Representation Theory of the Symmetric Group: Basic Elements
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REPRESENTATION THEORY OF THE SYMMETRIC GROUP: BASIC ELEMENTS BY GRAHAM GILL, FOR MAT1196F Since Cayley’s theorem implies that every finite group G is isomorphic to a subgroup of SjGj, understanding the representation theory of the finite symmetric groups is likely to yield productive tools and results for the representation theory of finite groups in general. In these notes we will examine the basic results for the representation theory of Sn. We will describe all irreducible representations of Sn and their characters, as well as giving a complete method for decomposing a representation of Sn into its irreducible factors, in terms of certain elements of the group algebra CSn known as Young symmetrizers. No such general method exists for general finite groups, so in a sense the fundamental problems of representation theory have been answered in the case of finite symmetric groups. The Group Algebra CG We first review some facts regarding the group algebra CG where G is a finite group. If A(G) is the algebra of all complex valued functions on G with multipli- cation given by the convolution X ¡ ¢ f ¤ f 0(g) = f (g0) f 0 g0¡1g ; g02G then CG is isomorphic as a complex algebra with A(G), via the map CG !A(G), 0 P g 7! e , where e : G ! C is the function g 7! ± 0 . If a = a g 2 CG, then g g g;g gP2G g a is sent to the function g 7! ag. Henceforth we will write a = a(g)g. Then ³ ¡ ¢´ g2G P 0 0 P P 0 0¡1 ab = 0 a(g)b(g )gg = 0 a(g )b g g g, so that as an element g;g 2G g2G g 2G¡ ¢ P 0 0¡1 in A(G), (ab)(g) := a ¤ b(g) = g02G a(g )b g g . In this way we may pass easily between CG and A(G). Under this isomorphism we also find that Z(CG), the centre of CG, corresponds with C(G), the set of class functions on G. On CG we have the involution à ! X ¤ X X a¤ = a(g)g := a(g)g¡1 = a (g¡1)g: g g g P ¡ ¢ ¤ ¤ 0¡1 0¡1 Then if a a = 0 we have a a(g) = g02G a (g )a g g = 0 and in particular, ¤ P ¤ 0 = a a(e) = g2G a (g)a (g), so that a = 0. An involution satisfying a a = 0 ) a = 0 is called nondegenerate, and the algebra possessing it is called symmetric. If a¤ = a then a is called Hermitian. For a Hermitian element a 2 CG, a2 = 0 ) a = 0. It follows that if I is a left ideal of CG and I2 = f0g then I = 0, because a 2 I ) (a¤a)2 2 I2, so that (a¤a)2 = 0 ) a¤a = 0 ) a = 0. Date: Fall 2005. 1 REPRESENTATION THEORY OF THE SYMMETRIC GROUP: BASIC ELEMENTS 2 If (¼; V ) is a group representation of the finite group G then it determines the algebra representation (^¼; V ) of CG, via X ¼^(a) = a(g)¼(g) 2 End(V ) g2G and it is easily checked that ¼^ is an algebra homomorphism. If (½; V ) is a repre- sentation of CG, then by restriction of ½ to G ½ CG we obtain a representation (~½; V ) of G, since ½(g) 2 GL(V ) for all g 2 G. Clearly ¼^~ = ¼ and ½~^ = ½, and the representation theory of CG and of G will exhibit corresponding structures. We will denote group representations using ¼ and algebra representations using ½ for clarity. The left regular representation (¼L; A(G)) of G is given by (¼L (g0) f)(g) = ¡ ¡1 ¢ f g0 g , which we may write for (¼L; CG) as ¼L(g)(a) = ga. Then the corre- sponding left regular representation of the algebra CG is its self-representation (½L =¼ ^L; CG) with ½L(a)(b) = ab. It follows then that a (vector) subspace W ⊆ CG is invariant under the left regular representation of CG if and only if W is a left ideal of CG. A restriction of the left regular representation of CG to a left ideal W is irreducible if and only if W is a minimal left ideal. Hence the left regular repre- sentation of CG is completely reducible if and only if CG = W1 © ¢ ¢ ¢ © Wk, with the Wi minimal left ideals, and this gives the decomposition of the representation into irreducibles. Thus, in order to decompose the left regular representation of CG into irreducibles it is equivalent to write CG as a direct sum of minimal left ideals. In A(G) we have the inner product X ¡1 (f1; f2) = jGj f1(g)f2(g) g2G which in CG becomes ha; bi = jGj¡1 b¤a(e). If W is a left ideal in CG it follows that W ? is also, and CG = W © W ?. Then we obtain the following lemma. Lemma 1. Every left ideal in CG is principal and generated by an idempotent. Proof. Let W be a left ideal of CG. Since CG = W ©W ?, we write e = ! +!0 with ! 2 W , !0 2 W ?. Then ! = !2 + !!0 = !2 + !0! and so !!0 = !0! 2 W \ W ? = f0g. It follows that ! = !2 is idempotent, as is !0 = (!0)2. Then (CG) ! ⊆ W , (CG) !0 ⊆ W ?, and if a 2 CG we have a = a! + a!0, so that W © W ? = CG = (CG) ! © (CG) !0 which shows that W = (CG) !, W ? = (CG) !0. ¤ In the notation of the lemma, we find also that W = fa 2 CGja! = ag, W ? = fa 2 CGja! = 0g. If ² 2 CG is idempotent, define ²0 = e ¡ ², which is also idempotent. We find ²²0 = ²0² = 0 and conclude that CG = (CG)² © (CG)²0, (CG)² = fa 2 CGja² = ag = fa 2 CGja²0 = 0g. However (CG)² and (CG)²0 will not in general be orthogonal. The search for the irreducible representations of Sn will reduce to the construc- tion of two families of idempotents in CSn, one of which satisfies an orthogonality property. The first step is more general, showing how the character of a subrep- resentation of the left regular representation of G may be expressed in terms of idempotents in CG. REPRESENTATION THEORY OF THE SYMMETRIC GROUP: BASIC ELEMENTS 3 Lemma 2. If (¼; W ) is the restriction of the left regular representation (¼L; CG) of G to a left ideal W of CG, i.e. ¼ is a subrepresentation, and if W = (CG) ² for some idempotent ², then we have the following formula for the character of ¼: X ³ ´ 0 ¡1 0 ¡1 ¼(g) = ² g g g : g02G Proof. Define P 2 EndC(CG) by P (a) = a². Define T : G ! EndC(CG) by T (g)(b) = gb². It follows that T (g) = ¼L(g) ± P , W is T (g)-invariant and the restriction of T (g) to W is ¼(g). (I.e. ¼ = T as maps G ! EndC(W ).) Let ²0 = e ¡ ² and let W 0 = (CG)²0. Then CG = W © W 0 and P is projection from CG onto W along W 0. T (g) restricts to the zero operator on W 0. Choosing bases of W 0 and W we see that tr T (g) = tr ¼(g) = ¼(g). Now consider the basis of CG given by the elements of G in some fixed order: th fg1; : : : ; gng. We have T (g)(gj) = ggj², and to discover the i coordinate in terms of this basis we calculate using the correspondence with A(G): ³ ´ ¡1 ¡ ¡1 ¡1 ¢ (ggj²)(gi) = ² (ggj) gi = ² gj g gi : Hence Xn X ³ ´ ¡ ¡1 ¡1 ¢ 0 ¡1 0 ¡1 ¼(g) = tr T (g) = ² gi g gi = ² g g g : ¤ i=1 g02G For any representation (¼; V ) of G we may regard V as a CG-module via a ¢ v := ¼^(a)(v). (This is not in general a faithful module action.) Under this action, idempotents in CG become projection operators in End(V ). If (¼1;V1) ;:::; (¼k;Vk) is a complete list of irreducible representations of G, then for each m = 1; : : : ; k the projections X ¡1 Pm = (dim Vm) jGj ¼m (g)¼(g) 2 End(V ) g2G give Mk ©rm ©rm V = PmV; where PmV ' Vm and ¼ jPmV ' ¼m ; m=1 k ©rm and so ¼ ' ©m=1¼m , for some nonnegative integer rm. This gives the decompo- sition of (¼; V ) into a direct sum of inequivalent subrepresentations, each of which is equivalent to a multiple (rm) of some irreducible. In the language of CG each Pm corresponds to the idempotent X ¡1 ²m = (dim Vm) jGj ¼m (g)g 2 CG: g2G (The fact that Pm is a projection implies ²m is idempotent follows by calculating Pm for (¼; V ) = (¼L; CG), the left regular representation of G, for then Pm(e) = ²m, 2 2 and ²m = Pm(e)Pm(e) = Pm(e) = Pm(e) = ²m.) To discover the irreducible factors of (¼; V ) and not only multiples of them we search for other idempotents of CG, and we see this process carried out for G = Sn below. A representation (½; V ) of CG, where (V; h¢j¢i) is a complex inner product space, is called symmetric if h½(a)(u)jvi = huj½ (a¤)(v)i for all a 2 CG, u; v 2 V . If V is finite dimensional, this condition means ½ (a¤) = ½(a)¤, the adjoint operator to ¤ ¤ ½(a). Thus in an orthonormal basis ¯ for V we have [½ (a )]¯ = [½(a)]¯. Then REPRESENTATION THEORY OF THE SYMMETRIC GROUP: BASIC ELEMENTS 4 it is easily checked that a representation (¼; V ) of G is unitary if and only if the representation (^¼; V ) of CG is symmetric, where V is finite dimensional.