Simple basis coordinates | Epsilon coordinates | Reflection w.r.t. root |
(-1, -1, -1, -1) | -e_{1}+e_{5} | s1s2s3s4s3s2s1 |
(0, -1, -1, -1) | -e_{2}+e_{5} | s2s3s4s3s2 |
(-1, -1, -1, 0) | -e_{1}+e_{4} | s1s2s3s2s1 |
(0, 0, -1, -1) | -e_{3}+e_{5} | s3s4s3 |
(0, -1, -1, 0) | -e_{2}+e_{4} | s2s3s2 |
(-1, -1, 0, 0) | -e_{1}+e_{3} | s1s2s1 |
(0, 0, 0, -1) | -e_{4}+e_{5} | s4 |
(0, 0, -1, 0) | -e_{3}+e_{4} | s3 |
(0, -1, 0, 0) | -e_{2}+e_{3} | s2 |
(-1, 0, 0, 0) | -e_{1}+e_{2} | s1 |
(1, 0, 0, 0) | e_{1}-e_{2} | s1 |
(0, 1, 0, 0) | e_{2}-e_{3} | s2 |
(0, 0, 1, 0) | e_{3}-e_{4} | s3 |
(0, 0, 0, 1) | e_{4}-e_{5} | s4 |
(1, 1, 0, 0) | e_{1}-e_{3} | s1s2s1 |
(0, 1, 1, 0) | e_{2}-e_{4} | s2s3s2 |
(0, 0, 1, 1) | e_{3}-e_{5} | s3s4s3 |
(1, 1, 1, 0) | e_{1}-e_{4} | s1s2s3s2s1 |
(0, 1, 1, 1) | e_{2}-e_{5} | s2s3s4s3s2 |
(1, 1, 1, 1) | e_{1}-e_{5} | s1s2s3s4s3s2s1 |
roots simple coords | epsilon coordinates | [,] | g_{-10} | g_{-9} | g_{-8} | g_{-7} | g_{-6} | g_{-5} | g_{-4} | g_{-3} | g_{-2} | g_{-1} | h_{1} | h_{2} | h_{3} | h_{4} | g_{1} | g_{2} | g_{3} | g_{4} | g_{5} | g_{6} | g_{7} | g_{8} | g_{9} | g_{10} |
(-1, -1, -1, -1) | -e_{1}+e_{5} | g_{-10} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | g_{-10} | 0 | 0 | g_{-10} | g_{-9} | 0 | 0 | -g_{-8} | g_{-7} | 0 | -g_{-5} | g_{-4} | -g_{-1} | -h_{4}-h_{3}-h_{2}-h_{1} |
(0, -1, -1, -1) | -e_{2}+e_{5} | g_{-9} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | g_{-10} | -g_{-9} | g_{-9} | 0 | g_{-9} | 0 | g_{-7} | 0 | -g_{-6} | 0 | g_{-4} | -g_{-2} | 0 | -h_{4}-h_{3}-h_{2} | -g_{1} |
(-1, -1, -1, 0) | -e_{1}+e_{4} | g_{-8} | 0 | 0 | 0 | 0 | 0 | 0 | -g_{-10} | 0 | 0 | 0 | g_{-8} | 0 | g_{-8} | -g_{-8} | g_{-6} | 0 | -g_{-5} | 0 | g_{-3} | -g_{-1} | 0 | -h_{3}-h_{2}-h_{1} | 0 | g_{4} |
(0, 0, -1, -1) | -e_{3}+e_{5} | g_{-7} | 0 | 0 | 0 | 0 | 0 | g_{-10} | 0 | 0 | g_{-9} | 0 | 0 | -g_{-7} | g_{-7} | g_{-7} | 0 | 0 | g_{-4} | -g_{-3} | 0 | 0 | -h_{4}-h_{3} | 0 | -g_{2} | -g_{5} |
(0, -1, -1, 0) | -e_{2}+e_{4} | g_{-6} | 0 | 0 | 0 | 0 | 0 | 0 | -g_{-9} | 0 | 0 | g_{-8} | -g_{-6} | g_{-6} | g_{-6} | -g_{-6} | 0 | g_{-3} | -g_{-2} | 0 | 0 | -h_{3}-h_{2} | 0 | -g_{1} | g_{4} | 0 |
(-1, -1, 0, 0) | -e_{1}+e_{3} | g_{-5} | 0 | 0 | 0 | -g_{-10} | 0 | 0 | 0 | -g_{-8} | 0 | 0 | g_{-5} | g_{-5} | -g_{-5} | 0 | g_{-2} | -g_{-1} | 0 | 0 | -h_{2}-h_{1} | 0 | 0 | g_{3} | 0 | g_{7} |
(0, 0, 0, -1) | -e_{4}+e_{5} | g_{-4} | 0 | 0 | g_{-10} | 0 | g_{-9} | 0 | 0 | g_{-7} | 0 | 0 | 0 | 0 | -g_{-4} | 2g_{-4} | 0 | 0 | 0 | -h_{4} | 0 | 0 | -g_{3} | 0 | -g_{6} | -g_{8} |
(0, 0, -1, 0) | -e_{3}+e_{4} | g_{-3} | 0 | 0 | 0 | 0 | 0 | g_{-8} | -g_{-7} | 0 | g_{-6} | 0 | 0 | -g_{-3} | 2g_{-3} | -g_{-3} | 0 | 0 | -h_{3} | 0 | 0 | -g_{2} | g_{4} | -g_{5} | 0 | 0 |
(0, -1, 0, 0) | -e_{2}+e_{3} | g_{-2} | 0 | 0 | 0 | -g_{-9} | 0 | 0 | 0 | -g_{-6} | 0 | g_{-5} | -g_{-2} | 2g_{-2} | -g_{-2} | 0 | 0 | -h_{2} | 0 | 0 | -g_{1} | g_{3} | 0 | 0 | g_{7} | 0 |
(-1, 0, 0, 0) | -e_{1}+e_{2} | g_{-1} | 0 | -g_{-10} | 0 | 0 | -g_{-8} | 0 | 0 | 0 | -g_{-5} | 0 | 2g_{-1} | -g_{-1} | 0 | 0 | -h_{1} | 0 | 0 | 0 | g_{2} | 0 | 0 | g_{6} | 0 | g_{9} |
(0, 0, 0, 0) | 0 | h_{1} | -g_{-10} | g_{-9} | -g_{-8} | 0 | g_{-6} | -g_{-5} | 0 | 0 | g_{-2} | -2g_{-1} | 0 | 0 | 0 | 0 | 2g_{1} | -g_{2} | 0 | 0 | g_{5} | -g_{6} | 0 | g_{8} | -g_{9} | g_{10} |
(0, 0, 0, 0) | 0 | h_{2} | 0 | -g_{-9} | 0 | g_{-7} | -g_{-6} | -g_{-5} | 0 | g_{-3} | -2g_{-2} | g_{-1} | 0 | 0 | 0 | 0 | -g_{1} | 2g_{2} | -g_{3} | 0 | g_{5} | g_{6} | -g_{7} | 0 | g_{9} | 0 |
(0, 0, 0, 0) | 0 | h_{3} | 0 | 0 | -g_{-8} | -g_{-7} | -g_{-6} | g_{-5} | g_{-4} | -2g_{-3} | g_{-2} | 0 | 0 | 0 | 0 | 0 | 0 | -g_{2} | 2g_{3} | -g_{4} | -g_{5} | g_{6} | g_{7} | g_{8} | 0 | 0 |
(0, 0, 0, 0) | 0 | h_{4} | -g_{-10} | -g_{-9} | g_{-8} | -g_{-7} | g_{-6} | 0 | -2g_{-4} | g_{-3} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -g_{3} | 2g_{4} | 0 | -g_{6} | g_{7} | -g_{8} | g_{9} | g_{10} |
(1, 0, 0, 0) | e_{1}-e_{2} | g_{1} | -g_{-9} | 0 | -g_{-6} | 0 | 0 | -g_{-2} | 0 | 0 | 0 | h_{1} | -2g_{1} | g_{1} | 0 | 0 | 0 | g_{5} | 0 | 0 | 0 | g_{8} | 0 | 0 | g_{10} | 0 |
(0, 1, 0, 0) | e_{2}-e_{3} | g_{2} | 0 | -g_{-7} | 0 | 0 | -g_{-3} | g_{-1} | 0 | 0 | h_{2} | 0 | g_{2} | -2g_{2} | g_{2} | 0 | -g_{5} | 0 | g_{6} | 0 | 0 | 0 | g_{9} | 0 | 0 | 0 |
(0, 0, 1, 0) | e_{3}-e_{4} | g_{3} | 0 | 0 | g_{-5} | -g_{-4} | g_{-2} | 0 | 0 | h_{3} | 0 | 0 | 0 | g_{3} | -2g_{3} | g_{3} | 0 | -g_{6} | 0 | g_{7} | -g_{8} | 0 | 0 | 0 | 0 | 0 |
(0, 0, 0, 1) | e_{4}-e_{5} | g_{4} | g_{-8} | g_{-6} | 0 | g_{-3} | 0 | 0 | h_{4} | 0 | 0 | 0 | 0 | 0 | g_{4} | -2g_{4} | 0 | 0 | -g_{7} | 0 | 0 | -g_{9} | 0 | -g_{10} | 0 | 0 |
(1, 1, 0, 0) | e_{1}-e_{3} | g_{5} | -g_{-7} | 0 | -g_{-3} | 0 | 0 | h_{2}+h_{1} | 0 | 0 | g_{1} | -g_{2} | -g_{5} | -g_{5} | g_{5} | 0 | 0 | 0 | g_{8} | 0 | 0 | 0 | g_{10} | 0 | 0 | 0 |
(0, 1, 1, 0) | e_{2}-e_{4} | g_{6} | 0 | -g_{-4} | g_{-1} | 0 | h_{3}+h_{2} | 0 | 0 | g_{2} | -g_{3} | 0 | g_{6} | -g_{6} | -g_{6} | g_{6} | -g_{8} | 0 | 0 | g_{9} | 0 | 0 | 0 | 0 | 0 | 0 |
(0, 0, 1, 1) | e_{3}-e_{5} | g_{7} | g_{-5} | g_{-2} | 0 | h_{4}+h_{3} | 0 | 0 | g_{3} | -g_{4} | 0 | 0 | 0 | g_{7} | -g_{7} | -g_{7} | 0 | -g_{9} | 0 | 0 | -g_{10} | 0 | 0 | 0 | 0 | 0 |
(1, 1, 1, 0) | e_{1}-e_{4} | g_{8} | -g_{-4} | 0 | h_{3}+h_{2}+h_{1} | 0 | g_{1} | -g_{3} | 0 | g_{5} | 0 | -g_{6} | -g_{8} | 0 | -g_{8} | g_{8} | 0 | 0 | 0 | g_{10} | 0 | 0 | 0 | 0 | 0 | 0 |
(0, 1, 1, 1) | e_{2}-e_{5} | g_{9} | g_{-1} | h_{4}+h_{3}+h_{2} | 0 | g_{2} | -g_{4} | 0 | g_{6} | 0 | -g_{7} | 0 | g_{9} | -g_{9} | 0 | -g_{9} | -g_{10} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
(1, 1, 1, 1) | e_{1}-e_{5} | g_{10} | h_{4}+h_{3}+h_{2}+h_{1} | g_{1} | -g_{4} | g_{5} | 0 | -g_{7} | g_{8} | 0 | 0 | -g_{9} | -g_{10} | 0 | 0 | -g_{10} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
(4/5, 3/5, 2/5, 1/5) | = | 4/5ε1−1/5ε2−1/5ε3−1/5ε4−1/5ε5 |
(3/5, 6/5, 4/5, 2/5) | = | 3/5ε1+3/5ε2−2/5ε3−2/5ε4−2/5ε5 |
(2/5, 4/5, 6/5, 3/5) | = | 2/5ε1+2/5ε2+2/5ε3−3/5ε4−3/5ε5 |
(1/5, 2/5, 3/5, 4/5) | = | 1/5ε1+1/5ε2+1/5ε3+1/5ε4−4/5ε5 |
(1, 0, 0, 0) | = | ε1−ε2 |
(0, 1, 0, 0) | = | ε2−ε3 |
(0, 0, 1, 0) | = | ε3−ε4 |
(0, 0, 0, 1) | = | ε4−ε5 |