List of tables

Table type $A_{1}$

Table `GL2`

The generic character table of $\mathrm{GL}_2(q)$.

The table was first computed in [Jor07], [Sch07].
See also: [Ste51].

Table `GU2`

The generic character table of $\mathrm{GU}_2(q)$.

The table was first computed in [Enn63].
See also: [Enn62].

Table `PGL2.1`

The generic character table of $\mathrm{PGL}_2(q)$, $q$ odd (See SL2.0 for the generic character table of $\mathrm{PGL}_2(q)$, $q$ even)

The table was first computed in [Jor07], [Sch07].
See also: [Ste51].

Table `PSL2.1`

The generic character table of $\mathrm{PSL}_2(q^2)$, $q^2$ congruent to $1$ modulo $4$. The possible values for q are given by $q^2 = p^m$ with m a non negative integer and $p$ a prime number.

The table was first (implicitly) computed in [Fro96].
See also: [Sch07].

Table `PSL2.3`

The generic character table of $\mathrm{PSL}_2(q^2)$, $q^2$ congruent to $3$ modulo $4$. The possible values for q are given by $q^2 = p^m$ with m a non negative integer and $p$ a prime number.

The table was first (implicitly) computed in [Fro96].
See also: [Sch07].

Table `SL2.0` (synonym: `PSL2.0`)

The generic character table of $\mathrm{SL}_2(q)$, $q$ even (See SL2.1 for the generic character table of $\mathrm{SL}_2(q)$, $q$ odd)

The table was first computed in [Sch07].

The generic character table of $\mathrm{SL}_2(q)$, $q$ odd. The possible values for $q$ are given by $q = q_0^2 = p^m$ with m a non negative integer and $p$ a prime number. Note that the variable $q_0$ represents $\sqrt{q}$ which is needed as some table entries involve this value. (See SL2.0 for the generic character table of $\mathrm{SL}_2(q)$, $q$ even).

The table was first computed in [Fro96].
See also: [Sch07].

Table type $A_{2}$

Table `GL3`

The generic character table of $\mathrm{GL}_3(q)$.

The table was first computed in [Ste51].
See also: [Gre55].

Table `PGL3.1`

The generic character table of $\mathrm{PGL}_3(q)$, $q$ congruent to $1$ modulo $3$. (See SL3.n1 for the generic character table of $\mathrm{PGL}_3(q)$, $q$ not congruent to $1$ modulo $3$.)

The table was first computed in [Ste51].
See also: [Gre55].

Table `PSL3.1`

The generic character table of $\mathrm{PSL}_3(q)$, $q$ congruent to $1$ modulo $3$. (See SL3.n1 for the generic character table of $\mathrm{PSL}_3(q)$, $q$ not congruent to $1$ modulo $3$.)

The table was first computed in [SF73].
The table was constructed by Jochen Gruber from the generic character table of $\mathrm{SL}_3(q)$, $q$ congruent to $1$ modulo $3$.
See also: [Ste51], [Gre55].

Table `SL3.1`

The generic character table of $\mathrm{SL}_3(q)$, $q$ congruent to $1$ modulo $3$. (See SL3.n1 for the generic character table of $\mathrm{SL}_3(q)$, $q$ not congruent to $1$ modulo $3$.)

The table was first computed in [SF73].
See also: [Ste51], [Gre55].

Table `SL3.n1` (synonym: `PSL3.n1`)

The generic character table of $\mathrm{SL}_3(q)$, $q$ not congruent to $1$ modulo $3$ (See SL3.1 for the generic character table of $\mathrm{SL}_3(q)$, $q$ congruent to $1$ modulo $3$.)

Note: The three groups $\mathrm{SL}_3(q)$, $\mathrm{PGL}_3(q)$ and $\mathrm{PSL}_3(q)$ are mutually isomorphic for these values of $q$.
The table was first computed in [SF73].
See also: [Ste51], [Gre55].

Table type ${}^2A_{2}$

Table `GU3`

The generic character table of $\mathrm{GU}_3(q)$.

The table was first computed in: [Enn63].
See also: [Enn62].

Table `PGU3.2`

The generic character table of $\mathrm{PGU}_3(q)$, $q$ congruent to $2$ modulo $3$. (See SU3.n2 for the generic character table of $\mathrm{PGU}_3(q)$, $q$ not congruent to $2$ modulo $3$.)

The table was derived by Meinolf Geck from: [Enn63].
See also: [Enn62].

Table `PSU3.2`

The generic character table of $\mathrm{PSU}_3(q)$, $q$ congruent to $2$ modulo $3$ (See SU3.n2 for the generic character table of $\mathrm{SL}_3(q)$, $q$ not congruent to $2$ modulo $3$.)

The table was first computed in [SF73].
The table was constructed by Jochen Gruber from the generic character table of $\mathrm{SL}_3(q)$, $q$ congruent to $1$ modulo $3$.
Corrections in [Gec90].
See also: [Enn63], [Enn62], [Ste51].

Table `SU3.2`

The generic character table of $\mathrm{SU}_3(q)$, $q$ congruent to $2$ modulo $3$ (See SU3.n2 for the generic character table of $\mathrm{SL}_3(q)$, $q$ not congruent to $2$ modulo $3$.)

The table was first computed in [SF73].
Corrections in [Gec90].
See also: [Enn63], [Enn62], [Ste51].

Table `SU3.n2` (synonym: `PGU3.n2`)

The generic character table of $\mathrm{SU}_3(q)$, $q$ not congruent to $2$ modulo $3$ (See SU3.2 for the generic character table of $\mathrm{SL}_3(q)$, $q$ congruent to $2$ modulo $3$.)

Note: The three groups $\mathrm{SU}_3(q)$, $\mathrm{PGU}_3(q)$ and $\mathrm{PSU}_3(q)$ are mutually isomorphic for these values of $q$.
The table was first computed in [SF73].
Corrections in [Gec90].
See also: [Enn63], [Enn62], [Ste51].

Table type uniGL

Table `uniGL2`

The unipotent characters for $\mathrm{GL}_2(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGL3`

The unipotent characters for $\mathrm{GL}_3(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGL4`

The unipotent characters for $\mathrm{GL}_4(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGL5`

The unipotent characters for $\mathrm{GL}_5(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGL6`

The unipotent characters for $\mathrm{GL}_6(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGL7`

The unipotent characters for $\mathrm{GL}_7(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGL8`

The unipotent characters for $\mathrm{GL}_8(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table type uniGU

Table `uniGU2`

The unipotent characters for $\mathrm{GU}_2(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGU3`

The unipotent characters for $\mathrm{GU}_3(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGU4`

The unipotent characters for $\mathrm{GU}_4(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGU5`

The unipotent characters for $\mathrm{GU}_5(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGU6`

The unipotent characters for $\mathrm{GU}_6(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGU7`

The unipotent characters for $\mathrm{GU}_7(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table `uniGU8`

The unipotent characters for $\mathrm{GU}_8(q)$.

This table is computed with general programs written by F. Lübeck. They compute the Deligne-Lusztig characters $R_T^G(1)$ and find the unipotent characters as linear combinations of them.

Table type ${}^2B_{2}$

Table `2B2` (synonym: `Sz`)

The generic character table of $^2\mathrm{B}_2(q^2)$, where $q = \sqrt{2}q_0$.

The table was first computed in [Suz62].

Table type $C_{2}$

Table `Sp4.0`

The generic character table of $\mathrm{Sp}_4(q)$, $q$ even

The table was first computed in [Eno72]
The table in the cited paper contains a lot of misprints. The table in the CHEVIE-library was recomputed by F. Lübeck (using Deligne-Lusztig theory).
The names for the class (respectively character) types used in the paper of Enomoto can be found as 3rd component of the list returned by the info functions when applied to the class type. Example:
```
julia> t = generic_character_table("Sp4.0");

julia> cls = t[:,3];  # get the third column

julia> info(cls)[3]
"A31"
```
This shows that the class type of the third column of the table is called A31 by Enomoto.

Table `uniCSp4.1`

The generic character table of $\mathrm{CSp}_4(q)$ with odd $q$ (unipotent characters).

The table was first computed in [Shi82].
The name of the i-th class type in the cited paper can be found as 3rd component of the list returned by the info function when applied to the i-th class type.
This release of CHEVIE only contains the unipotent characters. We have a preliminary version of the complete table, but it contains a few errors.
We are developing programs for computing (parts of) generic character tables and we will use the group $\mathrm{CSp}_4(q)$ as "test example". This is the reason that we did not try to find the errors in the above mentioned table for this release.

Table type $C_{3}$

Table `CSp6.1`

The generic character table of $\mathrm{CSp}_6(q)$, $q$ odd

The computation of this table is explained in [Lbe93].
The first twelve character types contain the unipotent characters. Specialize the parameter k1 to a multiple of $q-1$ e.g., k1=0. If you are only interested in the unipotent characters, you can use the table uniCSp6.1, which allows faster calculations.

Table `Sp6.0`

The generic character table of $\mathrm{Sp}_6(q)$, $q$ even

The computation of this table is explained in [Lbe93].
The irreducible characters of $\mathrm{Sp}_6(q)$ with even $q$ were already (independently) determined in [Loo77].
The first twelve character types are the unipotent characters. If you are only interested in the unipotent characters, you can use the table uniSp6.0, which allows faster calculations.

Table `uniCSp6.1`

The unipotent characters of $\mathrm{CSp}_6(q)$, $q$ odd

The computation of the whole table of this group is explained in [Lbe93].
This table is extracted from the table CSp6.1, which contains the unipotent characters in the first 12 character types (specialize the parameter k1 to $0$).
If you want to calculate only with the unipotent characters then use this table uniCSp6.1 (the calculations will run much faster). If you also want to use the other characters of CSp6.1, then produce the unipotent characters with the specialize function as explained above.

Table `uniSp6.0`

The unipotent characters of $\mathrm{Sp}_6(q)$, $q$ even

The computation of the whole table of this group is explained in [Lbe93].
The irreducible characters of $\mathrm{Sp}_6(q)$ with even $q$ were already (independently) determined in [Loo77].
This table is extracted from the table Sp6.0, which contains the unipotent characters in the first 12 character types.
If you want to calculate only with the unipotent characters then use this table uniSp6.0 (the calculations will run much faster). If you also want to use the other characters of Sp6.0, then don't use uniSp6.0.

Table type $D_{4}$

Table `uniCSpin8.1`

The unipotent characters of $\mathrm{CSpin}_8(q)$, $q$ odd

The table was first computed in [GP92].
The symbols parametrizing the unipotent characters are given in form of a pair of lists in (the 2nd part of) position 2 of the character information list returned by the info function.

Table `uniSpin8.0`

The unipotent characters of $\mathrm{Spin}_8(q)$, $q$ even

The table was first computed by Meinolf Geck (unpublished).
The symbols parametrizing the unipotent characters are given in form of a pair of lists in (the 2nd part of) position 2 of the character information list returned by the info function.

Table type ${}^2D_{4}$

Table `uni2D4.0`

The unipotent characters of $\mathrm{CO}_8^-(q)$, with even q.

This table was computed by F. Lübeck, most of it with general programs.

Table `uni2D4.1`

The unipotent characters of $\mathrm{CO}_8^-(q)$, with odd q.

This table was computed by F. Lübeck, most of it with general programs.

Table type ${}^3D_{4}$

Table `3D4.0`

The generic character table of $^3\mathrm{D}_4(2^n)$.

The table was first computed in [DM87].

Table `3D4.11`

The generic character table of $^3\mathrm{D}_4(q)$, $p>2$, congruent to $1$ modulo $4$.

The table was first computed in [DM87].

Table `3D4.13`

The generic character table of $^3\mathrm{D}_4(q)$, $p>2$, congruent to $3$ modulo $4$.

The table was first computed in [DM87].

Table type $F_{4}$

Table `uniuniF4p2`

The unipotent characters of $\mathrm{F}_4(2^n)$ on unipotent classes.

The table was computed by Marcelo and Shinoda. This is explained in [MS95].
PrintInfoClass shows the geometric unipotent classes in the notation of [Spa82] and also Shinoda's notation from the cited preprint.
PrintInfoChar shows the unipotent characters in the notation of [Car85]. (but for the characters of a Weyl group of type $\mathrm{B}_2$ we use the notation with pairs of partitions.)

Table type ${}^2F_{4}$

Table `2F4.1` (synonym: `Ree.1`)

The generic character table of $^2\mathrm{F}_4(q^2)$, where $\frac{q}{\sqrt{2}} = q_0$ is congruent to $1$ modulo $3$.

The unipotent characters were first computed in [Mal90].
The other irreducible characters were added by G. Malle

Table `2F4.2` (synonym: `Ree.2`)

The generic character table of $^2\mathrm{F}_4(q^2)$, where $\frac{q}{\sqrt{2}} = q_0$ is congruent to $2$ modulo $3$.

The unipotent characters were first computed in [Mal90].
The other irreducible characters were added by G. Malle

Table type $G_{2}$

Table `G2.01`

The generic character table of $\mathrm{G}_2(q)$, $q$ even, congruent to $1$ modulo $3$

The table was first computed in [EY86].
See also: [CR74], [Hi90].
Enomoto's and Yamada's notation for the irreducible characters is given in the fourth component of the character information list.
An equivalent to Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Enomoto's and Yamada's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.01");

julia> info(t[6])
4-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"
 "\\vartheta_1'"

Explanation of example: Character type six of G2.01 is called $\vartheta_1$ by Enomoto–Yamada and corresponds to $X_{18}$ in Chang–Ree.

Table `G2.02`

The generic character table of $\mathrm{G}_2(q)$, $q$ even, congruent to $2$ modulo $3$

The table was first computed in [EY86].
See also: [CR74], [Hi90].
Enomoto's and Yamada's notation for the irreducible characters is given in the fourth component of the character information list.
An equivalent to Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Enomoto's and Yamada's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.02");

julia> info(t[6])
4-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"
 "\\vartheta_1'"

Explanation of example: Character type six of G2.02 is called $\vartheta_1$ by Enomoto–Yamada and corresponds to $X_{18}$ in Chang–Ree.

Table `G2.101`

The generic character table of $\mathrm{G}_2(q)$, $q$ a power of $3$, congruent to $1$ modulo $4$.

The table was first computed in [Eno76].
See also: [CR74], [Hi90].
Enomoto's notation for the irreducible characters is given in the fourth component of the character information list.
An equivalent to Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Enomoto's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.101");

julia> info(t[6])
4-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"
 "\\vartheta_{10}"

Explanation of example: Character type six of G2.101 is called $\vartheta_{10}$ by Enomoto and corresponds to $X_{18}$ in Chang–Ree.

Table `G2.103`

The generic character table of $\mathrm{G}_2(q)$, $q$ a power of $3$, congruent to $3$ modulo $4$.

The table was first computed in [Eno76].
See also: [CR74], [Hi90].
Enomoto's notation for the irreducible characters is given in the fourth component of the character information list.
An equivalent to Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Enomoto's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.103");

julia> info(t[6])
4-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"
 "\\vartheta_{10}"

Explanation of example: Character type six of G2.103 is called $\vartheta_{10}$ by Enomoto and corresponds to $X_{18}$ in Chang–Ree.

Table `G2.111`

The generic character table of $G_2(q)$, $q$ odd, congruent to $1$ modulo $3$ and $4$.

The table was first computed in [CR74].
See also: [Cha68], [Hi90].
Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Chang's and Ree's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.111");

julia> info(t[6])
3-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"

Explanation of example: Character type six of G2.111 is called $X_{18}$ by Chang and Ree

Table `G2.113`

The generic character table of $G_2(q)$, $q$ odd, congruent to $1$ modulo $3$ and $3$ modulo $4$.

The table was first computed in [CR74].
See also: [Cha68], [Hi90].
Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Chang's and Ree's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.113");

julia> info(t[6])
3-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"

Explanation of example: Character type six of G2.113 is called $X_{18}$ by Chang and Ree

Table `G2.121`

The generic character table of $\mathrm{G}_2(q)$, $q$ odd, congruent to $2$ modulo $3$ and $1$ modulo $4$.

The table was first computed in [CR74].
See also: [Cha68], [Hi90].
Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Chang's and Ree's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.121");

julia> info(t[6])
3-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"

Explanation of example: Character type six of G2.121 is called $X_{18}$ by Chang and Ree

Table `G2.123`

The generic character table of $\mathrm{G}_2(q)$, $q$ odd, congruent to $2$ modulo $3$ and $3$ modulo $4$.

The table was first computed in [CR74].
See also: [Cha68], [Hi90].
Chang's and Ree's notation for the irreducible characters is given in the third component of the character information list.
Chang's and Ree's notation for the conjugacy classes is given in the third component of the class information list.

Example:

julia> t = generic_character_table("G2.123");

julia> info(t[6])
3-element Vector{Any}:
 [1, 5]
 ["G_2", "G_2[1]"]
 "X_{18}"

Explanation of example: Character type six of G2.123 is called $X_{18}$ by Chang and Ree

Table type ${}^2G_{2}$

Table `2G2` (synonym: `ree`)

The generic character table of the Ree groups $^2\mathrm{G}_2(q)$. The possible values for $q$ are given by $q^2 = 3^{2m+1}$ with m a non negative integer. So $q = \sqrt{3}q_0$ where $q_0 = 3^m$.

Most of the table was determined in [War66].
The values of the irreducible Deligne-Lusztig characters were computed by F. Lübeck.
The names of class types and character types used in the above cited article can be recovered as fourths components of the information given by PrintInfoClass and PrintInfoChar. These names are also used as names for the unipotent parts of the classes.