The E_{8} calculationA 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 E_{8}.Specifically the goal was to compute KazhdanLusztigVogan polynomials for the large block of the split real form of E_{8}. This block as 453,060 irreducible representations. For more information here are even more details. The enormous size and complexity of E_{8} 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 answerThe result of the E_{8} 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 q^{22} + 3,472 q^{21} + 38,791 q^{20} + 293,021 q^{19} + 1,370,892 q^{18} + 4,067,059 q^{17} + 7,964,012 q^{16} + 11,159,003 q^{15} + 11,808,808 q^{14} + 9,859,915 q^{13} + 6,778,956 q^{12} + 3,964,369 q^{11} + 2,015,441 q^{10} + 906,567 q^{9} + 363,611 q^{8} + 129,820 q^{7} + 41,239 q^{6} + 11,426 q^{5} + 2,677 q^{4} + 492 q^{3} + 61 q^{2} + 3 qIf 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 MP3format.
Some other facts about the answerSize of the matrix: 453,060Number 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:
152q^{22} + 3,472q^{21} +
38,791q^{20} + 293,021q^{19} +
1,370,892q^{18} + 4,067,059q^{17} +
7,964,012q^{16} + 11,159,003q^{15} +
11,808,808q^{14} +
9,859,915q^{13} + 6,778,956q^{12} +
3,964,369q^{11} + 2,015,441q^{10} +
906,567q^{9} + 363,611q^{8} +
129,820q^{7} + 41,239q^{6} +
11,426q^{5} + 2,677q^{4} +
492q^{3} + 61q^{2} + 3q
Polynomial with the largest value at 1 which we've found so far:
1,583q^{22} + 18,668q^{21} + 127,878q^{20} + 604,872q^{19} + 2,040,844q^{18} +
4,880,797q^{17} + 8,470,080q^{16} + 11,143,777q^{15} + 11,467,297q^{14} +
9,503,114q^{13} + 6,554,446q^{12} + 3,862,269q^{11} + 1,979,443q^{10} +
896,537q^{9} + 361,489q^{8} + 129,510q^{7} + 41,211q^{6} + 11,425q^{5} + 2,677q^{4} +
492q^{3} + 61q^{2} + 3q
