A Digest on Representation Theory of the Symmetric Group

Partitions can be graphically represented by Young frames, which are Young tableaux with empty boxes. The i-th part λi corresponds to the i-th row of the frame, consisting of λi boxes. Conversely, the Young frames of n boxes can be uniquely labelled by a partition λ ` n. We will therefore identify a Young frame with the partition labelling it. A Young tableau (YT) of N objects and of shape λ ` n is a Young frame λ in which the boxes are labelled by numbers (1, . . . , N ). A standard Young tableau (SYT) of shape λ ` n is a Young tableau of N = n objects such that the labels appear increasing in every row from left to right, and increasing in every column downwards; hence every number occurs exactly once. A Semistandard Young tableau (SSYT) of shape λ ` n is a Young tableau such that the labels appear non-decreasing in every row from left to right, and increasing in every column downwards. The number of SSYTs of N objects and of shape λ ` n (imposing the condition l(λ) ≤ N ) is given by sλ(1×N ); see below for an explanation. The number f of SYTs of shape λ ` n; r is ([6] p. 119) f = n! ∆(ν1, . . . , νr) ν1! . . . νr! , r = l(λ), (1)