Table of e8 representations

Question

I want to understand the representation theory for the (complex-valued) $e8$ exceptional Lie algebra.

An ideal answer to this question would contain a link to a text file (or any other format) containing a list of (some of the) irreps with their dimensionality and tensor products included.

For example, for $su(2)$ the list would be

-----------+-------------------+------------------------------------------------
irrep id   |  dimensionality   |  tensor products
-----------+-------------------+------------------------------------------------
[0]        |  1                |  [0] * [0] = [0]
-----------+-------------------+------------------------------------------------
[1/2]      |  2                |  [1/2] * [0] = [1/2], [1/2] * [1/2] = [0] + [1]
-----------+-------------------+------------------------------------------------
[1]        |  3                |  [1] * [0] = [1], [1] * [1/2] = [1/2] + [3/2],
           |                   |  [1] * [1] = [0] + [1] + [2]
-----------+-------------------+------------------------------------------------
...

Of course there are an infinite number of irreps, and the list should only contain the first few (sorted by dimensionality).

In case if there is no such data available, I would like a description of an algorithm which would generate the data.

Answer 1

The dimensions of the irreducible representations of $\mathfrak{e}_8$ are $$ 1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000 (twice), 12692520960..., $$ see the sequence A121732 in the OEIS. More data can be found in the tables of Tits. See also the atlas for $E_8$.