on almost all absolutely irreducible representations of the quasi-simple groups, compiled and partially computed by Hiss and Malle in [[1]], [[2]].

• Data on all properties of the sporadic simple groups (and their covers) relevant to FiSGO.

• Lower bounds to the degree of zero characteristic projective representations of the simple groups. Mainly due to Landazuri, Seitz, Tiep and Zalesskii.

• Lübeck’s data on the projective representations of finite simple groups of Lie type with non-exceptional Schur multiplier. Available in his website.

Description of the files

In the sequel, we describe the content of each file.

• BuiltinPrimes.txt: A txt file containing the first 10^5 prime numbers. Each line contains a single prime number. The prime numbers are stored in increasing order, with line 1 containing the first prime number, line 2 containing the second prime number, etc.

• Hiss_Malle_data.json: A JSON file containing the data on almost all absolutely irreducible representations of the quasi-simple groups. A discussion on the missing representations is given in the repository’s folder HissMalleTableFormats README. The file consists of an array of JSON objects, each of which contains the data on a single representation. The objects have the following fields:

  • "degree": Degree of the representation.

  • "name": Name of the quasi-simple group in LaTeX.

  • "field": Irrationalities of the Brauer characters.

  • "ind": Frobenius-Schur indicators. See [[1], \(\S\) 5].

  • "char": Field characteristics where the representation is defined. If null, then all characteristics are admited unless stated otherwise in "not_char".

  • "not_char": Field characteristics where the representation is NOT defined. In null, then all characteristics are admited unless stated otherwise in "char".

  • "code": FiSGO’s simple group code of \(G/Z(G)\). If the group is simple, then the code is simply that of the group. If the group is a cover of a simple group, the code of the associated simple group (\(G/Z(G)\)) is given.

• smallest_pirrep_degree_exceptions.json: A JSON file containing exceptions to the lower-bound formulas for the degree of zero characteristic projective representations of the simple groups, mainly due to Landazuri, Lübeck, Seitz, Tiep and Zalesskii. The file consists of an array of JSON objects, each of which contains the data on a single group (up to isomorphism). The objects have the following fields:

  • "code": An array of FiSGO’s simple group codes. All the codes conform the same simple group (up to isomorphism).

  • "degree": Lower bound on the degree of zero characteristic projective representations of the simple groups.

  • "irreps": Number of distinct irreducible representations of the given "degree".

• sporadic_groups_data.json: A JSON file containing the data on all properties of the sporadic simple groups (and their covers) relevant to FiSGO. The file consists of an array of JSON objects, each of which contains the data on a single group (up to isomorphism). The objects have the following fields:

  • "code": FiSGO’s simple group code.

  • "id": A unique identifier of the sporadic group.

  • "order": Order of the group.

  • "latex_name": List of possible notations for the group in LaTeX.

  • "multiplier": Schur multiplier of the group.

  • "pirreps": Array of JSON objects, each of which contains the data on the irreducible representations of a cover of the sporadic group. The objects have the following fields:

    • "name": A possible notation for the cover in LaTeX.

    • "degrees": Degrees of the irreducible representations. Given as a list of pairs, each pair contains the degree of an irreducible representation and the number of distinct irreducible representations of that degree.

• Lubeck/Lubeck_[ID].json: A collection of JSON files, one per Lie type simple group family. Each file contains all information on the degrees of projective representations of such groups up to rank 8, with the exception of those groups of Lie type with an exceptional Schur multiplier.

  The top level of each file consists of a JSON object with a single field, which depends on the number of parameters of the group:

  • Uniparametric: The field is simply the group ID string. Example: “RF”.

  • Biparametric: Each field contains the group ID followed by the dimensional parameter. Example: “CA-2”, “SD-5”.

  With the exception of the groups with ID: “RF”, “SZ” and “RG”; each of the above fields contains an array of JSON objects with the following fields:

  • "files": An array containing the names of the original files indexed in the current object. The names are the ones as downloaded from Lübeck’s website

  • "mod": Representations are grouped by the modularity of the \(q\) parameter of the group. This field contains the integer indicating such modularity. Example: If we care aboud \(q \mod 4\), then this field contains the number 4.

  • "mod_groups": Certain modular values produce the same representations, so they are grouped together. We label such groups as "0", "1", "2", ect., this field provides a correspondence between the group labels and the actual modular values. Example: if "mod" : 4 then a possible value for this field is {"0": [0, 2], "1": [1], "2": [3]}. This field is null whenever "mod" : 0.

  • "irreps": For each modularity group, an array of JSON objects is given, each corresponding to a different representation and with the following fields:

    • "degree": A rational polynomial with variable \(q\) given as a list of integer coefficients and a common divisor, which is a single integer that divides all integer coefficients. Example: [[1, -1, 2, -1, 1, 0], 2]. This polynomial represents the degree of the representation in terms of the group parameter \(q\).

    • "mult": Same as the previous field. This polynomial represents the multiplicity of the representation in terms of the group parameter \(q\).

  For the cases of the groups with ID: “RF”, “SZ” and “RG”; all fields are the same except for the ones inside "irreps". This case contained a difficulty: the polynomials were not rational, they contained multiples of \(\sqrt{2}\) for “SZ” and “RF”, and multiples of \(\sqrt{3}\) for “RG”. I.e. their fields of definition were \(\mathbb{Q}(\sqrt{2})\) and \(\mathbb{Q}(\sqrt{3})\) respectively. This is due to the parameter \(q\) being taken as \(q^2 = 2^{2*n+1}\) or \(q^2 = 3^{3*n+1}\) where \(n\) is the actual parameter used for these families.

  Nevertheless, the values these polynomials take are integer values, and \(q^2\) is an integer. Thus, we store two rational polynomials in the same form as above. The first polynomial corresponds to the even coefficients and the second to the odd ones quotient \(\mathbb{Q}(\sqrt{k})\) for \(k=2,3\) depending on the case. Example:

  • "degree": [[[1, 0, -1], 1], [[-1, 1], 1]]

  • "mult": [[[1, 0], 4], [[1], 4]]

  To evaluate this polynomial, note \(q^2\) is an integer and \(\sqrt{k}q\) is also an integer. Thus, if \(p(q)\) represents the original polynomial over \(\mathbb{Q}(\sqrt{k})\), this can be writen as follows:

  • Consider \(p_0(x)\), \(p_1(x)\) the rational polynomials corresponding to the even and odd coefficients. For the above example of the degree, \(p_0(x)=x^2-1\) and \(p_1(x)=-x+1\). Then \(p(q) = p_0(q^2) + \sqrt{k}q p_1(q^2)\). In this way, the evaluation is completely done over the rationals and we avoid precision problems.

• Lubeck/Lubeck_exceptional_mult.json: A JSON file containing the degrees and multiplicities of the linear irreducible representations of the Schur coverings of Lie type groups with exceptional Schur multiplier. Currently some groups are missing. The file contains a single JSON object with the normalized codes as fields. Each field provides an array of (degree, multiplicity) pairs. The trivial representation is omitted.

Source of the data

We now provide a detailed description of how and where the data was obtained. We give a description for each file:

• BuiltinPrimes.txt: Taken form the OEIS [[5]] sequence . The table was contributed by N. J. A. Sloane and can be found here.

• Hiss_Malle_data.json: All fields except "code" were obtained from Hiss and Malle [[2], Table 2]. All data has been treated in accordance to the description of the table in [[1], \(\S\) 6]. For further details, see the repository’s HissMalleTableFormats directory README. The "code" field is particular to FiSGO.

  • smallest_pirrep_degree_exceptions.json: The "code" field is particular to FiSGO. All other fields were obtained manually from the tables [[6], Table 1, Table 2] in a survey of Tiep and Zalesskii [[6]]. The original results can be found in [[7]], [[8]] and [[9]].

• sporadic_groups_data.json: The "order", "latex_name" and "multiplier" fields were obtained from Wikipedia [[3]]. Fields "id" and "code" are particular to FiSGO. The data in the "pirreps" field was obtained from the GAP [[4]] database of character tables CTblLib. The script used can be found in DataProcessingScripts/GAP_sporadic_extraction.g.

• Lubeck/Lubeck_[ID].json: All data is sourced from Lübeck’s website and should be credited to him.

• Lubeck/Lubeck_exceptional_mult.json: All data except for "SA-3-3_1" has been obtained using GAP. We have used GAP’s database of character tables for those whose Schur covering was present. We have used GAP’s database of perfect groups to find the remaining Schur covers, and used GAP standard functions to compute the character tables. Finally, the character table of "SA-3-3_1" has been computed using GAP standard functions using a permutation representation computed by A. Hulpke, it can be found in his GitHub

References

[1] (1,2,3)

Hiss, G., & Malle, G. (2001). Low-Dimensional Representations of Quasi-Simple Groups. LMS Journal of Computation and Mathematics, 4, 22–63.

[2] (1,2)

Hiss, G., & Malle, G. (2002). Corrigenda: Low-dimensional Representations of Quasi-simple Groups. LMS Journal of Computation and Mathematics, 5, 95–126.

[3]

Wikipedia contributors. (2025, August 22). List of finite simple groups. In Wikipedia, The Free Encyclopedia. Retrieved 21:49, August 30, 2025, from

[4]

The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.14.0; 2024. (https://www.gap-system.org)

[5]

OEIS Foundation Inc.(2025), The On-Line Encyclopedia of Integer Sequences, Published electronically at https://oeis.org.

[6] (1,2)

Tiep, P. H., & Zalesskii, A. E. (2000). Some aspects of finite linear groups: A survey. Journal of Mathematical Sciences, 100(1), 1893–1914.

[7]

Landazuri, V., & Seitz, G. M. (1974). On the minimal degrees of projective representations of the finite Chevalley groups. Journal of Algebra, 32(2), 418–443.

[8]

Lübeck, F. (2001). Smallest degrees of representations of exceptional groups of lie type. Communications in Algebra, 29(5), 2147–2169.

[9]

Tiep, P. H., & Zalesskii, A. E. (1996). Minimal characters of the finite classical groups. Communications in Algebra, 24(6), 2093–2167.