Representation theory of the symmetric groupsConstruction of all the irreducible representations.The conjugacy classes•Every element of Sn can be written as a product of (disjoint) cycles: For example (1326)(45)(7)(8).•If s Î Sn is written in cycle form, and t is some other element of Sn , then tst-1 is obtained from s by replacing each integer i in the cycle form of s by t(i).•Conversely, if s1 and s2 have the same cycle structure so that there is a permutation t relating the entry i in s1 to t(i) in s2 then s2 =ts1t-1 .• A conjugacy class in Sn is thus determined by [n1,… nn ] where n1is the number of one cycles, n2 is the number of two cycles etc.The number of elements in a conjugacy class.TExample: the conjugacy classes of S4 .PartitionsSetYoung diagramsirreducible representation of Sn to each diagram. Then we will knowthat we will have found all the irreducibles of Sn .A partial order on the Young diagrams.Young tabloids.Young tabloids, continued.The group Sn acts on Ml by permuting the elements in the boxes.This action is clearly transitive.The number of elements in Ml .IfThe representation of Sn on F(Ml).The representation of Sn on F (Mn-2,2).Goal:Young tableaux.We are assuming thatSo if we define the operatorwe want to prove thatThus(2)We have proved