List of Green functions

Table type $A_{0}$

Table `GL1`

The Green functions of $\mathrm{GL}_1(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type $A_{1}$

Table `GL2`

The Green functions of $\mathrm{GL}_2(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type $A_{2}$

Table `GL3`

The Green functions of $\mathrm{GL}_3(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

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

Table `GU3`

The Green functions of $\mathrm{GU}_3(q)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_3(q^2)$ from those of $\mathrm{GL}_3(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{3}$

Table `GL4`

The Green functions of $\mathrm{GL}_4(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type $A_{3};2$

Table `GL4e2`

The Green functions of $\mathrm{GL}_4(2^n):2$.

The table was first computed in [Mal93*1].

Table type ${}^2A_{3}$

Table `GU4`

The Green functions of $\mathrm{GU}_4(q)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_4(q^2)$ from those of $\mathrm{GL}_4(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{4}$

Table `GL5`

The Green functions of $\mathrm{GL}_5(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type $A_{4};2$

Table `GL5e2`

The Green functions of $\mathrm{GL}_5(2^n):2$.

The table was first computed in [Mal93*1].

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

Table `GU5`

The Green functions of $\mathrm{GU}_5(q)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_5(q^2)$ from those of $\mathrm{GL}_5(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{5}$

Table `GL6`

The Green functions of $\mathrm{GL}_6(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type $A_{5};2$

Table `GL6e2`

The Green functions of $\mathrm{GL}_6(2^n):2$.

The table was first computed in [Mal93*1].

Table type ${}^2A_{5}$

Table `GU6`

The Green functions of $\mathrm{GU}_6(q)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_6(q^2)$ from those of $\mathrm{GL}_6(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{6}$

Table `GL7`

The Green functions of $\mathrm{GL}_7(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type ${}^2A_{6}$

Table `GU7`

The Green functions of $\mathrm{GU}_7(q)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_7(q^2)$ from those of $\mathrm{GL}_7(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{7}$

Table `GL8`

The Green functions of $\mathrm{GL}_8(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type ${}^2A_{7}$

Table `GU8`

The Green functions of $\mathrm{GU}_8(q^2)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_8(q^2)$ from those of $\mathrm{GL}_8(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{8}$

Table `GL9`

The Green functions of $\mathrm{GL}_9(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type ${}^2A_{8}$

Table `GU9`

The Green functions of $\mathrm{GU}_9(q)$.

By a theorem of Hotta, Springer and Kawanaka we can get the Green functions of the unitary group $\mathrm{GU}_9(q^2)$ from those of $\mathrm{GL}_9(q)$ by substituting $q$ by $-q$. This is proved in [HS77] and [Kaw85].
See also: [DM87*1].
For the computation of the Green functions for $\mathrm{GL}_n(q)$ see for example:
GreenFunTab(GL2); PrintInfoTab(GL2green);

Table type $A_{9}$

Table `GL10`

The Green functions of $\mathrm{GL}_10(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

Table type $A_{10}$

Table `GL11`

The Green functions of $\mathrm{GL}_11(q)$.

These Green functions were introduced in: [Gre55].
See also: [Ste51].
This CHEVIE-table is computed by an algorithm from [LS78].
The program which generates the files with the Green functions of $\mathrm{GL}_n(q)$ and $\mathrm{GU}_n(q)$ is part of the CHEVIE-system. You can reproduce them with the CHEVIE commands:
GreenFunctionsA(n,filename); GreenFunctions2A(n,filename);
(see the corresponding help) These programs are written by U. Porsch and F. Lübeck.

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

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

The Green functions of $^2\mathrm{B}_2(q^2)$.

The table was first computed in [Suz62].

Table type $C_{2}$

Table `C2n2`

The Green functions of $\mathrm{CSp}_4(q)$, $q$ odd.

The generic character table of $\mathrm{CSp}_4(q)$, $q$ odd, and hence its Green functions were computed in [Shi82].
They can also be read off the table of $\mathrm{Sp}_4(q)$, $q$ odd, computed in [Sri68].
See also: [LS90].
The names of the unipotent classes are taken from the article of Shinoda.

Table `C2p2`

The Green functions of $\mathrm{Sp}_4(q)$, q even.

The table was first computed in [Eno72].
The names of the unipotent classes are taken from this paper.

Table type $C_{3}$

Table `C3n2`

The Green functions of $\mathrm{CSp}_6(q)$, $q$ odd.

These Green functions are computed in [LS90].

Table `C3p2`

The Green functions of $\mathrm{Sp}_6(2^n)$.

The table was published in [Mal93].
The unipotent classes were determined in [Shi74].
The notation for the unipotent classes is taken from that paper.

Table type $D_{4}$

Table `D4n2`

Missing data

Unfortunately no information about this table is available.

Table `D4p2`

The Green functions of $\mathrm{O}_8^+(q)$.

The table was published in [Mal93].
The notation for the unipotent classes is taken from [Spa82].

Table type $D_{4};2$

Table `D4e2`

The Green functions of $\mathrm{SO}_8^+(2^n)$.

The table was first computed in [Mal93*1].

Table type $D_{4};3$

Table `D4e3`

The Green functions of $\mathrm{O}_8^+(3^n):3$.

The table was first computed in [Mal93*1].

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

Table `2D4n2`

The Green functions of $\mathrm{O}_8^-(q)$ with odd $q$.

This table of generalized Green functions is computed by F. Lübeck using Lusztig's algorithm.,
The occuring Levi subgroups have the following relative (twisted) Weyl groups: Levi L type of N(L)/L

[ [ D, 0 ] ] [ [ D, 4, 2 ] ]

Position [-1,-1] contains the transformation matrix to the Foulkes functions and [0,-1] the corresponding labels.

Table `2D4p2`

The Green functions of $\mathrm{O}_8^-(2^n)$.

The table was published in [Mal93]
The notation for the unipotent classes is as in that paper.

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

Table `3D4n2`

The Green functions of $^3\mathrm{D}_4(q)$, $p>2$.

The table was first computed in [Spa82*1].

Table `3D4p2`

The Green functions of $^3\mathrm{D}_4(2^n)$.

The table was first computed in [Spa82*1].

Table type $D_{5}$

Table `D5n2`

Missing data

Unfortunately no information about this table is available.

Table `D5p2`

The Green functions of $\mathrm{O}_{10}^+(2^n)$.

The table was published in [Mal93].
The notation for the unipotent classes is taken from [Spa82].

Table type $E_{6}$

Table `E6n23`

The Green functions of $\mathrm{E}_6(q)$, $p>3$.

The table was first computed in [BS84].

Table `E6p2`

The Green functions of $\mathrm{E}_6(2^n)$.

The table was first computed in [Mal93].

Table `E6p3`

The Green functions of $\mathrm{E}_6(3^n)$.

Table type $E_{6};2$

Table `E6e2`

The Green functions of $\mathrm{E}_6(2^n):2$.

The table was first computed in [Mal93].
The notation for the unipotent classes is as in that paper.

Table type ${}^2E_{6}$

Table `2E6n23`

The Green functions of $^2\mathrm{E}_6(q)$, $p>3$.

The table was first computed in [BS84].

Table `2E6p2`

The Green functions of $^2\mathrm{E}_6(2^n)$.

The table was first computed in [Mal93].
The notation for the unipotent classes is as in that paper.

Table type $F_{4}$

Table `F4n23`

The Green functions of $\mathrm{F}_4(q)$, $p>3$.

The table was first computed in [Sho82].

Table `F4p2`

The Green functions of $\mathrm{F}_4(2^n)$.

The table was first computed in [Mal93].
The unipotent classes were determined in [Shi74].
The notation for the unipotent classes is taken from that paper.

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

Table `2F4`

The Green functions of $^2\mathrm{F}_4(q^2)$.

The Green functions can easily be obtained from a knowledge of the unipotent characters; these were first computed in [Mal90].

Table type $G_{2}$

Table `G2n23`

The Green functions of $\mathrm{G}_2(q)$, $p>3$.

The table was first computed in [CR74].
The notation for the unipotent classes is taken from that paper.

Table `G2p2`

The Green functions of $\mathrm{G}_2(2^n)$.

The table was first computed in [EY86].
The notation for the unipotent classes is taken from that paper.

Table `G2p3`

The Green functions of $\mathrm{G}_2(3^n)$.

The table was first computed in [Eno76].
The notation for the unipotent classes is taken from that paper.

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

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

The Green functions of $^2\mathrm{G}_2(q^2)$.

The table was first computed in [War66].