Kostant's Weight Multiplicity Formula and the Fibonacci and Lucas Numbers
aa r X i v : . [ m a t h . R T ] N ov KOSTANT’S WEIGHT MULTIPLICITY FORMULA AND THEFIBONACCI NUMBERS
PAMELA E. HARRIS
Marquette UniversityMSCS DepartmentP.O. Box 1881, Milwaukee WI [email protected]
Abstract.
It is well known that the dimension of a weight space for a finite dimensionalrepresentation of a simple Lie algebra is given by Kostant’s weight multiplicity formula. Weaddress the question of how many terms in the alternation contribute to the multiplicity ofthe zero weight for certain, very special, highest weights. Specifically, we consider the casewhere the highest weight is equal to the sum of all simple roots. This weight is dominantonly in Lie types A and B . We prove that in all such cases the number of contributing termsis a Fibonacci number. Combinatorial consequences of this fact are provided. Introduction
Let G be a simple linear algebraic group over C , T a maximal algebraic torus in G ofdimension r , and B , T ⊆ B ⊆ G , a choice of Borel subgroup. Then let g , h , and b denotethe Lie algebras of G , T , and B respectively. We let Φ denote the set of roots correspondingto ( g , h ), and Φ + ⊆ Φ is the set of positive roots with respect to b . Let ∆ ⊆ Φ + be the setof simple roots. The denote the set of integral and dominant integral weights by P ( g ) and P + ( g ) respectively. Let W = N orm G ( T ) /T denote the Weyl group corresponding to G and T , and for any w ∈ W , we let ℓ ( w ) denote the length of w .A finite dimensional complex irreducible representation of g is equivalent to a highestweight representation with dominant integral highest weight λ . We denote such a repre-sentation by L ( λ ). To find the multiplicity of a weight µ in L ( λ ), we use Kostant’s weightmultiplicity formula, [5]: m ( λ, µ ) = X σ ∈ W ( − ℓ ( σ ) ℘ ( σ ( λ + ρ ) − ( µ + ρ )) , (1.1)where ℘ denotes Kostant’s partition function and ρ = P α ∈ Φ + α . Recall that Kostant’spartition function is the non-negative integer valued function, ℘ , defined on h ∗ , by ℘ ( ξ ) =number of ways ξ may be written as a non-negative integral sum of positive roots, for ξ ∈ h ∗ .With the aim of describing the contributing terms of (1.1) we introduce the following. Definition 1.1.
For λ, µ dominant integral weights of g define the Weyl alternation set tobe A ( λ, µ ) = { σ ∈ W | ℘ ( σ ( λ + ρ ) − ( µ + ρ )) > } . Date : November 7, 2018. herefore, σ ∈ A ( λ, µ ) if and only if σ ( λ + ρ ) − ( µ + ρ ) can be written as a nonnegativeintegral combination of positive roots.In [3], we considered the adjoint representation of sl r +1 = sl r +1 ( C ), whose highest weightis the highest root, which is defined as the sum of the simple roots. In this case we proved: Theorem. If r ≥ and ˜ α is the highest root of sl r +1 , then |A ( ˜ α, | = F r , where F r denotesthe r th Fibonacci number.
In this note we specialize to so r +1 = so r +1 ( C ) and, in Section 2, prove: Theorem 1.1. If r ≥ and ̟ = P α ∈ ∆ α is a fundamental weight of so r +1 , then |A ( ̟ , | = F r +1 , where F r +1 denotes the ( r + 1) th Fibonacci number.
This result gives rise to some combinatorial identities associated to a Cartan subalgebraof so r +1 , which we present in Section 3. The non-zero weights, µ , of so r +1 are consideredin Section 4 from the same point of view.In Sections 5 and 6, we prove that the weight defined by the sum of the simple roots isnot a dominant integral weight of the Lie algebras sp r ( C ) (for r ≥ so r ( C ) (for r ≥ G , F , E , E , and E , and hence does not correspond to finite-dimensional representation.2. The zero weight space of so r +1 Let r ≥
2, and let G = SO r +1 ( C ) = { g ∈ SL r +1 ( C ) : g t = g − } be the special orthogonalgroup of (2 r + 1) × (2 r + 1) matrices over C . Let g = so r +1 ( C ) = { X ∈ M r +1 ( C ) : X t = − X } , and h = { diag [ a , . . . , a r , , − a r , . . . , − a ] | a , . . . , a r ∈ C } be a fixed choice of Cartansubalgebra. For 1 ≤ i ≤ r , define the linear functionals ε i : h → C by ε i ( H ) = a i , for any H = diag [ a , . . . , a r , , − a r , . . . , − a ] ∈ h .For each 1 ≤ i ≤ r −
1, let α i = ε i − ε i +1 and let α r = ε r . Then the set of simple andpositive roots are ∆ = { α , . . . , α r } and Φ + = { ε i − ε j , ε i + ε j : 1 ≤ i < j ≤ n } ∪ { ε i :1 ≤ i ≤ r } , respectively. The fundamental weights are defined by ̟ i = ε + · · · + ε i for1 ≤ i ≤ r −
1, and ̟ r = ( ε + ε + · · · + ε r ). Then observe that ̟ = ε = α + · · · + α r ,and ρ = ( r − ) ε + ( r − ) ε + · · · + ε r − + ε r . Let ( , ) denote the symmetric bilinear formon h ∗ corresponding to the trace form as in [2].For 1 ≤ i ≤ r − s α i ( ε k ) = ε σ ( k ) , where σ is the transposition ( i i + 1) ∈ S r , and s α r ( ε k ) = ε k provided k = r and s α r ( ε r ) = − ε r . For any 1 ≤ i ≤ r , let s i := s α i . Then theWeyl group, W , acts on h ∗ by signed permutations of ε , ε , . . . , ε r and is generated by thesimple root reflections s , . . . , s r . Proposition 2.1.
Let σ = s i s i · · · s i k ∈ W , for some nonconsecutive integers i , i , . . . , i k between and r . Then σ ( ̟ + ρ ) − ρ = ̟ − P j = kj =1 α i j is a nonnegative integral combinationof positive roots.Proof. Recall that for any 1 ≤ i, j ≤ r , s i ( ̟ j ) = ̟ j − δ ij α j , where δ ij = ( i = j s i ( α j ) = α j − α j ,α i )( α i ,α i ) α i , for any 1 ≤ i ≤ r . Observe that ( α i , α j ) = 0, whenever diag [ a , . . . , a n ] denotes a diagonal n × n matrix with entries a , . . . , a n . , j are nonconsecutive integers between 1 and r . Thus, given σ = s i s i · · · s i k ∈ W , with i , i , . . . , i k nonconsecutive integers between 2 and r , we have that σ ( ̟ + ρ ) − ρ = s i s i · · · s i k ( ̟ + ( ̟ + · · · + ̟ r )) − ρ = 2 ̟ + ̟ + · · · + ̟ r − ( α i + · · · + α i k ) − ρ = ̟ − j = k X j =1 α i j . Now since ̟ = α + · · · + α r , notice ̟ − P j = kj =1 α i j is a nonnegative integral combinationof positive roots. (cid:3) Theorem 2.1.
Let σ ∈ W . Then σ ∈ A ( ̟ , if and only if σ = s i s i · · · s i k , for somenonconsecutive integers i , . . . , i k between and r .Proof. ( ⇒ ) Let σ ∈ A ( ̟ , ℓ ( σ ). If ℓ ( σ ) = 0, then σ = 1. If σ = s i , for some 1 ≤ i ≤ r , then s i ( ̟ + ρ ) − ρ = s i (2 ̟ + ̟ + · · · + ̟ r ) − ( ̟ + · · · + ̟ r )= ( − α + α + · · · + α r if i = 1 α + · · · + α i − + α i +1 + · · · + α r otherwise . If i = 1, then we get a contradiction to σ ∈ A ( ̟ , σ = s i , where 2 ≤ i ≤ r . If ℓ ( σ ) = 2, then σ = s i s j for some distinctintegers 1 ≤ i, j ≤ r . Clearly i, j = 1, else σ / ∈ A ( ̟ , s i s j ( ̟ i + ρ ) − ρ = ̟ + ρ − α i − ( α j − α i , α j )( α , α i ) α i ) − ρ = ̟ − α i − α j + 2( α i , α j )( α i , α i ) α i . If i and j are consecutive integers between 2 and r , then j = i ± α i , α j ) = −
1. Hence s i s j / ∈ A ( ̟ , i and j are nonconsecutiveintegers between 2 and r .Now assume that for any σ ∈ A ( ̟ ,
0) with ℓ ( σ ) := n ≤ k , there exist nonconsecutiveintegers i , . . . , i n between 2 and r such that σ = s i s i · · · s i n . Let σ ∈ A ( ̟ , ℓ ( σ ) = k + 1. Hence there exist distinct integers i , i , . . . , i k +1 between 1 and r such that σ = s i s i · · · s i k +1 . Let σ = s i π , where π = s i · · · s i k +1 , with ℓ ( π ) = k . Since σ ∈ A ( ̟ , π ∈ A ( ̟ , i , . . . , i k +1 are nonconsecutiveintegers between 2 and r . By Proposition 2.1, π ( ̟ + ρ ) = ̟ + ρ − P j = k +1 j =2 α i j , and hence σ ( ̟ + ρ ) − ρ = s i π ( ̟ + ρ ) − ρ = s i ( ̟ + ρ − j = k +1 X j =2 α i j ) − ρ = ̟ + ρ − α i − j = k +1 X j =2 ( α i j − α i j , α i )( α i , α i ) α i ) − ρ = ̟ − j = k +1 X j =1 α i j + 2 j = k +1 X j =2 ( α i j , α i )( α i , α i ) α i . Observe that if i = 1, then σ ( ̟ + ρ ) − ρ / ∈ A ( ̟ , ≤ j ≤ r such that i j and i are consecutive integers. Then ( α i j , α i ) = − nd hence the coefficient of α i is negative. Thus σ / ∈ A ( ̟ , i , i , . . . , i k +1 are nonconsecutive integers between 2 and r .( ⇐ ) By Proposition 2.1, if σ = s i s i · · · s i k ∈ W , for some nonconsecutive integers i , . . . , i k between 2 and r , then σ ( ̟ + ρ ) − ρ is a nonnegative integral combination ofpositive roots and hence σ ∈ A ( ̟ , (cid:3) Definition 2.1.
The Fibonacci numbers are defined, in [7] , by the recurrence relation F = F = 1 , and F n = F n − + F n − , for n ≥ . We now give the following:
Proof of Theorem 1.1.
By Theorem 2.1, we know A ( ̟ ,
0) = { σ ∈ W | σ = s i · · · s i k , for some nonconsecutive integers 2 ≤ i , . . . , i k ≤ r } . An induction argument shows that forany n ≥
1, the number of sequences consisting of nonconsecutive integers between 1 and n is given by F n +2 . Thus |A ( ̟ , | = F r +1 . (cid:3) A q -analog The q -analog of Kostant’s partition function is the polynomial valued function, ℘ q , definedon h ∗ by ℘ q ( ξ ) = c + c q + · · · + c k q k , where c j = number of ways to write ξ as a non-negativeintegral sum of exactly j positive roots, for ξ ∈ h ∗ . The q -analog of Kostant’s weightmultiplicity formula is defined, in [6], as: m q ( λ, µ ) = P σ ∈ W ( − ℓ ( σ ) ℘ q ( σ ( λ + ρ ) − ( µ + ρ )).It is known that the multiplicity of the zero weight in the representation L ( ̟ ) is equal to1, see [1]. In this section, we give a combinatorial proof of this fact, by proving the following. Theorem 3.1.
Let r ≥ and let ̟ = P α ∈ ∆ α be a fundamental weight of so r +1 . Then m q ( ̟ ,
0) = q r . Observe that the subset of positive roots of so r +1 used to write σ ( ̟ + ρ ) − ρ , for any σ ∈ A ( ̟ , sl r +1 . Therefore, the following lemmasand propositions follow from Lemma 3.1 and Proposition 3.2 in [3]. Lemma 3.1.
The cardinality of the sets { σ ∈ A ( ̟ ,
0) : ℓ ( σ ) = k and σ contains no s r factor } and { σ ∈ A ( ̟ ,
0) : ℓ ( σ ) = k and σ contains an s r factor } are (cid:0) r − − kk (cid:1) and (cid:0) r − − kk (cid:1) , respec-tively. Also max { ℓ ( σ ) : σ ∈ A ( ̟ , and σ contains no s r factor } = ⌊ r − ⌋ and max { ℓ ( σ ) : σ ∈ A ( ̟ , and σ contains an s r factor } = ⌊ r − ⌋ . Proposition 3.1.
Let σ ∈ A ( ̟ , . Then ℘ q ( σ ( ̟ + ρ ) − ρ ) = ( q ℓ ( σ ) (1 + q ) r − − ℓ ( σ ) if σ contains no s r factor q ℓ ( σ ) (1 + q ) r − − ℓ ( σ ) if σ contains an s r factor . Now can now prove the closed formula for the q -multiplicity of the zero weight in L ( ̟ ). Proof of Theorem 3.1.
Observe that m q ( ̟ ,
0) = X σ ∈A ( ̟ , s r factor ( − ℓ ( σ ) ℘ q ( σ ( ̟ + ρ ) − ρ )+ X σ ∈A ( ̟ , s r factor ( − ℓ ( σ ) ℘ q ( σ ( ̟ + ρ ) − ρ ) . y Lemma 3.1, Proposition 3.1 and Proposition 3.3 in [3] it follows that X σ ∈A ( ̟ , s r factor ( − ℓ ( σ ) ℘ q ( σ ( ̟ + ρ ) − ρ ) = ⌊ r − ⌋ X k =0 ( − k (cid:18) r − − kk (cid:19) q k (1 + q ) r − − k = r X i =1 q i , and X σ ∈A ( ̟ , s r factor ( − ℓ ( σ ) ℘ q ( σ ( ̟ + ρ ) − ρ ) = ⌊ r − ⌋ X k =0 ( − k (cid:18) r − − kk (cid:19) q k (1 + q ) r − − k = − r − X i =1 q i . Therefore, m q ( ̟ ,
0) = ( q + q + · · · + q r − + q r ) − ( q + q + · · · + q r − ) = q r . (cid:3) Corollary 3.1.
Let r ≥ and let ̟ = P α ∈ ∆ α be a fundamental weight of so r +1 . Then m ( ̟ ,
0) = 1 .Proof.
Follows directly from Theorem 3.1, since m ( ̟ ,
0) = m q ( ̟ , | q =1 = 1. (cid:3) Non-zero weight spaces of so r +1 We now consider the non-zero dominant weights, µ , of so r +1 and compute the Weylalternation sets A ( ̟ , µ ). Throughout this section r ≥ . Theorem 4.1. If µ ∈ P + ( so r +1 ) and µ = 0 , then A ( ̟ , µ ) = ( { } if µ = ̟ ∅ otherwise. We begin by proving the following Propositions, from which Theorem 4.1 follows.
Proposition 4.1. If ̟ = P α ∈ ∆ α is a fundamental weight of so r +1 , then A ( ̟ , ̟ ) = { } .Proof. Since ̟ = α + · · · + α r , notice σ ( ̟ + ρ ) − ρ − ̟ is a non-negative integral sum ofpositive roots only if σ ( ̟ + ρ ) − ρ is. By Theorem 2.1 we know σ ( ̟ + ρ ) − ρ is a non-negativeintegral sum of positive roots if and only if σ = s i s i · · · s i k , for some nonconsecutive integers i , . . . , i k between 2 and r . Hence A ( ̟ , ̟ ) ⊂ A ( ̟ , σ ∈ A ( ̟ , ̟ ) with ℓ ( σ ) = k ≥
1, then there exist nonconsecutive integers i , . . . , i k between 2 and r such that σ = s i s i · · · s i k . By Proposition 2.1 we have that σ ( ̟ + ρ ) − ρ = ̟ − P kj =1 α i j . Thennotice σ ( ̟ + ρ ) − ρ − ̟ will not be a non-negative integral sum of positive roots, reachinga contradiction. Thus ℓ ( σ ) = 0 and σ = 1. (cid:3) Proposition 4.2.
Let µ ∈ P + ( so r +1 ) , and µ = 0 . Then there exists σ ∈ W such that ℘ ( σ ( ̟ + ρ ) − ρ − µ ) > if and only if µ = ̟ .Proof. ( ⇒ ) Let µ ∈ P + ( so r +1 ) with µ = 0, and assume σ ∈ W such that ℘ ( σ ( ̟ + ρ ) − ρ − µ ) >
0. By Proposition 3.1.20 in [2], we know that P + ( so r +1 ) consists of all weights µ = k ε + k ε + · · · + k r ε r , with k ≥ k ≥ · · · ≥ k r ≥
0. Here 2 k i , and k i − k j are integersfor all i, j . ow observe that σ ( ̟ + ρ ) − ρ − µ = σ (( r + ) ε + ( r − ) ε + ( r − ) ε + · · · + ε r − + ε r ) − (( r − ) ε + ( r − ) ε + · · · + ε r ) − ( k ε + · · · + k r ε r ). Let a i denote the coefficientof α i in σ ( ̟ + ρ ) − ρ − µ . Then a = − i + 1 − k if σ ( ε ) = ε i for 2 ≤ i ≤ r − r + i − k if σ ( ε ) = − ε i for 2 ≤ i ≤ r − k if σ ( ε ) = ε − r − k if σ ( ε ) = − ε .Since r ≥ a ∈ N , we have that σ ( ε ) = ε and a = 1 − k . If k = 0 , then k i = 0for all 1 ≤ i ≤ r , and so µ = 0, a contradiction. Hence k = 1. Since k i − k j ∈ Z for all i and j , and since 1 = k ≥ k ≥ k ≥ · · · ≥ k r ≥
0, we have that k i = 0 or 1, for all 2 ≤ i ≤ r . Wewant to show that k i = 0 for all 2 ≤ i ≤ r . It suffices to show k = 0. A simple computationshows that a = − i + 2 − k if σ ( ε ) = ε i for 3 ≤ i ≤ r − r + i + 1 − k if σ ( ε ) = − ε i for 3 ≤ i ≤ r − k if σ ( ε ) = ε − r + 3 − k if σ ( ε ) = − ε . Since r ≥ a ∈ N , we have that σ ( ε ) = ε , and hence k = 0. Thus µ = ε = ̟ .( ⇐ ) By Proposition 4.1, we know if µ = ̟ , then ℘ ( σ ( ̟ + ρ ) − ρ − ̟ ) > σ = 1. (cid:3) Theorem 4.2. If µ ∈ P ( so r +1 ) , then m ( ̟ , µ ) = ( if µ = 0 or µ ∈ W · ̟ ∅ otherwise.Proof. Recall that given µ ∈ P ( so r +1 ), there exists w ∈ W and ξ ∈ P + ( so r +1 ) such that w ( ξ ) = µ and also recall that weight multiplicities are invariant under W (Propositions3.1.20, 3.2.27 in [2]). Thus it suffices to consider µ ∈ P + ( so r +1 ). Corollary 3.1 gives m ( ˜ α,
0) = 1, while Theorem 4.1 implies m ( ̟ , ̟ ) = 1 and m ( ̟ , µ ) = 0, whenever µ ∈ P + ( so r +1 ) \ { , ̟ } . (cid:3) The classical Lie algebras sp r and so r In this section we consider the classical Lie algebras sp r ( C ) and so r ( C ) and prove. Theorem 5.1. If g is the classical Lie algebra sp r ( C ) (with r ≥ ) or so r ( C ) (with r ≥ )and ∆ denotes a set of simple roots, then the weight P α ∈ ∆ α is not a dominant integralweight of g .Proof. We follow the notation in [2]. In the case of sp r = sp r ( C ), with r ≥
3, for each1 ≤ i ≤ r −
1, let α i = ε i − ε i +1 and let α r = 2 ε r . Then the set of simple roots is∆ = { α , . . . , α r } . The fundamental weights are defined by ̟ i = ε + · · · + ε i , for 1 ≤ i ≤ r .Now notice α + · · · + α r = ε + ε r = ̟ − ̟ r − + ̟ r . Thus, the weight defined by the sumof the simple roots is not a dominant weight of sp r , for r ≥ so r = so r ( C ), with r ≥
4, for each 1 ≤ i ≤ r −
1, let α i = ε i − ε i +1 and let α r = ε r − + ε r . Then the set of simple roots is ∆ = { α , . . . , α r } . The fundamental weightsare defined by ̟ i = ε + · · · + ε i if 1 ≤ i ≤ r −
1, and ̟ r − = ( ε + · · · + ε r − − ε r ) and ̟ r − = ( ε + · · · + ε r − + ε r ). Now notice α + · · · + α r = ε + ε r − = ̟ − ̟ r − + ̟ r − + ̟ r . N = { , , , , . . . } . hus, the weight defined by the sum of the simple roots is not a dominant weight of so r ,for r ≥ (cid:3) Exceptional Lie algebras
In this section we consider the exceptional simple Lie algebras over C and prove. Theorem 6.1. If g is an exceptional simple Lie algebra of type G , F , E , E , or E and ∆ denotes a set of simple roots, then the weight P α ∈ ∆ α is not a dominant integral weightof g .Proof. In each case we will describe, as in [4], the underlying vector space V and the rootsystem Φ as a subset of V . In each case the root system will be a subspace of some R k = { P ki =1 a i e i } , where { e i : 1 ≤ i ≤ k } is the standard orthonormal basis and the a i ’s are realnumbers.The underlying vector space of the exceptional Lie algebra G is V = { v ∈ R | ( v, e + e + e ) = 0 } , and the root system is given byΦ = {± ( e − e ) , ± ( e − e ) , ± ( e − e ) }∪{± (2 e − e − e ) , ± (2 e − e − e ) , ± (2 e − e − e )) } .The set of simple roots is ∆ = { α , α } , where α = e − e and α = − e + e + e , and thefundamental weights, in terms of the simple roots, are ̟ = 2 α + α and ̟ = 3 α + α .Observe that α + α = − ̟ + ̟ , and hence not a dominant integral weight of G .The underlying vector space of the exceptional Lie algebra F is V = R , and the rootsystem is given by Φ = {± e i ± e j | i < j } ∪ {± e i } ∪ { ( ± e ± e ± e ± e ) } . The set of simpleroots is ∆ = { α , α , α , α } , where α = ( e − e − e − e ), α = e , α = e − e , and α = e − e . The fundamental weights, in terms of the simple roots, are ̟ = 2 α + 3 α + 2 α + α ̟ = 3 α + 6 α + 4 α + 2 α ̟ = 4 α + 8 α + 6 α + 3 α ̟ = 2 α + 4 α + 3 α + 2 α Observe that α + α + α + α = ̟ − ̟ + ̟ , and hence not a dominant integral weightof F .The underlying vector space of the exceptional Lie algebra E is V = { v ∈ R | ( v, e − e > = < v, e + e ) = 0 } , and the root system is given by Φ = {± e i ± e j | i < j ≤ } ∪{ P i =1 ( − n ( i ) e i ∈ V | P i =1 n ( i ) even } . The set of simple roots is ∆ = { α , α , α , α , α , α } ,where α = ( e − e − e − e − e − e − e + e ), α = e + e , α = e − e , α = e − e , = e − e , and α = e − e . The fundamental weights, in terms of the simple roots, are ̟ = 13 (4 α + 3 α + 5 α + 6 α + 4 α + 2 α ) ̟ = α + 2 α + 2 α + 3 α + 2 α + α ̟ = 13 (5 α + 6 α + 10 α + 12 α + 8 α + 4 α ) ̟ = 2 α + 3 α + 4 α + 6 α + 4 α + 2 α ̟ = 13 (4 α + 6 α + 8 α + 12 α + 10 α + 5 α ) ̟ = 13 (2 α + 3 α + 4 α + 6 α + 5 α + 4 α )Observe that α + α + α + α + α + α = ̟ + ̟ − ̟ + ̟ , and hence not a dominantintegral weight of E .The underlying vector space of the exceptional Lie algebra E is V = { v ∈ R | ( v, e + e ) = 0 } , and the root system is given by Φ = {± e i ± e j | i < j ≤ } ∪ {± ( e − e ) } ∪{ P i =1 ( − n ( i ) e i ∈ V | P i =1 n ( i ) even } . The set of simple roots is ∆ = { α , α , α , α , α , α , α } ,where α = ( e − e − e − e − e − e − e + e ), α = e + e , α = e − e , α = e − e , α = e − e , α = e − e , and α = e − e . The fundamental weights, in terms of thesimple roots, are ̟ = 2 α + 2 α + 3 α + 4 α + 3 α + 2 α + α ̟ = 12 (4 α + 7 α + 8 α + 12 α + 9 α + 6 α + 3 α ) ̟ = 3 α + 4 α + 6 α + 8 α + 6 α + 4 α + 2 α ̟ = 4 α + 6 α + 8 α + 12 α + 9 α + 6 α + 3 α ̟ = 12 (6 α + 9 α + 12 α + 18 α + 15 α + 10 α + 5 α ) ̟ = 2 α + 3 α + 4 α + 6 α + 5 α + 4 α + 2 α ̟ = 12 (2 α + 3 α + 4 α + 6 α + 5 α + 4 α + 3 α )Observe that α + α + α + α + α + α + α = ̟ + ̟ − ̟ + ̟ , and hence not adominant integral weight of E .The underlying vector space of the exceptional Lie algebra E is V = R , and the rootsystem is given by Φ = {± e i ± e j | i < j } ∪ { P i =1 ( − n ( i ) e i | P i =1 n ( i ) even } . The set ofsimple roots is ∆ = { α , α , α , α , α , α , α , α } , where α = ( e − e − e − e − e − e − e + e ), α = e + e , α = e − e , α = e − e , α = e − e , α = e − e , α = e − e , nd α = e − e . The fundamental weights, in terms of the simple roots, are ̟ = 4 α + 5 α + 7 α + 10 α + 8 α + 6 α + 4 α + 2 α ̟ = 5 α + 8 α + 10 α + 15 α + 12 α + 9 α + 6 α + 3 α ̟ = 7 α + 10 α + 14 α + 20 α + 16 α + 12 α + 8 α + 4 α ̟ = 10 α + 15 α + 20 α + 30 α + 24 α + 18 α + 12 α + 6 α ̟ = 8 α + 12 α + 16 α + 24 α + 20 α + 15 α + 10 α + 5 α ̟ = 6 α + 9 α + 12 α + 18 α + 15 α + 12 α + 8 α + 4 α ̟ = 4 α + 6 α + 8 α + 12 α + 10 α + 8 α + 6 α + 3 α ̟ = 2 α + 3 α + 4 α + 6 α + 5 α + 4 α + 3 α + 2 α Observe that α + α + α + α + α + α + α + α = ̟ + ̟ − ̟ + ̟ , and hence nota dominant integral weight of E . (cid:3) References [1] A.D. Berenshtein and A.V. Zelevinskii,
When is the multiplicity of a weight equal to ? , Funct. Anal.Appl. (1991), 259–269. ↑ Symmetry, Representations and Invariants , Springer, New York, 2009.MR2011a:20119 ↑
2, 5, 6[3] P.E. Harris,
On the adjoint representation of sl n and the Fibonacci numbers , C. R. Math. Acad. Sci. Paris (2011), 935-937. ↑
2, 4, 5[4] A. W. Knapp,
Lie groups beyond an introduction , 2nd ed., Progress in Mathematics, vol. 140, Birkh¨auserBoston Inc., Boston, MA, 2002. MR1920389 (2003c:22001) ↑ A formula for the multiplicity of a weight , Proc. Nat. Acad. Sci. U.S.A. (1958),588–589. MR20 ↑ Singularities, character formulas, and a q -analog of weight multiplicities , Ast´erisque (1983), 208–229. MR85m:17005 ↑ Fibonacci’s Liber abaci , Springer-Verlag, New York, 2002. MR2003f:01011 ↑4