Frank Lübeck |

## Tables of Weight Multiplicities

This page provides data which were computed during the preparation of the following paper.

**Lübeck, F.**,
*Small degree representations of finite Chevalley groups in
defining characteristic*,
LMS J. Comput. Math., **4** (2001), p. 135--169

Here is a slightly updated preprint version of July 2001:

[BibTeX-Entry]
[dvi-file
(with hyperlinks)]
[postscript-file]
[pdf-file]

**Additional data: ** groups of type D_{5} in
characteristic 2 (containing ATLAS group O_{10}(2)) - all 2-restricted weights,

groups of type D_{4} in
characteristic 3 (containing ATLAS groups O_{8}(3) and
O_{8}^{-}(3)) - all
3-restricted weights,

the upper bound in case D_{4} was raised from 2000 to
7168 (the largest degree of the Weyl group of type E_{8}),

groups of type C_{5} in
characteristic 2 (containing ATLAS group Sp_{10}(2)) - all 2-restricted weights.

### Content

In the paper for each type of simple, simply connected, connected reductive
algebraic group a bound is fixed and all irreducible representations of
these groups in their defining characteristic p and of degree at most the given
bound were determined. These representations are parameterized by
*highest weights*, only those with p-restricted highest weights are
listed. See the paper for more details.

For space reasons the paper lists only the parameterizing highest
weights and the dimensions of the modules. Here we give a more detailed
information and describe the weight multiplicities (*characters*)
for all the representations of small rank groups appearing in the paper.
To do this efficiently, we use the
fact that the weight multiplicities are the same for all weights in a fixed
orbit of the Weyl group in the weight lattice. Each such orbit contains a
unique dominant weight (i.e., with non-negative coefficients as linear
combination of the fundamental weights). We only give the weight
multiplicities for dominant weights together with the length of the
corresponding Weyl group orbit.

**Update:** (6/2006) The cases of type A_{l} were
extended to l < 21 and degree bounds (l+1)^4.

Here are the types of groups considered and the bounds. Click on the type to see the tables of weight multiplicities.

### The Tables

Lie type | group names | bound |
---|---|---|

A2 | SL_{3} | 400 |

A3 | SL_{4} | 500 |

A4 | SL_{5} | 1000 |

A5 | SL_{6} | 2500 |

A6 | SL_{7} | 2800 |

A7 | SL_{8} | 4096 |

A8 | SL_{9} | 6561 |

A9 | SL_{10} | 10000 |

A10 | SL_{11} | 14641 |

A11 | SL_{12} | 20736 |

A12 | SL_{13} | 28561 |

A13 | SL_{14} | 38416 |

A14 | SL_{15} | 50625 |

A15 | SL_{16} | 65536 |

A16 | SL_{17} | 83521 |

A17 | SL_{18} | 104976 |

A18 | SL_{19} | 130321 |

A19 | SL_{20} | 160000 |

A20 | SL_{21} | 194481 |

B2 | Spin_{5} | 300 |

B3 | Spin_{7} | 700 |

B4 | Spin_{9} | 1000 |

B4 | Spin_{9} | [all 2-restricted weights in char. 2] |

B5 | Spin_{11} | 2000 |

B6 | Spin_{13} | 4000 |

B7 | Spin_{15} | 5000 |

B8 | Spin_{17} | 7000 |

B9 | Spin_{19} | 8000 |

B10 | Spin_{21} | 10000 |

B11 | Spin_{23} | 12000 |

C3 | Sp_{6} | 1000 |

C4 | Sp_{8} | 2000 |

C5 | Sp_{10} | 2500 |

C5 | Sp_{10} | [all 2-restricted weights in char. 2] |

C6 | Sp_{12} | 4000 |

C7 | Sp_{14} | 6000 |

C8 | Sp_{16} | 10000 |

C9 | Sp_{18} | 10000 |

C10 | Sp_{20} | 10000 |

C11 | Sp_{22} | 12000 |

D4 | Spin_{8} | 7168 |

D4 | Spin_{8} | [all 3-restricted weights in char. 3] |

D5 | Spin_{10} | 3000 |

D5 | Spin_{10} | [all 2-restricted weights in char. 2] |

D6 | Spin_{12} | 4000 |

D7 | Spin_{14} | 5000 |

D8 | Spin_{16} | 10000 |

D9 | Spin_{18} | 15000 |

D10 | Spin_{20} | 18000 |

D11 | Spin_{22} | 20000 |

E6 | E_{6} | 50000 |

E7 | E_{7} | 100000 |

E8 | E_{8} | 100000 |

F4 | F_{4} | 12000 |

G2 | G_{2} | 500 |

Last updated: Mon Jul 29 10:22:24 2013 (CET)