| Let me try (and probably fail) to translate some of that. S_n is the collection of permutations on n symbols with a multiplicative structure. In python from itertools import permutations n = 6 S_n = list(permutations(range(n))) multiply = lambda alpha, beta: [alpha[x] for x in beta] This is the reason why Substitution-Permutation networks interleave permutations with non-permutations. Two permutations in a row is just a permutation. "Representation Theory" is the art of taking such multiplicative structures and associating every object with a matrix so that the matrix multiplication encodes the original objects multiplication. This is useful because mathematicians know so much about matrices. For example if you define matrix = lambda perm: [[1 if i == j else 0 for j in range(n)] for i in perm] matrix(alpha) * matrix(beta) == matrix(multiply(alpha, beta)) However, it is desirable to represent it as matrices in a more "efficient manner". For example S_3 can be written using only two dimensions. Representation theory deals with stuff like minimal representation. If you snake from the bottom of young's lattice upwards to a specific diagram you can record the information into a young tableau by writing in the cell when it was added. Young tableau are very mathematically rich, but here is an "application" of them. If you have a balanced parenthesis (the dyck langauge), [[[][]][]] and you record two list of when you open and when you close [0, 1, 2, 4, 7] [3, 5, 6, 8, 9] you get a two-by-n young tableau, since the grid is increasing horizontally and vertically. Other special shapes of young tableau can encode other interesting structures. Also see TAOCP V3 5.1.4, or google the RSK correspondence. edit 1: newlines
edit 2: fixed python |