Dynkin diagram of the E8 root system

The E8 calculation

A group of mathematicians started to develop algorithms and software to do these calculations (for any Lie group) in 2002. Fokko du Cloux took on the monumental task of writing the software, and by the fall of 2005 this software was ready. After calculations on smaller groups, they were ready to tackle E8.

Specifically the goal was to compute Kazhdan-Lusztig-Vogan polynomials for the large block of the split real form of E8. This block as 453,060 irreducible representations. For more information here are even more details.

The enormous size and complexity of E8 meant that his program needed a very large computer to run - one with more than 200 gigabytes of RAM. Starting in the summer of 2006, du Cloux, David Vogan and Marc van Leeuwen worked to make the program run on a smaller computer. After some experiments on other computers, by Birne Binegar and Dan Barbasch, the computations were run on the supercomputer Sage, provided by William Stein at the University of Washington. Sage has 64 gigabytes of memory and 16 processors.

The size of the answer

The result of the E8 calculation is a matrix, or grid, with 453,060 rows and columns. There are 205,263,363,600 entries in the matrix, each of which is a polynomial. The largest entry in the matrix is:
152 q22 + 3,472 q21 + 38,791 q20 + 293,021 q19 + 1,370,892 q18 + 4,067,059 q17 + 7,964,012 q16 + 11,159,003 q15 + 11,808,808 q14 + 9,859,915 q13 + 6,778,956 q12 + 3,964,369 q11 + 2,015,441 q10 + 906,567 q9 + 363,611 q8 + 129,820 q7 + 41,239 q6 + 11,426 q5 + 2,677 q4 + 492 q3 + 61 q2 + 3 q
If each entry was written in a one inch square, then the entire matrix would measure more than 7 miles on each side.

Even with a supercomputer it required very sophisticated mathematics and computer science to carry out the calculation. The computation was completed on January 8, 2007. Ultimately the computation took 77 hours of computer time, and 60 gigabytes to store the answer in a highly compressed form.

This is a huge amount of data. By way of comparison, a human genome can be stored in less than one gigabyte. For a more down to earth comparison, 60 gigabytes is enough to store 45 days of continuous music in MP3-format.

Some other facts about the answer

Size of the matrix:

Number of distinct polynomials: 1,181,642,979

Number of coefficients in distinct polynomials: 13,721,641,221

Maximal coefficient: 11,808,808

Polynomial with the maximal coefficient: 152q22 + 3,472q21 + 38,791q20 + 293,021q19 + 1,370,892q18 + 4,067,059q17 + 7,964,012q16 + 11,159,003q15 + 11,808,808q14 + 9,859,915q13 + 6,778,956q12 + 3,964,369q11 + 2,015,441q10 + 906,567q9 + 363,611q8 + 129,820q7 + 41,239q6 + 11,426q5 + 2,677q4 + 492q3 + 61q2 + 3q
Value of this polynomial at q=1: 60,779,787

Polynomial with the largest value at 1 which we've found so far: 1,583q22 + 18,668q21 + 127,878q20 + 604,872q19 + 2,040,844q18 + 4,880,797q17 + 8,470,080q16 + 11,143,777q15 + 11,467,297q14 + 9,503,114q13 + 6,554,446q12 + 3,862,269q11 + 1,979,443q10 + 896,537q9 + 361,489q8 + 129,510q7 + 41,211q6 + 11,425q5 + 2,677q4 + 492q3 + 61q2 + 3q
Value of this polynomial at q=1: 62,098,473

Main E8 page