Jordan blocks of unipotent elements on 2-restricted irreducible representations

Let G be a simple algebraic group over an algebraically closed field K of characteristic 2.

For G of rank at most 5, we list in the tables below for every unipotent element u of G and every rational irreducible 2-restricted KG-module V the Jordan block sizes of u on V.

With Steinberg's tensor product theorem, one can then compute the Jordan block sizes of u on any rational irreducible KG-module.

We also include some computations for simple G of type E₆ and type E₇.

All of the computations below were done using programs written in MAGMA.

Here are some remarks about the notation, which is mostly standard.

• Simple roots are ordered by the usual Bourbaki labeling, and we denote the ith fundamental highest weight by ϖ_i.
• Highest weight a₁ϖ₁ + ... + a_lϖ_l is also denoted by (a₁, ..., a_l).
• If the image of unipotent element in a given irreducible representation has Jordan block sizes 0 < d(1) < ... < d(t) and d(i) has multiplicity n_i, in the tables we denote this by d(1)^n₁, ..., d(t)^n_t.
• In this notation, the exponent n_i is omitted whenever n_i = 1.
• For groups of type A_l and the exceptional types, we use the usual Bala-Carter labeling of the unipotent classes.
• Let G be simple of type C_l, with natural module V and non-degenerate G-invariant alternating bilinear form β on V. In characteristic 2, the conjugacy class of a unipotent element u is labeled by the tuple (d(1)_{ε₁}^{n₁}, ..., d(t)_{ε_t}^{n_t}), where:
• The Jordan block sizes of u on V are 0 < d(1) < ... < d(t), with block size d(i) occurring with multiplicity n_i.
• ε_i equals 0 or 1 for all i, and ε_i = 0 if and only if β((u-1)^d(i)-1v,v) = 0 for all v in the kernel of (u-1)^d(i).
• In characteristic 2, a simple group H of type D_l embeds into a simple group G of type C_l as a subgroup generated by short root subgroups.
  • We label the unipotent conjugacy classes of H in the same way as in G.
  • Here H intersects precisely those unipotent conjugacy classes of G which act with an even number of Jordan block sizes on the natural module.
  • Some unipotent classes split into two conjugacy classes in H, in which case we distinguish them by a prime (') or double prime ('') superscript.
• In characteristic 2, there exist isogenies between groups of type B_l and type C_l, so the tables for type B_l and type C_l are identical.

Rank 2, p = 2

Type A₂

Type C₂

Type G₂

Rank 3, p = 2

Type A₃

Type C₃

Rank 4, p = 2

Type A₄

Type C₄, 1/2

Type C₄, 2/2

Type D₄

Type F₄, 1/3

Type F₄, 2/3

Type F₄, 3/3

Rank 5, p = 2

Type A₅, 1/2

Type A₅, 2/2

Type C₅, 1/6

Type C₅, 2/6

Type C₅, 3/6

Type C₅, 4/6

Type C₅, 5/6

Type C₅, 6/6

Type D₅, 1/3