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The Symmetric Group Appendix A The Symmetric Group Il faut, lorsque vous étudiez les propriétés du cercle, de la sphère et des sections coniques, que vous vous sentiez frères par l’esprit d’Euclide et d’Archimède et que, comme eux vous voyiez avec ravissement se développer le monde idéal des figures et des proportions dont les harmonies enchanteresses se retrouvent ensuite dans le monde réel. Jean Jaurès, Toulouse (1914) The purpose of this appendix is essentially to fix the notations and to recall some elementary facts about the representations of the symmetric group that are needed in this book. The main result is the Schur Lemma which says that in characteristic zero any homogeneous polynomial functor is completely determined by a representation of the symmetric group. For more details the reader can consult any book on the subject, for instance [Mac95, Sag01]. A.1 Action of Groups A.1.1 Group Algebra, Representation The group algebra K[G] of a group G over a commutative algebra K is the free module with basis the elements of G. The product is induced by the product in the group: aigi aj gj = (aiaj )(gigj ). i j i,j The tensor product of modules over K[G] is often denoted by −⊗G − instead of −⊗K[G] −. J.-L. Loday, B. Vallette, Algebraic Operads, 567 Grundlehren der mathematischen Wissenschaften 346, DOI 10.1007/978-3-642-30362-3, © Springer-Verlag Berlin Heidelberg 2012 568 A The Symmetric Group A representation of the group G is a left module M over K[G]. The action of g ∈ G on m ∈ M is denoted either by g · m or by gm if there is no ambiguity. The algebra K[G] is called the regular representation of G when viewed as a left module over itself. A.1.2 Invariants and Coinvariants To any representation M of G is associated its space of invariants MG and its space of coinvariants MG defined as follows: MG := {m ∈ M | g · m = m, ∀g ∈ G}, MG := M/{g · m − m | g ∈ G, m ∈ M}. G The notation follows the usual convention since M is contravariant in G and MG is covariant in G. There is a natural map from invariants to coinvariants: G M M MG. Whenever G is finite and #G is invertible in the ground ring, then this map is an [ ]→ 1 · [ ] isomorphism. Indeed an inverse is given by m #G g∈G g m, where m is the class of m ∈ M in MG. A.1.3 Restriction and Induction Let H be a subgroup of G.Therestriction of a representation of G to H is simply the same space but viewed as a module over H .IfM is a right H -module, then the induced representation is the following representation of G: G := ⊗ K[ ] IndH M M H G , where K[G] is viewed as a left module over K[H ] through the multiplication in K[ ] G ⊗ G . Observe that the space IndH M can be identified with the space M K K[H \G]. A.1.4 Equivariance Let M and N be two (left) G-modules. A linear map f : M → N is said to be G- equivariant,orsimplyequivariant, if, for any g ∈ G, m ∈ M one has f(g· m) = g ·f(m). In other words f is a morphism in the category of G-modules. The space of A.2 Representations of the Symmetric Group Sn 569 G-equivariant maps is denoted by HomG(M, N) since it is made of the morphisms which are invariant for the action of G by (g · f )(m) := g−1 · f(g· m).IfL is another G-module, one has a natural isomorphism G ∼ Hom M,Hom(N, L) = Hom(M ⊗G N,L). A.2 Representations of the Symmetric Group Sn A.2.1 The Symmetric Group Sn By definition the symmetric group Sn is the group of automorphisms of the set {1, 2,...,n}. Its elements are called permutations. The image of the permutation σ is denoted either by (σ (1),...,σ(n))or, sometimes, by [σ(1)...σ(n)]. The neutral element is denoted either by 1 or 1n or even idn. Let us denote by si the permutation which exchanges the elements i and i + 1 and leaves the others fixed. It is called a transposition and sometimes denoted by (i i + 1) (cycle notation). The transposi- tions generate Sn. In fact Sn is presented by the set of generators {s1,...,sn−1} and the set of relations ⎧ ⎨ 2 = = − si 1, for i 1,...,n 1, = | − |≥ ⎩sisj sj si, for i j 2, sisi+1si = si+1sisi+1, for i = 1,...,n− 2. This is called the Coxeter presentation of the symmetric group. In general there is no preferred way of writing a permutation in terms of the Coxeter generators. However, for any permutation σ there is a minimum number of generators in such a writing. It is called the length of the permutation and is denoted by (σ ).For instance we have ([321]) = 3 since [321]=s1s2s1. A.2.2 Irreducible Representations A representation of Sn is said to be irreducible if it is not isomorphic to the direct sum of two nonzero representations. It can be shown that the isomorphism classes of representations of the symmetric group Sn are in one-to-one correspondence with = the partitions of the integern, that is the sequence of integers λ (λ1,...,λr ) such ≥···≥ ≥ = that λ1 λr 1 and i λi n. It is often helpful to represent such a sequence by a Young diagram. For instance the Young diagram of (4, 2, 1) is 570 A The Symmetric Group The irreducible representation associated to λ is denoted by Sλ and called the Specht module. Any finite dimensional representation M of Sn is isomorphic to the sum of its isotypic components Mλ, one for each irreducible representation: M = Mλ λ where Mλ is isomorphic to the sum of a finite number of copies of Sλ. This number is called the multiplicity of Sλ in M. For instance the multiplicity of Sλ in the regular representation is equal to the dimension of Sλ. From this result follows the formula: dim Sλ 2 = n!. λ EXAMPLES. n = 1. The unique irreducible representation of S1 is one-dimensional. n = 2. There are two partitions of 2: λ = (2) and λ = (1, 1): The representation S2 is the one-dimensional trivial representation, so S2 = Kx and [21]·x = x. The representation S1,1 is the one-dimensional signature represen- tation, so S1,1 = Kx and [21]·x =−x. n = 3. There are three partitions of 3: λ = (3), λ = (2, 1) and λ = (1, 1, 1): The associated representations are the trivial representation for λ = (3) and the signature representation for λ = (1, 1, 1).Forλ = (2, 1) it is the hook representation which can be described as follows. Consider the three-dimensional space Kx1 ⊕ Kx2 ⊕ Kx3 on which S3 acts by permutation of the indices. The kernel of the linear map Kx1 ⊕Kx2 ⊕Kx3 → K,xi → 1 is clearly invariant under the action of S3.This is the two-dimensional hook representation S2,1. Observe that the three elements x1 − x2,x2 − x3,x3 − x1 lie in the kernel, but are not linearly independent since 2,1 their sum is zero. Any two of them form a basis of S . The vector x1 + x2 + x3 3 is invariant under the action of S3, hence it is isomorphic to S and there is an isomorphism ∼ 2,1 3 Kx1 ⊕ Kx2 ⊕ Kx3 = S ⊕ S . A.2 Representations of the Symmetric Group Sn 571 n = 4. There are five partitions of 4: λ = (4), λ = (3, 1), λ = (2, 2), λ = (2, 1, 1) and λ = (1, 1, 1, 1): The associated irreducible representations are as follows: S4 is the one-dimensional trivial representation, 3,1 ⊕i=4K → K → S is the hook representation, that is the kernel of i=1 xi ,xi 1, S2,2 is described below, 2,1,1 ⊕i=4K → K → S S is the kernel of i=1 xi ,xi 1, where we let 4 act on the four- dimensional space by permutation of the indices and multiplication by the signa- ture, S1,1,1,1 is the one-dimensional signature representation. In the set of four elements {a,b,c,d} we consider the non-ordered subsets of non-ordered pairs. So we have only three elements: {a,b}, {c,d} ; {a,c}, {b,d} ; {a,d}, {b,c} . We let S4 act on the ordered set {a,b,c,d} as usual, hence the three-dimensional vector space K3 spanned by the aforementioned subsets of pairs is a representation 3 of S4. Consider the map K → K which sends each generator to 1. The kernel is a 2,2 two-dimensional representation of S4.ThisisS . A.2.3 The Schur Lemma The following result, called the Schur Lemma, says that, for an infinite field, any → ⊗ ⊗n homogeneous polynomial functor of degree n is of the form V M(n) Sn V for some Sn-module M(n). It justifies the choice of S-modules to define an operad in characteristic zero. Lemma A.2.1. If K is an infinite field, then any natural transformation θ : V ⊗n → ⊗n V is of the form θn · (v1,...,vn) for some θn ∈ K[Sn]. Proof. Let GL(V ) be the group of isomorphisms of the space V (general linear group). Let α ∈ GL(V ) act on End(V ) by conjugation: α · f := αf α −1. 572 A The Symmetric Group We let GL(V ) act on End(V ⊗n) diagonally, and we denote by End(V ⊗n)GL(V ) the space of invariants.
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