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Part III Representation Theory Lectured by Stuart Martin, Lent Term 2013

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Part III Representation Theory Lectured by Stuart Martin, Lent Term 2013 Part III Representation Theory Lectured by Stuart Martin, Lent Term 2013 Please send any comments and corrections to [email protected]. Contents 0 Preliminaries 2 1 Semisimple algebras 4 2 Irreducible modules for the symmetric group 8 3 Standard basis of Specht modules 12 4 Character formula 14 5 The hook length formula 21 6 Multilinear algebra and algebraic geometry 24 Interlude: some reminders about affine varieties 29 7 Schur{Weyl duality 31 8 Tensor decomposition 34 9 Polynomial and rational representations of GL(V ) 37 Interlude: some reminders about characters of GLn(C) 40 10 Weyl character formula 41 11 Introduction to invariant theory and first examples 44 12 First fundamental theorem of invariant theory 50 Example sheet 1 53 Example sheet 2 57 Last updated Thursday 21 March 2013 1 0 Preliminaries In this course we will study representations of (a) (finite) symmetric groups (b) (infinite) general linear groups all over C, and apply the theory to \classical" invariant theory. We require no previous knowledge, except for some commutative algebra, ordinary representation theory and algebraic geometry, as outlined below. Commutative algebra References: Part III course, Atiyah{Macdonald. It is assumed that you will know about rings, modules, homomorphisms, quotients and chain conditions (the ascending chain condition, ACC, and descending chain condition, DCC). Recall: a chain, or filtration, of submodules of a module M is a sequence (Mj : 0 j n) such that ≤ ≤ M = M0 > M1 > > Mn > 0 ··· The length of the chain is n, the number of links. A composition series is a maximal chain; equivalently, each quotient Mj−1=Mj is irreducible. Results: 1. Suppose M has a composition series of length n. Then every composition series of M has length n, and every chain in M can be extended to a composition series. 2. M has a composition series if and only if it satisfies ACC and DCC. Modules satisfying both ACC and DCC are called modules of finite length. By (1), all composition series of M have the same length, which we denote by `(M) and call the length of M. 0 3.( Jordan–H¨older) If (Mi) and (Mi ) are composition series then there exists a one-to-one cor- 0 0 respondence between the set of quotients (Mi−1=Mi) and the set of quotients (Mi−1=Mi ) such that the corresponding quotients are isomorphic. Remark. For a vector space V over a field K, the following are equivalent: (i) finite dimension (ii) finite length (iii) ACC (iv) DCC and if these conditions are satisfied then `(V ) = dim(V ). (Ordinary) representation theory (of finite groups) References: Part II course, James{Liebeck, Curtis{Riener Vol. 1, §1 below. Work over C or any field K where char(K) - G . Let V be a finite-dimensional vector space. j j 2 GL(V ) is the general linear group of K-linear automorphisms of V . n If dimK (V ) = n, choose a basis e1; : : : ; en of V over K to identify it with K . Then θ 2 GL(V ) Aθ = (aij) Mn(K), where $ 2 n θ(ej) = aijei i=1 X and, in fact, Aθ GLn(K). This gives rise to a group isomorphism GL(V ) = GLn(K). 2 ∼ Given G and V , a representation of G on V is a homomorphism ρ = ρV : G GL(V ) ! G acts linearly on V if there is a linear action G V V , given by (g; v) g v, where (a)( action)(g1g2) v = g1 (g2 v) and e v = v × ! 7! · · · · · (b)( linearity) g (v1 + v2) = g v1 + g v2, g (λv) = λ(g v) · · · · · If G acts linearly on V then the map G GL(V ) given by g ρg, where ρg(v) = g v, defines a representation of G on V . Conversely,! given a representation7!ρ : G GL(V ) we have· a linear action of G on V given by g v = ρ(g)v. ! · In this case, we say V is a G-space or G-module. By linear extension, V is a KG-module, where KG is the group algebra { more to come later. For finite groups you should know If S is an irreducible KG-module then S is a composition factor of KG (Maschke's theorem) If G is finite and char(K) - G, then ever KG-module is completely reducible; or equivalently, every submodule is a summand. The number of inequivalent (ordinary) irreducible representations of G is equal to the number of conjugacy classes of G. If S is an irreducible CG-module and M is any CG-module, then the number of composition factors of M isomorphic to S is equal to dim HomCG(S; M). Algebraic geometry References: Part II course, Reid, §6 below. You should know about affine varieties, polynomial functions, the Zariski topology (in particular, Zariski denseness) and the Nullstellensatz. 3 1 Semisimple algebras Conventions All rings are associative and have an identity, which we denote by 1 or 1R. Modules are always left R-modules. (1.1) Definition A C-algebra is a ring R which is also a C-vector space, whose notions of addition coincide, and for which λ(rr0) = (λr)r0 = r(λr0) for all r; r0 R and λ C. There are obvious notions of subalgebras and algebra homomor- phisms. 2 2 Usually, our algebras will be finite-dimensional over C. (1.2) Examples and remarks (a) C is a C-algebra, and Mn(C) is a C-algebra. If G is a finite group then the group algebra CG = αgg : αg C is a C-algebra with f g 2 g pointwise addition: g αgg + g βgg = g(αg + βg)g P multiplication: g P hh0=g αhPβh0 g P Recall the correspondenceP P between representations of G over C and finite-dimensional CG-modules. (b) If R; S are C-algebras then the tensor product R S = R C S is a C-algebra. [More on can be found in Atiyah{MacDonald p. 30.] ⊗ ⊗ ⊗ (c) If 1R = 0R then R = 0R is the zero ring. Otherwise, for λ, µ C, we have λ1R = µ1R if and f g 2 only if λ = µ, so we can identify λ C with λ1R R. 2 2 With the identification C , R we can view C as a subalgebra of R. ! (d) An R-module M becomes a C-space via λm = (λ1R)m for λ C; m M. 2 2 Given any R-module N, HomR(M; N) is a C-subspace of HomC(M; N). In particular, it is a C-space, with structure (λϕ)(m) = λϕ(m) = '(λm) λ C; m M; φ HomR(M; N) 8 2 2 2 (e) If M be an R-module. Then EndR(M) = HomR(M; M) is a C-algebra with multiplication given by composition of maps: (' )(m) = '( (m)) m M; φ, EndR(M) 8 2 2 V V V In particular, if is a C-space then EndC( ) is an algebra. Recall that EndC( ) ∼= Mn(C), where n = dimC(V ). (f) Let R be a C-algebra and X R be a subset. Define the centraliser of X in R by ⊆ cR(X) = r R : rx = xr x X f 2 8 2 g It is a subalgebra of R. The centre of R is cR(R) = Z(R). (g) Let R be C-algebra and M be a R-module. Then the map αR End (M) given by α(r)(m) = rm ! C for r R, m M is a map of C-algebras. 2 2 Now, EndR(M) = cEnd (M)(αR), and so αR cEnd (M)(EndR(M)). C ⊆ C (h) An R-module M is itself naturally an EndR(M)-module, where the action is evaluation, and the action commutes with that of R. The inclusion α as in (g) says that elements of R act as EndR(M)-module endomorphisms of M. Given an R-module N, HomR(M; N) is also an EndR(M)-module, where the action is composition of maps. 4 (1.3) Lemma Let R be a finite-dimensional C-algebra. Then there are only finitely many isomor- phism classes of irreducible R-modules. Moreover, all the irreducible R-modules are finite-dimensional. Proof. Let S be an irreducible R-module and pick 0 = x S. Define a map R S by r rx. This map has nonzero image, so its image is S by irreducibility.6 2 In particular, ! 7! dim (S) dim (R) < C ≤ C 1 But S must occur in any composition series of R, so by the Jordan–H¨oldertheorem, there are only finitely many isomorphism classes of irreducible modules. (1.4) Lemma (Schur's lemma) Let R be a finite-dimensional C-algebra. Then (a) If S T are nonisomorphic irreducible R-modules then HomR(S; T ) = 0. (b) If S is an irreducible R-module then EndR(S) ∼= C. Proof (a) If f : R S is a map of algebras then ker(f) R and im(f) S. By irreducibility, ker(f) = 0 ! ≤ ≤ or R and im(f) = 0 or S. If ker(f) = 0 then R ∼= im(f) = S, contradicting nonisomorphism; so we must have ker(f) = R and im(f) = 0, i.e. f = 0. (b) Clearly D = EndR(S) is a division ring. Since S is finite-dimensional, so is D. Thus, if d D, 2 then the elements 1; d; d2;::: are linearly dependent. Let p C[X] be a nonzero polynomial with p(d) = 0. 2 Since C is algebraically closed, p factors as p(X) = c(X a1) (X an) − ··· − for some 0 = c C and ai C.
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