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Representations of the Symmetric Group Sn 49 III.5 Representations of the symmetric group Sn 49 III.5 Representations of the symmetric group Sn Let us now investigate the representations of the group S of the permutations of n N objects, n 2 ⇤ using the ideas sketched in § III.4.2b. For definiteness, we shall assume that the group algebra, on which the regular representation operates, is a complex vector space, thus to be denoted CSn. The multiplication on CS is (built from) the composition of permutations: for σ,⌧ S , ⌧?σ= ⌧ σ, n 2 n ◦ which will be more shortly written ⌧. Gathering first a few results obtained in previous sections, we know that the group Sn is of order S = n! (property I.18) and that its conjugacy classes are in one-to-one correspondence with the | n| integer partitions of n, which are conveniently depicted with the help of Young diagrams (§ I.4.3 b). III.5.1 Irreducible representations of Sn According to § III.4.2 b, the reduction of the regular representation of Sn can be done using primitive idempotents, i.e. adequate linear combinations of the group elements σ S that gener- 2 n ate invariant subspaces of CSn. In this specific case, the primitive idempotents are called Young elements. Two of them (for n 2(19)) are easily found, namely ≥ 1 the symmetrizer: S σ (III.55a) ⌘ n! σ Sn X2 1 the antisymmetrizer: A "(σ)σ, (III.55b) ⌘ n! σ Sn X2 with "(σ) the signature of the permutation σ. Using the identities ⌧S = S and ⌧A = "(⌧)A for all ⌧ S , one indeed checks that S and A are idempotent and that their product is 0 (the zero 2 n vector of the group algebra). Additionally, one sees that both elements generate one-dimensional invariant subspaces, corresponding to the trivial and the signature (or alternating) representations, respectively. For S2, these are the only two irreducible representations. For n 3, there are further irreducible representations, corresponding to other Young elements. ≥ We shall just state a few results regarding those, without proof, and illustrate them on the examples (20) of S3 or S4. For that purpose, let us first introduce two definitions. :::::::III.5.1 a::::::::::::::::Young tableaux Definition III.56. A Young tableau of order n is a Young diagram with n boxes in which the numbers 1, . , n have been written in any arbitrary order. Definition III.57. A Young tableau is called standard if the numbers occur in increasing order in each row and in each column. For instance, the following are four Young tableaux of order 7 corresponding to the same Young diagram; only the third and fourth tableaux are standard: 3 1 8 7 1 3 6 7 1 3 6 7 1 2 3 4 4 2 , 4 5 , 2 4 , 5 6 . 6 2 5 7 As further examples, we list in Fig. III.1 resp. III.2 all standard Young tableaux with 3 resp. 4 boxes, arranging them according to the Young diagrams. To the Young diagrams (of any order n) with a single row or a single column, there corresponds only one standard Young tableau, respectively. In contrast, when there are at least two rows or (19) The reduction of S1 is straightforward: it has a single irrep., the trivial representation. (20) Note that the irreducible representations of S3,whichisisomorphictoD3,wereactuallyalreadyintroducedin Sec. II.3.2. 50 Representations of finite groups 1 1 2 1 3 1 2 3 , 2 . • • 3 2 • 3 Figure III.1 – Standard Young tableaux for S3. 1 2 3 1 2 4 1 3 4 1 2 3 4 , , • • 4 3 2 1 1 2 1 3 1 4 1 2 1 3 2 , 3 , 2 , 2 . • 3 4 2 4 • • 3 4 4 3 4 Figure III.2 – Standard Young tableaux for S4. columns in the diagram, there are several standard tableaux. To know the number of the latter without enumerating them, let us introduce the hook length of a given box in a Young diagram, which is the number of boxes (including itself) either directly to its right or directly below. We illustrate this definition on an example in Fig. III.3, where the number in each box of the right diagram is precisely the hook length of the box. 6 4 2 1 Hook lengths for the diagram : 3 1 1 Figure III.3 – Example of hook lengths (right) for the Young diagram depicted left. Using the hook lengths, the number of standard Young tableaux for a given Young diagram of order n is given by the hook length formula: n! d↵ = , (III.58) (hook lengths) where the product in the denominator runsQ over all boxes of the diagram, while the subscript ↵ merely labels that diagram. For the example of Fig. III.3, one would find 7!/(6 4 2 1 3 1 1) = 35, · · · · · · different standard Young tableaux, which the reader probably does not want to enumerate. One quickly checks that the formula (III.58) gives the right results for the diagrams with only one row or only one column, as well as for those presented in Figs. III.1–III.2. :::::::III.5.1 b::::::::::::::::::::::::::::::::::::::::::::::::::::Young elements; irreducible representations of Sn For each standard Young tableau of order n, corresponding to a Young diagram (↵), one defines an element of the group algebra CS as follows. Let resp. denote the subset of S consisting n P Q n of the permutations that leave every row resp. every line of the standard Young tableau invariant. Denoting P σ and Q "(σ)σ, ⌘ ⌘ σ σ X2P X2Q both elements of CSn, their product Y = QP (III.59) is again an element of CSn which is a primitive idempotent up to a factor: Y is a Young element. For instance, the standard Young diagram built from the Young diagram with only a single row resp. a single column leads to the symmetrizer (III.55a) resp. to the antisymmetrizer (III.55b). Remark: resp. is sometimes called the row resp. column stabilizer of the standard Young P Q tableau under consideration. Both are actually subgroups of Sn. III.5 Representations of the symmetric group Sn 51 To each Young diagram (↵) correspond d↵ standard tableaux, with d↵ given by Eq. (III.58), which lead to as many Young elements. In turn, each of these Young elements generates (by left multiplication with the group algebra) a subspace of dimension d↵ that is invariant under the regular representation. More precisely, each Young diagram (↵) leads to d↵ copies of the same irreducible (↵) representation, namely of a representation of dimension d↵ — which we shall naturally denote Dˆ . Going to the standard Young diagrams with 3 boxes as given in Fig. III.1, this means we should find 4 Young elements S , Y , Y 0, A , respectively. S and A generate two one-dimensional invariant spaces with different irreducible representations, which we already know. In turn, Y and Y 0 generate two invariant spaces of dimension 2, corresponding to two copies of the same irreducible representation — which is the standard representation of § II.3.2 c. 1 2 Consider for example the standard Young diagram 3 . The corresponding row stabilizer is = Id, (1 2) and the column stabilizer = Id, (1 3) ,leadingto P { } Q { } P = Id + (1 2) and Q =Id (1 3), − leading to the Young element Y =[Id (1 3)][Id + (1 2)],thatis,afterfactorizingoutand − performing the products Y = Id + (1 2) (1 3) (1 2 3). − − The left multiplication of Y with each element of S3 gives (somewhat tediously) Id Y = Y (III.60a) (1 2 3) Y = (1 2 3) + (1 3) (2 3) (1 3 2) (III.60b) − − (1 3 2) Y = (1 3 2) + (2 3) (1 2) Id = Y (1 2 3) Y (III.60c) − − − − (1 2) Y = (1 2) + Id (1 3 2) (2 3) = Y + (1 2 3) Y (III.60d) − − (2 3) Y = (2 3) + (1 3 2) (1 2 3) (1 3) = (1 2 3) Y (III.60e) − − − (1 3) Y = (1 3) + (1 2 3) Id (1 2) = Y . (III.60f) − − − All 6 products can thus be expressed as linear combinations of Y and (1 2 3) Y, which shows that the subspace generated by multiplying Y left with CS3 is of dimension 2. For every element σ of S3,onecanalsocomputetheproductσ(1 2 3) Y :usingEqs.(III.60), one can then construct a 2 2-matrix representing σ in the basis consisting of Y and (1 2 3) Y, ⇥ i.e. precisely a two-dimensional representation of S3. Eventually, note that Eqs. (III.60) allow one to compute Y 2 =3Y, which shows that Y is indeed idempotent up to a factor, as well as SY = AY =0. Regarding S4, the standard Young diagrams depicted in Fig. III.2 lead to two representations of dimension 1 — the trivial and alternating representations —, two inequivalent irreducible repre- sentations of dimension 3 — which are both contained three times in the regular representation —, and one irreducible representation of dimension 2, contained twice in the regular representation. :::::::III.5.1 c::::::::::::::::::::::::::::Conjugate Young diagrams Definition III.61. Given a Young diagram of order n,theconjugate diagram (or transposed diagram) is that obtained by exchanging the roles of the rows and columns. A self-conjugate diagram is then a Young diagram which coincides with its conjugate diagram. Examples: Considering the Young diagrams of order 4, the diagrams ⇤ and resp. and form two pairs of conjugate diagrams, while the diagram is self-conjugate. 52 Representations of finite groups More generally, for any n the diagrams consisting of a single row and that consisting of a single ⇤ column, corresponding respectively to the Young symmetrizer and antisymmetrizer, are conjugate to each other.
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