FiSGO.SimpleGroups.ExcetionalChevalleyE7

class ExcetionalChevalleyE7(q, validate=True)

Bases: UniParamSimpleGroup

Class representing the exceptional Chevalley group \(E_7(q)\), a simple group of Lie type. See Wikipedia.

The q parameter is a prime power and can be given as an integer or a pair of integers. Example: q = 9 and q=(3,2) represent the same parameter as 9 = 3**2. The latter format is preferred when dealing with large powers.

On object initialization, if validate is True (default), it will attempt to check if the introduced parameters are valid. Parameter validation can be globally disabled by changing GLOBAL_VALIDATE to False.

Parameters:

• q (int | tuple[int, int]) – Positive integer or tuple of positive integers. Prime power, finite field parameter of the group.

• validate (bool) – Boolean. If True, it will attempt to validate the given parameters on object initialization; if False, it will skip any checks.

Methods

GAP_name	Returns a string with the name of the simple group in GAP4/Atlas like notation.
code	Returns the code of the simple group.
compute_multiplier	Computes the size Schur multiplier of the simple group (these are known values).
compute_order	Computes the order of the group.
from_code	Given a valid simple group code, returns the corresponding simple group object.
from_id	Given an uniparametric simple group ID and its parameter par, returns the corresponding simple group object.
hiss_malle_pirreps	This function looks for projective representations of degree up to 250.
isomorphisms	Returns a list of group codes whose groups are isomorphic to self.
latex_name	Returns a list of strings containing possible notations for the simple group in a LaTeX format.
lubeck_pirreps	This function computes projective representations of Lie type groups of rank at most 8.
multiplier	Returns the order of simple group's Schur multiplier.
normalized_code	There are two formats for the q parameter of the groups (power of a prime number): either as a single number or as a pair of numbers.
order	Returns the order of the group.
p_sylow_power	Given a prime number p, it returns exponent n of the size of its p-Sylow subgroup(s), p**n.
par_value	Returns the value of the group parameter as an integer.
smallest_pirrep_degree	Using the bounds of Seitz, Landazuri, Tiep and Zalesskii, returns the degree of the smallest non-trivial projective irreducible complex representation of the simple group.

GAP_name()

Returns a string with the name of the simple group in GAP4/Atlas like notation. This name can be used to look up the group character table in GAP if already available in a package such as AtlasRep or CTblLib.

Return type:

str

Returns:

Name of the group in GAP4/Atlas like notation.

code()

Returns the code of the simple group.

Return type:

str

Returns:

Code of the simple group.

compute_multiplier()

Computes the size Schur multiplier of the simple group (these are known values).

Returns:

Order of the simple group’s Schur multiplier.

compute_order()

Computes the order of the group.

Returns:

Order of the group.

classmethod from_code(code)

Given a valid simple group code, returns the corresponding simple group object.

A simple group code is a string formed by an ID and the group parameters:

• ID: identifies to which family of simple groups the code refers to. It consists of two characters, all valid IDs are listed in SimpleGroups.simple_group_ids().

• Parameters: Any group has up to two parameters. Except for sporadic groups, these parameters are called ‘n’ or ‘q’. ‘n’ refers to a positive integer, while ‘q’ refers to a prime power. A ‘q’ parameter can be given either as the number or as a pair [prime number]_[power].

• Sporadic group names: All valid names for sporadic groups are listed in SimpleGroups.sporadic_group_names().

A code is formed by joining the code and its parameters using ‘-’ as the separator. The syntax is as follows:

• 0-parametric group: “[ID]”, Example: “TT”.

• 1-parametric group: “[ID]-[parameter]”, Examples: “E6-9”, “E6-3_2”, “SZ-6”.

• 2-parametric group: “[ID]-[n parameter]-[q parameter]”, Examples: “CA-1-2”, “SA-2-9”, “SA-2-3_2”.

• Sporadic group: “SP-[group name]”, Examples: “SP-M11”, “SP-Fi24’”.

Caution

The name of the Fischer group 24’ is “Fi24’”, if printed, it will show “Fi24’” as the character “’” is being formated, this may create confusion.

Parameters:

code (str) – Code corresponding to some simple group.

Returns:

The simple group object corresponding to the given code.

classmethod from_id(id_, par)

Given an uniparametric simple group ID and its parameter par, returns the corresponding simple group object.

For information on which IDs are valid and uniparametric, see the documentation of SimpleGroups.simple_group_ids.

Parameters:

• id – ID of an uniparametric simple group.

• par – The n or q parameter of the group.

Returns:

An object of a derived class of UniParamSimpleGroup.

hiss_malle_pirreps(char=0, all_pirrep_data=False, allow_duplicates=False)

This function looks for projective representations of degree up to 250.

The representations are obtained from the tables given by Gerard Hiss and Gunter Malle in [HM1] and [HM2]. By default, it returns a list with all degrees (less than 251) of the simple group’s characteristic zero pirreps.

Information of positive characteristic absolutely irreducible represenations can be obtained by changing the char parameter. Furthermore, if char is set to None, it returns all the information available for the simple group as a list of dicts, which can be parsed into a JSON file.

For a fixed characteristic, all information on the pirreps can be obtained by setting all_pirrep_data to True. Again, as a list of dicts which can be parsed into a JSON file.

Finally, it is possible for different covers of a simple group to produce different projective representations of the same degree. As such, it is possible that the returned list of degrees may contain duplicates. If this is desired, for instance, to detect this phenomenon, set allow_duplicates to True.

Parameters:

• char (int | None) – Characteristic in which to look for projective (absolutely) irreps. If None, it will look for all projective absolutely irreducible represenations, regardless of the characteristic, and return all available information as a list of dicts. By default, char is 0.

• all_pirrep_data (bool) – False by default. If True, it will return all the information available for each representation and return a list of dicts. If False, it will return only the degrees of the simple group’s pirreps. This parameter is ignored if char is None.

• allow_duplicates (bool) – False by default. If True, it will allow duplicates in the returned list of degrees. This parameter is ignored if all_pirrep_data is True or if char is None.

Return type:

list[int | dict]

Returns:

By default, a list of degrees less than 251 of the simple group’s pirreps in characteristic 0. See the parameters’ description for more details on changing the function’s output.

[HM1]

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

[HM2]

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

isomorphisms()

Returns a list of group codes whose groups are isomorphic to self. Returns an empty list if the group is not isomorphic to any other group apart from itself.

Return type:

list[str]

Returns:

Codes of groups isomorphic to itself.

latex_name()

Returns a list of strings containing possible notations for the simple group in a LaTeX format. To properly visualize the string in LaTeX format, the strings need to be printed.

The first string of the list corresponds to the Wikipedia List of finite simple groups recommended names.

Return type:

list[str]

Returns:

List of possible notations for the simple group formatted in LaTeX.

lubeck_pirreps()

This function computes projective representations of Lie type groups of rank at most 8. The multiplicity corresponds to that of the linear irreducible characters of the Schur covering.

Returns:

If available, returns a list of pairs containing the projective representations of the group alongside their multiplicities in the Schur covering.

multiplier()

Returns the order of simple group’s Schur multiplier.

If it has not been calculated before (i.e. has not been internally stored yet), it calculates it and returns it.

Return type:

int

Returns:

Order of the simple group’s Schur multiplier.

normalized_code()

There are two formats for the q parameter of the groups (power of a prime number): either as a single number or as a pair of numbers. The normalized code corresponds to the pair of numbers. This function returns the normalized code. If the group does not have a q parameter, it returns the same as self.code().

Example: “CA-1-4” or “CA-1-2_2” are both valid codes for the Chevalley A group with parameters n=1 and q=4, the normalized code is “CA-1-2_2”.

Return type:

str

Returns:

Normalized code of the simple group.

order()

Returns the order of the group.

If the order has not been calculated before (i.e. has not been internally stored yet), it calculates the order and returns it.

Return type:

int

Returns:

Order of the group.

p_sylow_power(p)

Given a prime number p, it returns exponent n of the size of its p-Sylow subgroup(s), p**n.

Parameters:

p (int) – Prime number.

Return type:

int

Returns:

Number n such that the p-Sylow subgroup(s) of the simple group have order p**n.

par_value()

Returns the value of the group parameter as an integer. If the parameter is of q-type and stored as a pair of integers, it returns the computed value.

Example:

>>> UniParamSimpleGroup((3,2)).par_value()
9

Returns:

The value of the group parameter.

Return type:

int

smallest_pirrep_degree()

Using the bounds of Seitz, Landazuri, Tiep and Zalesskii, returns the degree of the smallest non-trivial projective irreducible complex representation of the simple group. Furthermore, it also returns the number of different representations of that degree.

Caution

In the case of the alternating groups, only the smallest degree is currently implemented, the number of different representations is given as None.

Return type:

tuple[int, int]

Returns:

The degree of the smallest non-trivial complex projective representation and the number of different representations of that degree.