Lecture 10 - Representation Theory II: Heuristics

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1 Lecture 10 - Representation Theory II: Heuristics February 15, Weights 1.1 Weight space decomposition We switch notation from last time. We let Λ indicate a highest weight, and λ to be an arbitrary weights themselves. If g is simple of rank n with CSA h, then h is an abelian vector space of linear operators on any g-module V, and so decomposes into weight spaces relative to h. If λ : h C is a linear functional, the m-weight space is V λ = { v V } h.v = λ(h)v (1) provided this space is non-trivial. Now let Φ be the Lie algebra s root system in h, and let be a choice of base. Roots α Φ have associated sl 2 -subalgebras span{h α, x α, y α }, for which V is a finite dimensional module. If V is finite dimensional, then each weight m of V takes integral values on the h α. Recall that dualizing h α (via the Killing form) yields the element so that 2α (α, α) h (2) λ(h α ) = 2 (λ, α) (α, α) = λ, α Z. (3) Because = {α 1,..., α n } is a basis for h, a weight λ is uniquely specified as a row vector: λ = (λ 1,..., λ n ) Z n. (4) 1

2 1.2 Highest weights and generation of modules Given, let B( ) and N( ) be the associated Borel subalgebra and its derived subalgebra: B( ) = h α>0 g α N( ) = α>0 g α = [B( ), B( )]. (5) A highest weight is a weight Λ with a non-zero weight space V Λ such that x α.v Λ = 0 for all α > 0. That is, N( ).V Λ = {0}. (6) If V is an irreducible g-module, then V Λ is 1-dimensional, and V Λ generates V under the action of various applications of the y α. That is, letting v 0 V Λ, we have V Λ = span C { y k 1 α 1... y kn α n.v 0 k1,..., k n Z + }. (7) where V Λ is the notation for the irreducible g-module of highest weight Λ. Now if v µ is any weight-vector and α i is any root, we have that sl 2 = span{h αi, x αi, y αi } operates on v µ to create an sl 2 -submodule, called the α i -weight string through µ. Given any root α j we have h αj.y αi.v µ = [ h αj, y αi ].vµ + y αi.h αj.v µ = α i (h αj )y αi.v µ + y αi.(µ(h αj )v µ ) = (µ α i )(h αj )y αi.v µ = µ α i, α j y αi.v µ. (8) so that y αi.µ (if non-zero!) is also a weight vector, of weight µ α i. We want to determine the vector expression of µ + α i. To this end, recall the Cartan matrix C = (C ij ) of g, with components Then if then x αi.v µ is a weight vector of weight C ij = α i, α j. (9) µ = (µ 1,..., µ n ) (10) µ + α i = (µ 1 + C i1,..., µ j + C ij,..., µ n + C in ) (11) That is, y αi.v µ is the weight vector whose weight is µ minus the i th -row of the Cartan matrix. 2

3 1.3 Weight strings Given gl 2 span C {h α, x α, y α } and the sub-representation of V it generates by action on the weight-space V µ, the full weight string is µ + qα,..., µ,..., µ rα (12) which can be expressed geometrically as points in h. We know that the reflection in α reverses the string, and since the reflection of any vector v about the hyperplane determined by α is v v, α α, we have (µ rα) µ rα, α α = µ + qα rα ( µ, α 2r) α = qα µ, α = r q (13) Therefore is µ is the highest root in the α-string (that is, x α kills v µ ), then 1 + µ, α gives the length of the string. This provides an algorithm for determining the weight-space decomposition of representations. 1.4 Remark on using simple roots Given highest weight Λ the module V Λ is clearly generated by Ug. Given any basis of g we know the structure of Ug, and of course the most natural basis comes from a choice of base. If = {α 1,..., α n } is a base, then monomials of the form span Ug. y αi1... y αin h αj1... h αjm x αk1... x αkp (14) There are two advantages to using such a presentation. The first is that the y αi and the x αj commute, unless α i = α = α j, in which case a copy of h α is thrown off. But then h α commutes with both the x αi and y αj after throwing off α, α i x αi or α, α i y αi. Thus the only real relations that emerge from re-arranging the monomial comes from switching the x s with each other and switching the y s with each other. The other advantage is that we easily see that an irreducible module V Λ of highest weight is generated entirely by the y s acting on a vector v 0 V Λ V Λ. This produces the following proposition. Proposition 1.1 If g is a simple Lie algebra (finite dimensional, over C) and if V Λ is a finite-dimensional irreducible module of highest weight Λ, then the weight space V Λ V Λ of highest weight is spanned by a single vector v 0, and V Λ = span C { y αi1... y αin. v 0 αij }. (15) 3

4 Pf. With Ug.V Λ = V Λ along with the facts that x α.v Λ = 0 and h α.v Λ = Λ, α V Λ, this follows from Poincare-Birkhoff-Witt theorem which says that U g is spanned by monomials of the form (14). 2 sl(3, C) Representations We finally explore some examples. We have two simple roots, h 1 and h 2. The sl(3) Cartan matrix is ( 2 1 ) 1 2 (16) The irreducible representations of sl 3 are determined by highest weights, which have the form µ = (a 1, a 1 ) where a 1, a 2 are non-negative integers. As before µ(h i ) = µ, h i = a i. (17) 2.1 The (0, 0) representation This is the trivial 1-dimensional representation. 2.2 The (1, 0) representation Let v be the highest weight vector with weight µ = (1, 0). Then the h 1 -string has length 2 and the α 2 -string is trivial (length 1). Then applying y 1 we have that x 1.v has weight µ α 1 = (1 C 11, 0 C 12 ) = ( 1, 1) (18) Now consider the x 2 -action on this vector: since [x 2, y 1 ] = 0 (from the fact that the x i are simple roots) we have x 2.y 1.v = y 1.x 2.v = 0 (19) so that the α 2 -string through ( 1, 1) has length 2. Applying y α2, we get µ α 1 α 2 = ( 1 C 21, 1 C 22 ) = (0, 1) (20) 4

5 Finally there is no additional α 1 -string through (0, 1), so the weight-space decomposition is W eight Dimension Height Expression (1, 0) 1 1 v 0 ( 1, 1) 1 0 y 1.v 0 (21) (0, 1) 1 1 y 2.y 1.v The (0, 1) representation The first weight-string is the length-2 α 2 -string produced by y 2, giving us the single additional weight (1, 1). From this we get a length-2 α 1 -string, which gives the single weight ( 1, 0). Then we are done. W eight Dimension Height Expression (0, 1) 1 1 v 0 (1, 1) 1 0 y 2.v 0 (22) ( 1, 0) 1 1 y 1 y 2.v The (1, 1) representation From (1, 1) we both an α 1 - and an α 2 -string, each of length 1, and obtain weights ( 1, 2) and (2, 1). We get two more strings ( 1, 2) (0, 0) (1, 2) α 2 string (2, 1) (0, 0) ( 2, 1) α 1 string. (23) Note that [y 1, y 2 ] = y 1 + y 2, so that y 1.y 2.v y 2.y 1.v, which shows that the (0, 0)-weight space must have dimension 2. Now (1, 2) and ( 2, 1) each produce a string of length 1, given by (1, 2) ( 1, 1) α 1 string ( 2, 1) ( 1, 1) α 2 string (24) 5

6 However the ( 1, 1) weight space has dimension 1. We have W eight Dimension Height Expression (1, 1) 1 2 v 0 ( 1, 2), (2, 1) 1, 1 1 y 1.v 0, y 2.v 0 (0, 0) 2 0 y 2 y 1.v 0, y 1 y 2.v 0 (25) ( 2, 1), (1, 2) 1, 1 1 y 2 y 1 y 2.v 0, y 1 y 2 y 1.v 0 ( 1, 1) 1 2 y 2 y 1 y 1 y 2.v 0 Thus the (1, 1)-representation is eight-dimensional. It is just the adjoint representation. 2.5 The (2, 0) representation It as an exercise to check the following table, which will be used below. W eight Height Dimension (2, 0) 2 1 (0, 1) 1 1 (1, 1), ( 2, 2) 0 1 (26) This is a six-dimensional representation. ( 1, 0) 1 1 (0, 2) Fundamental Representations Given a Lie algebra g with base = {α 1,..., α n }, a fundamental representation is any irreducible representation of highest weight (0,..., 0, 1, 0,..., 0) (27) Because weights sum under taking tensor products, we have that (V (10...0) ) k1 (V (01...0) ) k2... (V (00...1) ) kn, (28) 6

7 while not irreducible, has highest weight (k 1, k 2,..., k n ). (29) The fundamental representations thus generate all representations. In the case of First note that the V (1,0) and V (0,1) representations generate all representations: (V (1,0)) k (V (0,1)) l = V (k,l)... (30) is a representation with highest weight (k, l), so splits off a factor of V (k,l). But also we have (V (1,0) ) 2 = V (2,0) V (0,1) (31) so that V (1,0) generates V (0,1). To see (31), note that the weights occurring in V (1,0) V (1,0) are all 9 of the 2-fold sums (one must count multiplicities) of the weights from Table (21). In this way we obtain two copies of the weight (0, 1), although the weight space V (0,1) in V (2,0) has dimension 1. The fundamental representations are V (1,0) and V (0,1), although V (1,0) alone generates all representations. Note that V (1,0) and V (0,1) are exchanged by the generator of the outer automorphism group of A 2. The algebras sl n+1 have the n fundamental representations V (1,0,...,0),..., V (0,0,...,1). (32) Of these, the first and last are isomorphic, the second and second to last are isomorphic, and so on. However, the single representation generates them all. (1, 0,..., 0) (33) Under some terminology (moreso in the physics literature), (1, 0,..., 0) alone is called the fundamental representation, and the representations in (32) are called the basic representations. The A n are special. Other algebras require two or more representations to generate all others. 7