CHAPTER VIII

Characters of the Irreducible Modules

The main result of this chapter is the formula, due to Weyl, for the character of any finite-dimensional irreducible module for a split semi-simple Lie algebra over a field of characteristic 0. If the base field is the field of complex numbers and R is a finitedimensional irreducible representation, then the character of R is the function on a Cartan subalgebra defined by

where, as usual, exp a = 1 + a + a²/2! + …. If is a connected semi-simple compact Lie group, then (1) gives the character in the ordinary sense of an irreducible representation of . Thus in this case corresponds to a maximal torus and it is known that any element of is conjugate to an element in this torus. Then (1) gives the values of the characters for elements of the torus. We know that h^R acts diagonally, that is, a basis can be chosen so that the matrix of h^R is

where Λ(h), M(h), … are the weights of in the representation. Then the matrix exp h^R is diag {exp Λ(h), exp M(h), …}; hence

where n_M is the multiplicity of the weight M(h), that is, the dimensionality of the weight space _M.

One obtains a purely algebraic form of the definition of the character x(h) by replacing the exponentials exp M(h) of (3) by formal exponentials which are elements of a certain group algebra. Weyl’s formula gives an expression for the character x_Λ of the finite-dimensional irreducible module with highest weight Λ as a quotient of two quite simple elementary alternating expressions in the exponentials.

Weyl derived his formula originally by using integration on compact groups. An elementary purely algebraic method for obtaining the result is due to Freudenthal and we shall follow this in our discussion. A preliminary result of Freudenthal’s gives a recursion formula for the multiplicities n_M. Weyl’s formula can be used to derive a formula for the dimensionality of the irreducible module with highest weight Λ. It can also be used to obtain the irreducible constituents of the tensor product of two irreducible modules.

1. Some properties of the Weyl group

In this section we derive some properties of the Weyl group which are needed for the proof of Weyl’s formula and for the determination of the automorphisms of semi-simple Lie algebras over an algebraically closed field of characteristic 0 (Chapter IX). As usual, denotes a finite-dimensional split semi-simple Lie algebra over a field Φ of characteristic zero, e_i, f_i, h_i, i = 1,2, …, l, are canonical generators for as in Chapters IV and VII, and is the splitting Cartan subalgebra spanned by the h_i. Let * be the conjugate space of , ^*₀ the rational vector space spanned by the roots of in . If α is a non-zero root, then the Weyl reflection is the mapping

in ^*₀. Here (ξ, η) is the positive definite scalar product in ^*₀ defined as in §4.1. The mapping S_α is characterized by the properties that it is a linear transformation in ^*₀ which maps a into —α and leaves fixed every vector orthogonal to α. The reflection S_α is an orthogonal transformation relative to (ξ, η) and S_α permutes the weights of in any finite-dimensional module for . The S_α generate the Weyl group W which is a finite group. If T is an orthogonal transformation such that αT is a root for some root α ≠ 0, then a direct calculation shows that

In particular, this holds for every T ∈ W. We note also that if α is a root then — α is a root and it is clear from (4) that S_α = S_{_α}.

The roots α_i, i = 1,2, …, l, such that α_i(h_j) = A_ji, (A_ij) the Cartan matrix, form a simple system. The notion of a simple system of roots is that given in § 4.3 and depends on a lexicographic ordering of the rational vector space ^*₀. We recall that l roots form a simple system if and only if every root α = ±(Σk_iα_i) where the k_i are non-negative integers. This criterion implies that if π = {α₁, α₂, …, α_i} is a simple system and T is a linear transformation in ^*₀ which leaves the set of roots invariant, then πT = {α₁T, …, α_lT} is a simple system of roots. The set P of positive roots in the lexicographic ordering which gives rise to the simple system π is the set of non-zero roots of the form Σk_iα_i, k_i 0. On the other hand, if the ordering is given, so that P is known, then π is the set of elements of P which cannot be written in the form β + γ, β, γ, ∈ P. Thus P and π determine each other. Any simple system of roots defines a set of canonical generators e_i, f_i, h_i(cf. §4.3).

LEMMA 1. Let π = {α₁, α₂, …, α_l} be a simple system of roots and let a be a positive (negative)root. Then αS_αi > 0 (< 0) if α ≠ -α; and αS_αi < 0 (> 0) if α = α (α = — α _i).

Proof. If α is a negative root, then — α is a positive root and the assertion on α will follow from that on — α. Hence it suffices to assume α > 0. We have α = Σk_jα_j, k_j 0 and

If α ≠ α_i, then since no multiple except 0, ± α of a root is a root, α ≠ k_iα_i and so k_j ≠ 0 for some j ≠ i. Then the expression for αS_αi has a positive coefficient for some α_j and hence αS_αi > 0. If α = α_i, then αS_αi = - α _i < 0. This completes the proof.

Our results on W will be given in two theorems. The first of these is

THEOREM 1. If π = {α₁, α₂, …, α_i} is a simple system of roots, then the Weyl group W (relative to ) is generated by the reflections S_αi, α_i ∈ π. If a is any non-zero root, then there exists α ∈ π and S ∈ W such that α_jS = α.

Proof: Let W′ be the subgroup of W generated by the S_αi. We show first that any positive root α has the form α_jS, α_j ∈ π, S ∈ W. We recall that if α = Σk_iα_i then Σk_i is the level of α and we shall prove the result by induction on the level. The result is clear if the level is one since this condition is equivalent to α ∈ π. If Σk_i > 1, and there exists an α_i such that (α, α_i) > 0. Otherwise (α, α_i) 0 for all i and since α = Σk_iα_i, k_i 0 contrary to (α, α) 0. Choose α_i so that (α, α) > 0. Then β = αS_αi. > 0 by Lemma 1 and we have β = Σ_j≠ikk_jα_j + (k_i – [2(α, α_i)])α_i. Since (α, α_i) > 0 it is clear that the level of β is lower than that of α. Hence the induction shows that β = α_jS′ where S ∈ W′, α_j ∈ π, S′ ∈ W′. Then α = α_jS′S_αi as required. Since α = α_j S where S ∈ W′, S_α = S^-1S_αjS ∈ W′. Since S_α, = S_-α this shows that every S_α ∈ W′ and so W′ = W. It remains to show that if α is any non-zero root then α = α_iS, α ∈ π, S ∈ W. This has been shown for α > 0. Since αS_α = — α the result is clear also for α < 0.

THEOREM 2. Let π and π′ be any two simple systems of roots. Then there exists one and only one element S ∈ W such that πS = π′.

Proof. Let P and P′, respectively, denote the sets of positive roots determined by π and π′. It is clear that P and P′ contain the same number q of elements, since this is half the number of non-zero roots. It is clear also that P = P′ if and only if π = π′ and if ≠ P′ then π P′ and π′ P. Let r be the number of elements in the intersection P P′. If r = q, P = P′ and S = 1 satisfies πS = π′, so the result holds in this case. We now employ induction on q — r and we may assume r < q, or P ≠ P′. Then there exists α_i ∈ π such that and hence . If , by Lemma 1. Hence . Also α_i = (−α_i)S_αi ∈ P P′ S_αi. Hence P P′ S_αi contains at least r + 1 elements. The simple system corresponding to P′S_αi is π′S_αi so the induction hypothesis permits us to conclude that there exists a T ∈ W such that πT = π ′S_αi. Then S = TS_αi satisfies πS = π′. This proves the existence of S. To prove uniqueness it suffices to show that if S ∈ W satisfies πS = π, or equivalently, PS = P, then S=1. We give an elementary but somewhat long proof of this here and in an exercise (Exercise 2, below) we shall indicate a short proof which is based on the existence theorem for irreducible modules. We now write S_i = S_αi and by Theorem 1, S = S_i1, S_i2 … S_im, i_j = 1, 2, …, l. We cannot have m = 1 since α_i1S_i1 = - α_i1 π and if m = 2, then α_i1S_i = — α_i1 and since (— α_i1)S_i1 = - α_i1S_i1S_i2 > 0, i₂ = i₁ by Lemma 1. Thus . We now suppose m > 2 and we may assume that if T = S_j1S_j2 … S_jr, r < m, and πT = π, then T = 1. Let S′ = S_i1S_i2 … S_im-1, so that S = S′S_im. Since S ≠, S_im, S′ ≠ 1 and since S′ is a product of m — 1 S_j, PS′ ≠ P by the induction hypothesis. Then there exists an α_j ∈ π such that α_jS′ < 0. Since α_jS′S_im > 0, Lemma 1 implies that α_jS′ = -α_im. Similarly, if β is any positive root such that βS′ < 0, then βS′S_im > 0 implies that βS′ = — α_im = α_jS′ hence β = α_i. Thus S′ has the following two properties: α_jS′ = — α_im and βS′ > 0 for every positive β ≠α_j. Set S_i0 = 1. Then α_jS_i0 > 0 and

Hence there exists a k, 1 k m — 1, such that α_jS_i0S_i1 … S_ik-1 > 0 α_jS_i0 … S_ik-1 > 0 and (α_jS_i0 … S_ik-1)S_ik < 0, α_jS_i0 … S_ik-1 = α_ik. Hence if we set T = S_i0S_i1 … S_ik-1, then T^-1S_jT = S_ik or S_jT = TS_ik. Then, if T′ = S_ik+1 … S_im-1 for k < m — 1 and T′ = 1 for k = m — 1, S′ = TS_ikT′ = S_jTT′. Hence TT′ = S_jS′. If α is a positive root ≠ α_j, then β = αS_j is a positive root ≠ α_j. Hence αTT′ = αS_jS′ = β S′ > 0. Also α_jTT = α_jS_jS′ = (— α_j)S′ = α_im > 0. Hence πTT′ = π and since TT′ is a product of m — 1 S_j’s the induction hypothesis gives TT′ — 1. Then S_jS′ = 1 and so S′ = S_j and S = S_i1S_j. Hence S = 1 by the case m = 2 which we considered before.

2. Freudenthal’s formula

Let be a finite-dimensional irreducible module for with highest weight Λ = Λ(h) a dominant integral linear function on . We know that is a direct sum of weight spaces relative to and that the weights are integral linear functions on of the form Λ — Σk_iα_i, k_i a non-negative integer. If M = M(h) is any integral linear function on we define the multiplicity n_Mof M in to be 0 if M is not a weight and otherwise, define n_M = dim _M, where 9ftis the weight space in of corresponding to the weight M. We recall that for the highest weight Λ we have n_Λ = 1. We shall now derive a recursion formula due to Freudenthal which expresses n_M in terms of the n_M, for M′ > M in the lexicographic ordering of ₀^* determined by the simple system of roots π = {α₁, α₂, …, α_l}.

Let α be a non-zero root of and choose so that (e_α, e_-α) = — 1 where (x, y) = tr ad x ad y, the Killing form on . Then we know (§4.1) that [e_αe_-α] = h_α where, in general for ρ ∈ ^*, h_ρ is the element of such that (h, hρ) = ρ(h). As in §4.2, let ^(α) be the subalgebra + Фe_α + Фe_-α. Then we know that is completely reducible as (α)-module (Theorem 4.1). We consider a particular decomposition of as a direct sum of irreducible ^(α)-submodules and we shall refer to these as the irreducible ^(α)-constituents of . Let be one of these. Then we know that has a basis (y₀, y₁, …, y_m) such that

We know also that

where (ρ, σ) = σ(h_σ) = σ (h_ρ), as in §4.1. Equations (7) imply that

We note also that is a direct sum of weight spaces, that the weights are M, M — α, M — 2α, …, M — mα and the weight spaces _M-pα = _M-pα are one-dimensional.

Let M be a weight of in such that M + α is not a weight. Then the α-string of weights in containing M is M, M — α, …, M — mα where m = 2(M, α)/(α, α). Let 0 p m. Then M — pα is a weight and _M-Tα is a direct sum of the M — pα weight spaces of those irreducible ^(α) -constituents which have this as weight. If 0 p (M, α)/(α, α), then these are the irreducible ^(α)-constituents having maximal weight M, M – α, …, M – pα. Let m_j, 0 j (M, α)/(a, a), denote the number of irreducible ^(α)-constituents of highest weight M — jα. Then it is now clear that

so that,

If 0 j p (M,α)/(α, α), then the weight space corresponding to M — pα in an irreducible ^(α) -constituent having highest weight M — jα is spanned by the vector y_p-j in the notation of (7). The dimensionality of this module is 2(M—jα, a)/(α, a) + 1 = (m — 2j) + 1.

Hence we may replace i by p — j and m by m— 2j in (9) to obtain

We can use this to compute tr , the trace of the restriction to tr . We remark that it is clear that

so that is invariant under . By (12), the contribution to tr of the m_j irreducible ^(α) -constituents with highest weight M — jα is

hence,

Since m/2 = (M, α)/(α, α). we have the formula

for 0 p (M, α)/(α, α).

Next let (M, α)/(α, α)< p m. If we apply the Weyl reflection S_α to M — pα we obtain since m = 2(M, α)/(α, α).We have and we see that M — pα is a weight for the irreducible ^(α)-constituents having the following highest weights: and only these. The reasoning used to establish (13) now gives

for (M, α)/(α, α;) <p m. We recall that if M is a weight, the MS_α is a weight and n_M= n_MSα (Theorem 4.1). Hence . On the other hand, . Hence

This and (14) imply that (13) is valid also for p > (M, α)/(α, α).We recall that M was any weight of in such that M + α is not a weight. Hence M — pα can represent any weight of in . We now change our notation and write M for M — pα. If we recall that n_M′ = 0 if M′ is not a weight, then we can re-write (13) as

for any weight M and any root α ≠ 0.

We consider next the Casimir element defined by the Killing form. This is where (u_i), (uⁱ) are dual bases for relative to (x,y). We know that [a^RΓ] = 0 for all a є . Since R is absolutely irreducible it follows from Schur’s lemma that , (cf. Jacobson, Lectures in Abstract Algebra II, p. 276). We choose dual bases in the following way: (h₁, …, h_l), (h¹, …, h^l) are dual bases in relative to (x,y). Then since (e_α, e_—α) = — 1 and e_α is orthogonal to and to every the following are dual:

Then

If we take the trace of the induced mappings in _M, M a weight, we obtain

Since h^R is the scalar multiplication by M(h) in _M we have , so the first term on the right hand side of (18) is . We proceed to show that (M, M). Thus we write . Then and and .

Hence . It follows that If we use this result and (16) in (18), we obtain

The terms n_M(M, α) and n_M(M, —α) cancel in this formula so we may replace by . The resulting formula is valid also if M is not a weight. In this case n_M=0. If, for a particular α,no M + jα, j 1, is a weight then n_{M + jα} — 0 and the particular sum . If M + jα is a weight for some j 1, then no M — kα is a weight for k 1 since the α-string containing M+jα does not contain M. Hence the set {M+kα | k 1}contains the complete α-string containing M + jα. Then , since

Thus in all cases and (19) holds. The argument shows also that

holds for any integral linear function M on . This implies that , so if we substitute in (19) with we obtain

Setting this gives the following formula:

If M = Λ the highest weight of in , then n_M = 1 and n_M+kα = 0 for α > 0, k 1. Hence (21) gives

and substitution in (21) gives Freudentha’s recursion formula:

We shall now show that this gives an effective procedure for calculating n_M beginning with n_Λ = 1. For this we require a couple of lemmas.

LEMMA 2. Let . Then δ(h_i) = 1, i= 1,2, …,l, if e_i, f_i, h_i is a set of canonical generators for . Also if S ≠ 1 is in W, then δ — δS is a non-zero sum of distinct positive roots.

Proof: If α is a positive root we know (Lemma 1) that αS_i ≡ αS_αi = α-α(h_i)α_i > 0 unless α = α_i in which case α_iS_i = —α_i.Hence

Also, (δS_i,α_i) = (δ,α_iS_i) = (δ,- α_i). Hence (δ- α_i, α _i)= (α,- α_i) and 2(δ, α_i) = (α_i, α_i). Thus δ(h_i) = 2(δ, α_i)/(α_i, α _i) = 1, i = 1,2,…,l. Since any S α W maps roots into roots we evidently have δS = δ-Σβ, where the summation is taken over the β = -αS > 0. If there are no such β, then αS > 0 for all α > 0. This implies S = 1, by Theorem 2, contrary to hypothesis.

LEMMA 3. Let Λ be the highest weight of in . Then

for any weight M ≠ Λ of in .

Proof: We shall prove that there exists a M' > M such that . This will prove the result by an evident ascending chain argument. Assume first there exists an i such that M(h_i) < 0, h_i in the set of canonical generators. Then we take M’ = MS = M-M(h_i)α_i > M and we have

since M(h_i) = 2(M, α_i)/(α_i, α _i). Since (α, α_i) > 0 this shows that Next suppose M(h_i) 0, i= 1, 2,..l. M +α_i is a root and x ≠ 0 satisfies xh = Mx, then xe_i = 0 for all i and x is canonical generator of an e-extreme -module. This must coincide with , so M is the highest weight, contrary to hypothesis. Now let α_i be one of the simple roots such that M’ = M+ α_i is a weight. Then M' > M and

Since M(h_i) 0, (M,α) 0. Also (δ,α_i) > 0 and (α_i, α_i) >0, so again we have

It is now clear that (22) can be used to determine the weights and their multiplicities. Thus we begin with n_A = 1. Suppose that for a given non-negative integers with at least one k_i > 0, we already know n_M' for every integers, 0 k′_i k′_i, M' ≠ M. Then every term on the right hand side of (22) is known. Moreover, if (Λ+α, Λ+α) = (M+α, M+α), then, by Lemma 3, M is not a weight and n_M = 0. Otherwise, the coefficient of n_M in (22) is not zero so we can solve for n_M using the formula.

3. Wey's character formula

In order to obtain a general formulation of Weyl's formula valid for any field it is necessary to replace the exponentials which appear in this formula by “formal” exponentials. This notion can be made precise by introducing the group algebra over the base field of the group of integral linear functions on . We recall that an element M ∈ * is called integral if M(h_i) is an integer for i = 1,2, …, l, where e_i, f_i, h_i, are canonical generators for . The set ^- of integral linear functions is a group under addition, and it is clear that is the direct sum of the cyclic groups generated by elements λ_i of such that λ_i(h_j) = δ_ij. We now introduce an algebra over α with basis in 1:1 correspondence with the elements of in which the multiplication table is

Then is the group algebra over Φ of and e(0) = 1 is the identity element of . The elements of are the formal exponentials to which we alluded before. We now define the character X of a finite-dimensional module to be the formal exponential

where n_M is the multiplicity of M ∈ as defined before: n_M = 0 if M is not a weight and n_M = dim _M the dimensionality of the weight space if M is a weight. The summation in (25) is taken over all M ∈ . This is a finite sum since n_M ≠ 0 for only a finite number of M.

Let x_i = e(λ_i), λ_i (h _j) = δ_ij. Since any integral linear function can be written in one and only one way as M = Σm_iλ_i where the m_i are integers, the base element . We have called M dominant if M(h_i) 0 for i = 1, 2, …,l. This is equivalent to m_i 0. The set of linear combinations of the e(M), M dominant, is a subalgebra of which is the same as the set of linear combinations of the monomials . The set of these monomials obtained from the sequences of integers (m₁,…, m_l) with m_i 0 are linearly independent. Hence the subalgebra we have indicated can be identified with the commutative polynomial algebra Φ[x₁, x₂, …, x_l] in the algebraically independent elements x₁ …, x_l. Every element of has the form lf where the r_i 0 and f∈ Φ[x₁ …, x_l]. It is well known that Φ[x₁ …, x_l] is an integral domain. It follows that is a commutative integral domain.

With each element S in the Weyl group W we associate the linear mapping in such that e(M)S = e(MS). Since

it is clear that S is an automorphism in . The set of S in is a group of automorphisms isomorphic to W. An element α ∈ is called symmetric if aS = a,S ∈ W, and alternating if aS = (det S)a, S ∈ W where det S is the determinant of the orthogonal transformation S in . Thus det S = ± 1 and if S = S_α the Weyl reflection determined by the root α then det S_α =—1. In particular, det S_i =- 1 for S_i =S_αi. Since the S_i generate W, a is symmetric if and only if a S_i = a, i — 1, 2, …, l, and a is alternating if and only if aS_i = a for all i. The set of symmetric elements is a subalgebra; the set of alternating elements is a subspace. The product of two alternating elements is symmetric and the product of an alternating element and a symmetric element is alternating. Since n_Ms = n_M for S ∈ W, it follows that the character is a symmetric element. We note next that the element

where δ = (1/2)(Σ_{α > 0} α) the integral linear function defined in § 2, is an alternating element of . Thus we have seen (proof of Lemma 2) that δS_i = δ – α_i so (-δ)S_i = -δ + α_i and e(-δ)S_i = e(-δ)e(α_i). Also we know that S_i permutes the positive roots ≠ α_i and sends α_i into —α_i. Hence

and

We set

This element is alternating. It turns out that we can obtain a formula for f and this will give the desired formula for χ.

Let σ = Σ_S∈W(det S)S as linear operator in . We have for any T ∈ W that σT = Σ_S(det S)ST = (det T)^-1 Σ_s(det ST)ST = (det T)σ and similarly Tσ = (det T)σ. If a is any element of , then (aσ)T = (det T)(aσ). Hence aσ is an alternating element. If a is alternating to begin with, then aS = (det S)a and so aσ = wa where w is the order of the Weyl group. It follows that (1/w)σ is a projection operator of onto the space of alternating elements. Hence any alternating element has the form aσ, a ∈ , and consequently such an element is a linear combination of the elements e(M)σ, M an integral linear function. Since Sσ = (MS)σ it is clear that e(M)σ can be replaced by e(M)σ and so we may express any alternating element as a linear combination of elements e(M)σ where M is highest among its conjugates MS under the Weyl group. In particular, we may suppose that M MS_i = M — M(h_i)α_i, that is, we may suppose that M(h_i) 0 for i = 1,2,…, l. Suppose M(h_i) = 0 for some i, or equivalently MS_i = M. Then e(M)σ = — e(M)S_iσ = -σ(MS_i) σ= -e(M)σ and e(M)σ = 0. We therefore conclude that every alternating element is a linear combination of the elements e(M)σ where M(h_i) > 0, i=1, 2, …, l.

We now apply this argument to the element Q defined in (26). If we multiply out the product in (26) we see that Q is a linear combination of elements e(M) where M = δ — ρ and ρ is a sum of a subset of positive roots. Thus where ∈_α = ±1 and any conjugate MS of M under the Weyl group again has the form where ρ′ is a sum of positive roots and ∈′_α = ±1. If we apply the projection operator (1/w)σ to the expression indicated for Q we obtain an expression for Q as linear combination of elements e(M)σ where M is of the form δ – ρ, ρ a sum of positive roots. We have seen also that we may assume that M satisfies M(h_i) > 0, i = 1,2, …, l. These conditions imply that M = δ. Thus, M = δ – ρ, ρ = Σk_iα_i where the k_i are nonnegative integers, then . Since is an integer. Hence we must have ρ(h_i) 0. On the other hand, . Hence (ρ, ρ) =0 and ρ = 0. Thus we have shown that . Now . By Lamma 2, . Hence where M = δ - ρ, ρ ≠ 0 and a sum of positive roots. Also it is clear from (26) that Q = e(δ) + Σ ± e(M). Since Q = ηe(δ)σ a it follows that η = 1. We have therefore proved the following

LEMMA 4. Let Q = e(- –)II_{α > 0}(e(α) - 1). Then

We shall next introduce the vector space and we shall define certain linear mappings of this space and of the algebra . The elements of have the form . The algebra composition in provides a linear mapping of ⊗ into such that a ⊗ b → ab. This gives a linear mapping of so that . It follows that if we set then this product of Σ_ρi ⊗ a_i and b is single-valued and coincides with the image of Σ_ρi ⊗ a_i ⊗ b in * ⊗ . It is clear that the product (Σ_ρi ⊗ a_i)b = Σ_ρi ⊗ a_ib turns *⊗ into a right -module. We recall that we have the bilinear form (ρ, σ) = (h_ρ, h_σ) on * which defines the linear mapping * ⊗ * into so that ρ ⊗ π → (ρ, π). If we combine this with the linear mapping of ⊗ into we obtain the linear mapping of (* ⊗ ) ⊗ (* ⊗ ) into ⊗ = such that (ρ ⊗ a) ⊗ (τ ⊗ b) → (ρ, τ)ab. This defines a -bilinear mapping of * ⊗ such that the value (ρ ⊗ a, τ ⊗ b) = (ρ, τ)ab. Since (ρ, σ) is symmetric and is commutative this is a symmetric bilinear form. Also if c ∈ , then and (ρ ⊗ a, τ ⊗ b)c = (ρ, τ)abc and similarly which implies that (x, y), x, y ∈ * ⊗ , is -bilinear.

Next we define a linear mapping, the gradient, of into * ⊗ : a → aG such that e(M)G = M ⊗ e(M) and a linear mapping, the Laplacian, of into : a →aα such that e(M)Δ = (M, M)e(M). We have

The linearity then implies that

We have

This implies that

We now return to the formulas which we developed for the multiplicity n_M of the integral linear function M in a finite-dimensional irreducible module of highest weight Λ, γ the element of (rational number) determined by the Casimir operator. We consider again (19):

We multiply both sides by e(M) and sum on M. This gives

which we multiply through by

by (26). This gives

The coefficient of e(M + α) in the right-hand side of (31) is

Hence (31) can be written in the form

Hence we have

and canceling Q ≠ 0 in the integral domain , we obtain

If we set f = χQ, as before, we obtain

Since Q = Σ_S∈Wdet S(e(δS)) and (δS, δS) = (δ, δ) we have QΔ = (δ, δ)Q. Since γ = (Λ + δ, Λ + δ) — (δ, δ) (eq. below (21)) these substitutions convert (33) to the following fundamental equation for f

The element f = χQ is an alternating element and consequently this element is a linear combination of elements of the form e(M)σ. Moreover, we can limit the M which are needed here by looking at the form of χ and Q. Thus we have χ = Σn_Me(M) where we now consider the summation as taken just over the weights M of the representation. Also we have seen that Q = Σ_S∈W(det S)e(δS). If we multiply we obtain χQ as a linear combination of terms e(M + δS) where M is a weight: M a weight, S ∈ W. Now

so that e(M + δS) is in the characteristic space of the characteristic root (MS^-1 + δ, MS^-1+ δ) of Δ. Since f belongs to the characteristic root (Λ + δ, Λ + δ) for α and characteristic spaces belonging to distinct roots are linearly independent it follows that f is a linear combination of e(M + δS) such that

By Lemma 3 (MS^-1 + δ, MS^-1 + δ) = (Λ + δ, Λ + δ) for the weight MS′ implies that MS′ = Λ. Hence we see that f is a linear combination of the terms e(Λ + δ)S. If we apply the projection operator (1/w)σ to f we see that . Since . Hence the coefficient of e(Λ + δ) in our expression for f is η. On the other hand, the coefficient of this term in χQ is n_Λ = 1. Hence η = 1 and we have proved

Weyl’s Theorem. Let be the irreducible module for with highest weight Λ. Then the character χ_Λ = Σn_Me(M) of in is given by the formula

where δ = (1/2) Σ_α>0α, α a root.

This theorem means that the expression on the right is divisible by Q = Σ_s (det S)e(δS) in and the quotient is the character χ of the representation.

It is easy to see that this result gives Weyl’s original formula

in the complex case. Weyl employed his result to obtain by a limiting process the dimensionality of = Σn_M = χ_Λ(0). We proceed to obtain the same result by a somewhat similar device.

We introduce the algebra 〈t〉 of formal power series in an indeterminate t with coefficients in . We recall that the mapping of power series into their constant terms is a homomorphism ζ of 〈t〉 onto . We can also define homomorphisms of into 〈t〉 by employing exponentials: exp z = 1 + z +(z²/2!) + … which is defined for any z ∈ 〈t〉 with zero constant term. We have the relation exp (z₁ + z₂) = (exp z₁)(exp z₂); hence, if λ, μ, ρ ∈ *, then exp (λ, ρ)t exp (μ, ρ)t = exp (λ + μ, ρ)t. In particular, this holds for λ = M, μ = M′, integral linear functions on , which implies, in view of (24), that we have a homomorphism ζ_ρ of into 〈t〉 such that . Now consider χ_Λ ζ_ρ ζ. Since χ_Λ = Σn_Me(M) we have χ_Λ ζ _ρ = Σn_Me xp (M, ρ)t and since the constant term of an exponential is 1, χ_Λ ζ _ρζ = Σn _M= dim . We shall obtain the formula for dim by applying ζ_ρζ to (35), taking ρ = δ = (1/2) Σ_α > 0 α.

Let σ = Σ_S∈W (det S)S as before and let M, M′ be integral linear functions. Then

Hence

since δ = (1/2)Σ_{α > 0} α. Applying this to (35) we obtain

Now

where k is the number of positive roots. Hence if we divide both sides of (38) by t^k and then apply the homomorphism ζ which picks out the constant term, we obtain

or

4. Some examples

We now indicate how dim can be calculated from the Dynkin diagram for . Let w_i = 1, 2 or 3 be the weight of the vertex α_i in the diagram and let α_r be one of the roots in the simple system such that w_r = 1. We replace the scalar product (α, β) by (α, β)′ = 2(α, β)/(α_r, α_r). Then we evidently have dim . We write Λ = Σm_iλ_i, m_i 0, α = Σk_iα_i, k_j 0. Then since and δ(h_j) = 1, j = 1, …, l, δ = Σλ_i Hence we require (Σm_i + 1)λ_i, Σk_jα_j)′ = Σ_i,j(m_i + 1)k_j (λ_i, α_j)′ and Σ_i,jkk_j(λ_i, α_j)′ Now

Hence and (δ, α)′ = Σw_ik_i and

where the product is taken over all the sequences (k₁, k₂, …, k_l k_i 0, such that Σk_jα_j is a root. We have seen that this set as well as the w_i can be determined from the Dynkin diagram.

G₂. Here w₁ = 3, w₂ = 1 and the roots are α₁, α₂, α₁ + α₂, α₁ + 2α₂, α₁ + 3α₂, 2α₁ + 3α₂. Then (41) gives

if the highest weight of is m₁λ ₁+ m₂λ₂. For Λ = λ₁ and Λ = λ₂ we obtain, respectively, 14 and 7, which we had obtained previously.

B_l, l 2. Here w₁ = w₂ = … w_l-1 = 2, w_l = 1 and the weights are

These contribute the following factors to the dimensionality formula:

Let Λ = λ_k, 1 k l – 1, so that m_k = 1, m_i = 0 if i ≠ k. The product taken over the first set of factors in (44) is

The second set of factors gives

and the last is

Multiplication of these results gives

We recall that this result was what was required to complete our proof (§ 7.6) of the irreducibility of the module of k-vectors relative to B_l.

5. Applications and further results

We shall now call a character of a finite-dimensional irreducible representation of a primitive character. Such a character has the form , where the summation is taken over M < Λ in the lexicographic ordering in ₀^* or in . Since the e(Λ) constitute a basis for the group algebra of it is clear that χ _Λ1= χ _Λ2 implies Λ₁ = Λ₂ and this implies that the associated representations are equivalent. Conversely, equivalence of finite-dimensional irreducible representations implies equality of the characters. It is clear also from the expression we have indicated for a primitive character that distinct primitive characters are linearly independent. If is any finite-dimensional module for , is a direct sum of, say, m₁ irreducible modules with character χ_Λ1 = m₂ with character χ_Λ2 ≠ χ_Λ1 etc. Then the character χ of has the form

Since this expression is unique we see that if χ and the primitive characters are known then the m_i can be determined. This gives the isomorphism classes and multiplicities of the irreducible constituents of . As a corollary, we see that these classes and multiplicities are independent of the particular decomposition of into irreducible constituents.

The characters can be used to determine the isomorphism classes and multiplicities of the tensor product of two irreducible modules. Suppose (x₁, …, x_m) is a basis for a module such that x_ih = Λ_i((h)x_i and (y₁, …, y_n) is a basis for a module such that y_jh = M_j(h)y_j. Then the mn products x_i ⊗ y_j form a basis for ⊗ and we have

Hence we have the following expressions for the characters of , and ⊗ . Thus we see that the character of ⊗ is the product in of the characters of and . One obtains the structure of ⊗ by writing the product of two characters as a sum of primitive characters. In the case of the Lie algebra A₁ there is a classical formula for this called the Clebsch-Gordon formula. We shall give an extension of this which is due to R. Steinberg. This is based on an explicit formula for the multiplicity n_M of the weight M in the irreducible module with highest weight Λ. This formula is due to Kostant; the simple proof we shall give is due to Cartier and to Steinberg (independently).

We introduce first a partition function P(M) for M ∈ , which is analogous to the (unordered) partition function of number theory. If M ∈ , P(M) is the number of ways of writing M as a sum of positive roots, that is, P(M) is the number of solution (k_α, k_β,…,k_ρ), of Σ_α0k_α& α = M where the k_α are non-negative integers and {α, β, … ρ} is the set of positive roots. We have P(0) = 1 and P(M) = 0 unless M = Σm_iα_i, m_i-non-negative integer and defined as before. Then . It is convenient to replace the group algebra by the field of power series of the form is an infinite series with coefficients in in the elements n_i non-negative integral. In we can consider the “generating function” Σ_M∈ P(M)e(M) which is defined since P(M) = 0 unless M = Σmα_i, m_i 0. It is clear that we have the identity

Since (1 – e(α))^-1 = 1 + e(α) + e(2α) + …, we have the identity

We re-write Weyl’s formula (35) as

We replace M by — M in this and multiply the result through by e(δ) to obtain

By Lemma 4,

Hence

Hence if we multiply both sides of (48) by , we obtain, by (47), that

Comparison of coefficients of e(— M) gives Kostant’s formula for the multiplicity n_M in the irreducible module with highest weight Λ, namely,

We now consider the formula for χ_Λ1χ_Λ2 where is the character of the irreducible module _i with highest weight Λ_i. We have seen that χ_Λ1χ _Λ2= Σm_Λχ_Λ, where m_Λ is the multiplicity in ₁ ⊗ ₂ of the irreducible module with highest weight Λ. The summation is taken over all the dominant Λ. If we multiply through by Σ_S∈W (det S)e(δS) and apply Wey’s formula we get

The summation on the right-hand side can be taken for all if we define m_Λ = 0 for non-dominant Λ. Applying the formula (50) for we get

Hence

If we put on the left we get

Hence

It is easy to see that if Λ is dominant, then (Λ + δ)S and hence (Λ + δ)S - δ is not dominant if S ≠ 1. Hence if Λ is dominant then m_{(Λ+ δ})_{S- δ} = 0 if S ≠ 1 and so we obtain the formula

for the multiplicity of the module with highest weight Λ in ₁ ⊗ ₂

Exercises

The notations and conventions are as in Chapters VII and VIII.

1. Let ρ ∈ ₀^*. Give a direct proof that ρ ρS for every S ∈ W if and only if ρ(h_i) 0, i = 1, 2, …, l.

2. (Seligman). Prove the uniqueness assertion in Theorem 2 by using the fact that there exists a finite-dimensional irreducible module with highest weight Λ satisfying: Λ(h_i) are distinct and positive. Note that ΛS = Λ if πS = S, so that . This leads to Λ(h_i) = Λ(h_j) if α_iS = α_j.

3. Call a weight Λ in a frontier weight if for every root α ≠ 0 either Λ + α or Λ - α is not a weight. Show that if is finite-dimensional irreducible then any two frontier weights are conjugate under the Weyl group.

4. Let the base field be the field of real numbers. If a is a root let P_α be the hyperplane in defined by α(h) = 0. A chamber is defined to be a connected component (maximal connected subset) of the complement of ∪_α≠0P_α, in . Show that every chamber C is a convex set. A set of roots Σ is a defining system for C if C is the set of elements h satisfying α(h) > 0 for all α ∈ Σ. Defining systems which are minimal are called fundamental systems. Show that these are just simple systems of roots determined by the lexicographic orderings in *.

5. Show that the group algebra (of the group of integral functions on ) is a domain with unique factorization (into elements).

6. Prove that if P ∈ is alternating, then P is divisible by Q = Σ_S(det S)e(δS).

7. Let η be the automorphism of such that e(Λ)^η = e(— Λ). Show that if χ is the character of a finite-dimensional module then χ^η is the character of the contragredient module *.

8. Let be a finite-dimensional irreducible module whose character satisfies χ^η = χ. Assume the base field algebraically closed. Show that the image ^R under the representation R in is a subalgebra of an orthogonal or a symplectic Lie algebra of linear transformations in .

9. Let be the split three-dimensional simple Lie algebra with canonical basis e, f, h. Show that the character of the (m + 1)-dimensional irreducible module _m+1 for is x^m + x^m-1 + … + x^m where x = e(λ), λ(h) = 1. Use this to obtain the irreducible constituents of _m-1 ⊗m_n+1.

10 (Dynkin). Let and ⊗ be finite-dimensional irreducible modules for . Prove that ⊗ is irreducible if and only if for every l in any simple ideal of either l = 0 or l = 0.

11. Use Weyl’s formula to show that the dimensionalities of the four basic irreducible modules for F₄ are: 26, 52, 273, 1274. Use Freudenthal’s formula to obtain the character of the 26-dimensional basic module.

12. Use Weyl’s formula to prove dim = 2^l if has highest weight λ_l for B_l.

13 (Steinberg). Let be the finite-dimensional irreducible module with highest weight Λ. Show that M is a weight of if Λ — MS is a sum of positive roots for every S ∈ W.

14 (Kostant). Prove the following recursion formula for the partition function P(M):