Representations of finite general linear groups

This repository contains small SageMath computations of representations of finite general linear groups. The original focus is the Steinberg representation of GL_3(F_q), with concrete scripts for q = 2, 3 and a parameterized script for general prime powers q. It also includes projective-line and representation-family scripts for GL_2(F_q).

The original GL_3(F_2) computation corresponds to the expository paper: https://jacksonwalters.com/docs/notes/steinberg_representation_GL(3,2).pdf

The representations are realized geometrically from the spherical building of F_q^3. In this rank-2 case, the building is the incidence graph whose vertices are:

• the 1-dimensional subspaces of F_q^3;
• the 2-dimensional subspaces of F_q^3;
• edges given by inclusions line <= plane.

The Steinberg representation is computed as the top homology of this building:

St = H_1(Delta; C) = ker(partial: C_1(Delta; C) -> C_0(Delta; C)).

For GL_3(F_2), the computation finds an 8-dimensional Steinberg module. For GL_3(F_3), the same construction gives a 27-dimensional Steinberg module. In general this rank-2 construction gives dim St = q^3.

Files

• gl_3_q/steinberg_representation_gl_3_2.py is the original GL_3(F_2) SageMath script, now with concise default output.
• gl_3_q/steinberg_representation_gl_3_3.py is the GL_3(F_3) SageMath script.
• gl_3_q/steinberg_representation_gl_3_q.py is the general GL_3(F_q) script. It accepts q as a command-line argument and keeps larger computations opt-in or automatic only for small enough inputs.
• gl_3_q/fano_plane_representations_gl_3_2.py specializes to GL_3(F_2) as the automorphism group of the Fano plane. It constructs the point, line, flag, reduced point/line, boundary-image, and Steinberg representations as explicit matrices, and records the two complex 3-dimensional irreducible characters over Q(zeta_7).
• gl_3_q/representation_utils_gl_3_q.py contains shared GL_3(F_q) helpers: family counts, multiplicative characters, geometric cosets for B\G and P_{2,1}\G, and a vector-valued induction engine.
• gl_3_q/representations_gl_3_q.py prints the full GL_3(F_q) family table and points to the modular construction scripts.
• gl_3_q/unipotent_representations_gl_3_q.py constructs determinant characters, the middle unipotent representation from the projective-plane augmentation module, and Steinberg twists.
• gl_3_q/principal_series_gl_3_q.py constructs the irreducible principal series attached to three distinct split-torus characters.
• gl_3_q/repeated_split_representations_gl_3_q.py constructs the two irreducible constituents attached to a split character triple (chi, chi, mu): the small parabolic induction and the large constituent built from the GL_2(F_q) Steinberg module.
• gl_3_q/intermediate_series_gl_3_q.py constructs the intermediate series induced from a GL_2(F_q) cuspidal representation, reusing the direct Whittaker model in gl_2_q/direct_whittaker_cuspidal_gl_2_q.py.
• gl_3_q/cuspidal_parameters_gl_3_q.py organizes the genuine GL_3(F_q) cuspidal parameters from regular characters of the cubic nonsplit torus F_{q^3}^*. This module records parameters and degree counts; explicit matrices for these cuspidals are intentionally left for a later construction.
• gl_2_q/projective_line.py contains reusable P^1(F_q) geometry and projective-line action helpers for GL_2(F_q).
• gl_2_q/steinberg_representation_gl_2_q.py constructs the GL_2(F_q) Steinberg representation as reduced H_0(P^1(F_q)).
• gl_2_q/representations_gl_2_q.py organizes the standard complex representation families of GL_2(F_q), constructs determinant twists of Steinberg, and constructs principal series by explicit induction from the Borel subgroup.
• gl_2_q/whittaker_basis_gl_2_q.py changes the cuspidal projector model into a Whittaker basis indexed by F_q^*, giving compact (q - 1) x (q - 1) cuspidal matrices while retaining the Gelfand-Graev construction as a correctness oracle.
• gl_2_q/direct_whittaker_cuspidal_gl_2_q.py constructs the same cuspidal Whittaker model directly from the nonsplit-torus character formula and Bruhat decomposition, avoiding the large Gelfand-Graev projector space.

What the scripts do

The scripts:

1. Construct the points and planes in F_q^3.
2. Build the incidence edges of the spherical building.
3. Form the boundary map C_1 -> C_0.
4. Compute ker(partial), giving the Steinberg representation.
5. Construct apartment cycles from bases of F_q^3.
6. Define or test the natural GL_3(F_q) action on the building and cycles.
7. Count apartment cycles up to orientation.
8. Compute the span of apartment cycles.
9. Compute or verify the character of the resulting representation.

The GL_3(F_3) and general GL_3(F_q) scripts compute the Steinberg character from fixed vertex and edge counts instead of building a q^3 x q^3 matrix for every group element. For a connected graph,

chi_St(g) = # fixed edges - # fixed vertices + 1.

Running the computation

Install SageMath (tested with SageMath 10.8), then run from this directory:

sage gl_3_q/steinberg_representation_gl_3_2.py
sage gl_3_q/steinberg_representation_gl_3_3.py
sage gl_3_q/steinberg_representation_gl_3_q.py 3
sage gl_3_q/fano_plane_representations_gl_3_2.py
sage gl_3_q/representations_gl_3_q.py 3 --scripts
sage gl_3_q/unipotent_representations_gl_3_q.py 3
sage gl_3_q/repeated_split_representations_gl_3_q.py 3
sage gl_3_q/intermediate_series_gl_3_q.py 3
sage gl_3_q/principal_series_gl_3_q.py 4 --character yes
sage gl_3_q/cuspidal_parameters_gl_3_q.py 3 --parameters
sage gl_2_q/steinberg_representation_gl_2_q.py 5
sage gl_2_q/representations_gl_2_q.py 5

For another prime power, pass q:

sage gl_3_q/steinberg_representation_gl_3_q.py 5

The general script suppresses bulky data by default. Useful options are:

sage gl_3_q/steinberg_representation_gl_3_q.py 3 --character-table
sage gl_3_q/steinberg_representation_gl_3_q.py 7 --apartment-span yes
sage gl_3_q/steinberg_representation_gl_3_q.py 7 --character yes
sage gl_3_q/steinberg_representation_gl_3_q.py 3 --weyl-orbits
sage gl_3_q/fano_plane_representations_gl_3_2.py --character-table
sage gl_3_q/fano_plane_representations_gl_3_2.py --incidence
sage gl_3_q/fano_plane_representations_gl_3_2.py --generators
sage gl_3_q/unipotent_representations_gl_3_q.py 3 --character-table
sage gl_3_q/principal_series_gl_3_q.py 4 --characters 0 1 2 --generators
sage gl_3_q/repeated_split_representations_gl_3_q.py 3 --repeated 0 --single 1
sage gl_3_q/intermediate_series_gl_3_q.py 3 --cuspidal 1 --character-twist 0
sage gl_3_q/cuspidal_parameters_gl_3_q.py 4 --character 1
sage gl_2_q/steinberg_representation_gl_2_q.py 5 --character-table
sage gl_2_q/representations_gl_2_q.py 5 --all-principal-series
sage gl_2_q/representations_gl_2_q.py 4 --cuspidal-parameters
sage gl_2_q/representations_gl_2_q.py 5 --cuspidal 1
sage gl_2_q/representations_gl_2_q.py 3 --all-cuspidals
sage gl_2_q/whittaker_basis_gl_2_q.py 5 --cuspidal 1 --generators
sage gl_2_q/direct_whittaker_cuspidal_gl_2_q.py 7 --cuspidal 1 --generators

GL_2(F_q) scripts

For GL_2(F_q), the spherical building is the projective line P^1(F_q). The Steinberg representation is the reduced zero-th homology:

St = \widetilde H_0(P^1(F_q); C) = ker(sum: C[P^1(F_q)] -> C).

Since #P^1(F_q) = q + 1, this gives dim St = q, and the projective-line permutation module decomposes as

C[P^1(F_q)] = 1 + St.

The broader GL_2(F_q) representation script records the standard complex irreducible families:

• determinant characters, dimension 1;
• determinant twists of Steinberg, dimension q;
• principal series Ind_B^G(chi_1 tensor chi_2) with chi_1 != chi_2, dimension q + 1;
• cuspidal series from nonsplit-torus character orbits, dimension q - 1.

The script checks the family count by verifying that the sum of squares of the listed dimensions is |GL_2(F_q)|. It constructs the projective-line module, Steinberg twists, and principal series explicitly.

Cuspidal representations are constructed by a Gelfand-Graev projector. For a regular character theta of the nonsplit torus

T = F_{q^2}^*,

the script forms

Gamma = Ind_U^G(psi),

where U = {[[1, x], [0, 1]] : x in F_q} and psi is a nontrivial additive character of F_q. It then applies the central idempotent

e_theta = (dim pi_theta / |G|) sum_g chi_theta(g^{-1}) Gamma(g).

The image of e_theta has dimension q - 1; restricting Gamma(g) to this image gives explicit cuspidal matrices. This exact projector method is useful for small q, but it uses a Gelfand-Graev space of dimension |GL_2(F_q)|/q, so larger values of q become slower than the principal-series construction.

The Whittaker-basis script keeps this projector model as a source of truth and then decomposes the cuspidal image under the upper-unipotent subgroup. For each y in F_q^*, it applies the Fourier idempotent

P_y = (1/q) sum_x psi(-yx) pi(n(x)),

where n(x) = [[1, x], [0, 1]]. Each image is one-dimensional, and these lines form a basis in which

pi(n(x)) e_y = psi(yx) e_y.

This gives a genuine Whittaker/Kirillov-style basis indexed by F_q^*, while still checking the resulting traces against the nonsplit-torus cuspidal character formula.

The direct Whittaker script avoids constructing Gamma. It uses the explicit Borel action on basis vectors e_y, together with Bruhat decomposition

g = n(a/c) diag(-det(g)/c, -c) w n(d/c)        if c != 0,

for g = [[a, b], [c, d]]. The Weyl matrix for w = [[0, 1], [-1, 0]] is recovered by Fourier inversion from the identity

tr(pi(diag(a, 1) w n(s))) = sum_y W[a*y, y] psi(y*s).

Those traces are supplied by the nonsplit-torus cuspidal character formula, so the construction stays in dimension q - 1 throughout.

Fano-plane representations for GL_3(F_2)

Since F_2^* is trivial, GL_3(F_2) = PGL_3(F_2) acts faithfully on the Fano plane. The Fano-plane script builds this action on:

• the 7 projective points;
• the 7 projective lines;
• the 21 incident point-line flags.

It then forms the same building boundary map used in the Steinberg scripts,

partial: C[flags] -> C[points] + C[lines],

and verifies the equivariant decomposition

C[flags] = im(partial) + St,
dim C[flags] = 21 = 13 + 8.

The point and line permutation representations each split as

C[points] = 1 + chi_6,
C[lines]  = 1 + chi_6,

where chi_6 is the irreducible 6-dimensional augmentation representation. The flag representation has character decomposition

C[flags] = 1 + 2*chi_6 + St.

The script checks these decompositions using character inner products over the six conjugacy classes of GL_3(F_2). It also adds the two conjugate 3-dimensional characters with order-7 values

alpha = zeta_7 + zeta_7^2 + zeta_7^4,
alpha_bar = zeta_7^3 + zeta_7^5 + zeta_7^6,

and verifies the character identities

Sym^2(chi_3) = chi_6,
chi_3 * chi_3bar = 1 + St.

Expected output

Default output is concise: it reports counts, dimensions, apartment checks, and character irreducibility checks without printing large basis vectors or full character tables.

For GL_3(F_2):

• number of points: 7;
• number of planes: 7;
• number of incidence edges: 21;
• dimension of ker(partial): 8;
• number of unique apartments up to orientation: 28;
• Weyl group size: 6;
• number of Weyl-group orbits on apartments: 7;
• orbit sizes: [3, 6, 1, 6, 6, 3, 3];
• dimension of the GL_3(F_2)-orbit span of an apartment cycle: 8;
• Steinberg character values by class order/size: 8, 0, -1, 0, 1, 1;
• inner product of the Steinberg character with itself: 1.

For the Fano-plane representation script:

• group order: 168;
• Fano-plane counts: 7 points, 7 lines, 21 flags;
• building boundary rank: 13;
• dim St = 8;
• dim im(partial) = 13;
• point and line permutation modules split as 1 + chi_6;
• flag permutation character splits as 1 + 2*chi_6 + St;
• character inner products satisfy <chi_6, chi_6> = 1, <St, St> = 1, and <chi_6, St> = 0;
• the two 3-dimensional characters satisfy <chi_3, chi_3> = 1, <chi_3, chi_3bar> = 0, Sym^2(chi_3) = chi_6, and chi_3*chi_3bar = 1 + St.

For GL_2(F_5):

• group order: 480;
• projective-line size: 6;
• dim St = 5;
• C[P^1] = 1 + St;
• character inner product <St, St> = 1;
• representation-family counts: 4 determinant characters, 4 Steinberg twists, 6 principal series, and 10 cuspidal representations;
• sum of squares of listed dimensions: 480;
• each constructed principal series has character inner product 1;
• --cuspidal 1 builds a 96-dimensional Gelfand-Graev representation, projects to rank 4, and verifies the resulting cuspidal character has inner product 1.

For GL_3(F_3):

• group order: 11232;
• number of points: 13;
• number of planes: 13;
• number of incidence edges: 52;
• boundary rank: 25;
• dimension of ker(partial): 27;
• number of unique apartments up to orientation: 234;
• rank of the apartment-cycle span: 27;
• Weyl group size: 6;
• number of Weyl-group orbits on apartments: 49;
• number of conjugacy classes: 24;
• inner product of the Steinberg character with itself: 1.

The GL_3(F_3) script ran end-to-end in under 20 seconds on the machine used to create this repository.

For general q, the script checks the formulas:

# points = # planes = q^2 + q + 1
# edges = (q + 1)(q^2 + q + 1)
dim St = #edges - #vertices + 1 = q^3
# apartments up to orientation = q^3(q + 1)(q^2 + q + 1) / 6

Mathematical context

For a finite group of Lie type, the Steinberg representation can be constructed from the top reduced homology of the associated spherical building. Here the groups are GL_3(F_q), so the building is one-dimensional, and its chambers are flags

line <= plane <= F_q^3.

Apartments come from choices of ordered bases of F_q^3. Each basis determines a hexagonal cycle in the incidence graph, and the alternating sum of its oriented chambers gives an apartment cycle. The scripts check that the span of these apartment cycles has the expected Steinberg dimension.

Scalar matrices act trivially on the building, so this geometric action factors through PGL_3(F_q).

Notes

This is exploratory research code rather than a packaged library. The calculation is intentionally explicit so the construction of the building, boundary map, group action, apartment cycles, and character can all be inspected directly.

The repository is distributed under the MIT license; see the repository-level LICENSE file.