On finding the fundamental dominant weights of a root system

  ON FINDING THE FUNDAMENTAL DOMINANT WEIGHTS OF A ROOT SYSTEM By Hendra Gunawan

  Department of Mathematics, Institut Teknologi Bandung Jalan Ganesha 10 Bandung 40132, Indonesia Abstract.

  This note offers a formula which can be used as an alternative method of finding the fundamental dominant weights of a root system, as suggested in [3]. We explain how the formula actually works and verify the fundamental dominant weights of root systems of type A – G.

  Introduction Almost all symbols and terminology we use here are the same as in [4, Ch. III]. Let Φ

  , . . . , α be a root system of rank l. Fix ∆ = {α 1 } to be a base of Φ. Relative to ∆, denote l l

• P by Φ the set of positive roots, whose members are of the form α = k (α)α where the

  j j j=1 P

  1 k α.

  (α)’s are all nonnegative integers. Denote by δ the special element, namely δ = j

  2 α ∈Φ

• Now let {λ

  1 , . . . , λ } be the dual basis, for which hλ , α i = δ , j, k ∈ {1, . . . , l}. Then we l j k jk l

  P know that δ = λ , showing that δ is a dominant weight. j j=1

  Let us fix any j ∈ {1, . . . , l} and put Φ = {α ∈ Φ : k (α) = 0}. Then Φ = j

• Φ ∪ (−Φ ) forms a root system of rank l − 1. Φ is obviously the corresponding set

  P

  1 of positive roots relative to the base ∆ = {α : j 6= j }. Let δ = α and set j

  2

• α ∈Φ

  δ 1 = δ − δ . (Note that δ depends on the choice of j , and so does δ 1 .) One may observe that δ 1 is actually the projection of δ on λ . Moreover, as announced in [3], we have the j following result.

  δ

  j hδ i

1 Theorem. λ = .

  1 ,α j0

  The theorem offers a formula which can be used as an alternative method of finding the fundamental dominant weight λ for any given j ∈ {1, . . . , l}. Using this formula, we can j

  AMS Classification Numbers: 17B10, 17B20 find one particular fundamental dominant weight without worrying about the others. It is —in this case— less labor than employing the Cartan matrix (where we have to invert the matrix). Note, however, that the formula is useful only when we know δ and δ (so we can find δ

  1 ).

  We give here a shorter proof of the theorem. We also explain how to find δ 1 , and , j then compute and verify the fundamental dominant weights λ ∈ {1, . . . , l}, of root j systems of type A – G.

  1. Proof of the Theorem The theorem follows from the lemma below, which we have stated before.

  1 Lemma. δ 1 = hδ, λ iλ . j j

  2

  1 ∗ Proof.

  For each j ∈ {1, . . . , l}, set λ = λ − hλ , λ iλ . We observe that for j, k 6= j , j j j j j

  2

  1 ∗

  , α , α , λ , α , α , hλ i = hλ i − hλ ihλ i = hλ i = δ k j k j j j k j k jk j

  2 ∗ that is, {λ : j 6= j } is the dual basis relative to ∆ (see [3] for more details). From this j l

  P ∗ ∗

  λ and the fact that λ = 0, we have δ = , and accordingly j j j=1

  δ 1 = δ − δ l l

  X X ∗

  λ λ = j − j j=1 j=1 l

  X

  1 = hλ , λ iλ j j j

  2 j=1 l

1 P

  = h λ , λ iλ j j j

  2 j=1

  1 = hδ, λ iλ , j j

  2 proving the lemma. ⊔ ⊓ Now we prove the theorem. Proof of the Theorem.

  Using the above lemma, we have

  1

  1 , α , α hδ

  1 j i = hδ, λ j ihλ j j i = hδ, λ j i.

  2

  2 Hence we obtain that δ , α

  1 = hδ 1 j iλ j or

  δ

  1 λ = , j

  , α hδ 1 j i as stated. ⊔ ⊓

2. Finding δ

  1 We use the following technique to find δ 1 . (In [2], the same technique is used for a different purpose.) Suppose, for example, Φ is a root system of type E 6 , whose Dynkin diagram is as follows.

  2 ◦

  ◦ − − − − ◦ − − − − ◦ − − − − ◦ − − − − ◦

  1

  3

  4

  5

  6 Figure 1. The Dynkin diagram of E

  6 Remove the j -th vertex from the diagram to obtain a new diagram, which in general will be a union of Dynkin diagrams. For example, if we remove the 4th vertex, then we get a diagram for A

  2 ∪ A 1 ∪ A 2 . The resulting diagram is the Dynkin diagram of the root system Φ . Keeping in mind the roots associated with the vertices, we determine the weight δ , using available formulae. We then easily compute δ

  1 = δ − δ . For our example, by removing the 4th vertex we get

  1 1 α + δ = α 2 + α 3 + α 5 + α 6 .

2 We know that

  δ ,

  = 8α 1 + 11α 2 + 15α 3 + 21α 4 + 15α 5 + 8α

  6 and therefore

  21 δ α .

  1 2 + 14α 3 + 21α 4 + 14α 5 + 7α

• 1 = 7α

  6

  2 For root systems of type A – G, we use the (ordered) bases as in [4, pp. 64-65]. With respect to these bases, explicit formulae for the weight δ are obtainable (see, for example, [1]).

  2.1 A l (l ≥ 1). Here we have δ =

  1

  1 =

  1

  2 l P j=1 c j

  (δ 1 )α j with c j

  (δ 1 ) = ½ j(2l − j ), if j < j ; j (2l − j ), if j ≥ j .

  2.3 C l (l ≥ 3). Here we have δ =

  1

  2 l −1 P j=1 j

  (2l − j + 1)α j +

  4 l

  (δ ) =    j

  (l + 1)α l . In general, removing the j -th vertex from the Dynkin diagram will yield a diagram for A j −1

  ∪ C l −j

  . So we get δ =

  1

  2 l P j=1 c j

  (δ )α j with c j

  (δ ) =        j

  (j − j), if j < j ; 0, if j = j ; (j − j )(2l − j − j + 1), if j

  < j < l ;

  1

  (j − j), if j < j ; 0, if j = j ; (j − j )(2l − j − j), if j > j ; and hence we obtain δ

  (δ )α j with c j

  1

  Subtracting δ from δ gives δ 1 =

  2 l P j=1 j

  (l − j + 1)α j . In general, removing the j -th vertex from the Dynkin diagram will yield a diagram for A j −1

  ∪ A l −j

  . Accordingly we get δ =

  1

  2 l P j=1 c j

  (δ )α j with c j

  (δ ) =    j

  (j − j), if j < j ; 0, if j = j ; (j − j )(l − j + 1), if j > j .

  1

  2 l P j=1 c j

  2 l P j=1 c j (δ 1 )α j with c j

  (δ 1 ) = ½ j(l − j + 1), if j < j ; j (l − j + 1), if j ≥ j .

  2.2 B l

  (l ≥ 2). Here δ =

  1

  2 l P j=1 j (2l − j)α j

  . In general, removing the j -th vertex from the Dynkin diagram will give a diagram for A j −1

  ∪ B l −j

  . Thus we get δ =

  1

  2 (l − j )(l − j + 1), if j = l. l P

  1 c

  Therefore, δ 1 = j (δ 1 )α j with

  2 j=1  j (2l − j + 1), if j < j ;  j c (δ 1 ) = (2l − j + 1), if j ≤ j < l; j

  1  j

  (2l − j + 1), if j = l.

  2 l −2 P

  1

  1

2.4 D (l ≥ 4). Here δ = (l − 1)(α + j (2l − j − 1)α l + α ). Removing the j -th

  l j l −1 l

  2

  4 j=1 vertex will give a diagram for A ∪ D when j ≤ l − 2, or A when j = l − 1 or l. j −1 l −j l −1 l

  P

  1 Thus we have δ = c (δ )α with j j 2 j=1

   j (j − j), if j < j ;

    

  0, if j = j ; c (δ ) = j

  < j < l (j − j )(2l − j − j − 1), if j − 1;

   

  1 

  (l − j )(l − j − 1), if j ≥ l − 1,

  2 when j ≤ l − 2, or

   j (l − j), if j < l − 1;

   c (δ ) = 0, if j = l − 1; j

   l − 1, if j = l, when j = l − 1, or

  ½ j(l − j), if j < l; c (δ ) = j 0, if j = l. l

  P

  1 c when j = l. Calculating δ − δ , we get δ

  1 = (δ 1 )α with j j 2 j=1

   j (2l − j − 1), if j < j ;

   j c (2l − j − 1), if j ≤ j < l − 1; j (δ 1 ) =

  1  j (2l − j − 1), if j ≥ l − 1,

  2 when j ≤ l − 2, or

   j (l − 1), if j < l − 1;

  

  1 l c (δ ) = (l − 1), if j = l − 1; j

  1

  2

  1 

  (l − 1)(l − 2), if j = l,

  2 when j = l − 1, or

   j (l − 1), if j < l − 1; 

  1 c (δ 1 ) = (l − 1)(l − 2), if j = l − 1; j

  2

  1  l (l − 1), if j = l,

  2 when j = l. For convenience, we abbreviate l

  P j=1 k j

  25,

  18,

  2 ,

  45

  18, 27,

  2 ,

  27

  2 , 26, 13) 7 (8, 11, 15, 21, 15, 8, 0) (9,

  65

  2 , 26, 39,

  39

  2 ) (13,

  1

  2 ) 6 (4, 5, 7, 9, 5, 0,

  25

  2 ,

  2 )

  75

  30, 45,

  2 ,

  45

  2 ) (16, 24, 32, 48, 36, 24, 12) 5 (2, 2, 3, 3, 0, 1, 1) (15,

  3

  2 , 2,

  3

  2 , 1, 0,

  1

  22, 33, 44, 33, 22, 11) 4 (1,

  2 ,

  33

  2 ) (

  27

  In type E

  4,

  7

  2 ,

  91

  3, 0, 5, 6, 6, 5, 3) (

  2 ,

  1

  2 ) 3 (

  51

  2 , 51,

  153

  2 , 102,

  255

  2 , 68, 85,

  85

  2 ) (

  2 , 6,

  8 , we have δ = (46, 68, 91, 135, 110, 84, 57, 29). By removing the j -th vertex from the Dynkin diagram we obtain the following list. j δ δ

  2 ,

  1 1 (0,

  21

  2 ,

  21

  2 ,

  20, 18, 15, 11, 6) (46, 115

  161

  15

  2 ,

  115, 92, 69, 46, 23) 2 (

  7

  2 , 0, 6,

  15

  2 , 8,

  5

  2 ,

  α j by (k

  2 ,

  1

  14, 21, 14, 7) 5 (2, 2, 3, 3, 0,

  2 ,

  21

  1, 0, 1, 1) (7,

  2 ,

  1

  2 , 9, 15, 18, 12, 6) 4 (1,

  15

  2 , 2, 0, 3, 3, 2) (

  1

  2 ) 3 (

  11

  11,

  33

  15

  11, 11,

  2 ,

  11

  2 ) (

  5

  4,

  2 ,

  9

  0, 4,

  2 ,

  5

  1 1 (0, 5, 5, 9, 7, 4) (8, 6, 10, 12, 8, 4) 2 (

  2.5 E 6 , E 7 , and E 8 . In type E 6 , we have δ = (8, 11, 15, 21, 15, 8), as mentioned before. Below are the δ ’s and δ 1 ’s obtained by removing the j -th vertex from the Dynkin diagram. j δ δ

  1 , . . . , k l ).

  2 ) (6, 9, 12, 18, 15,

  2 ) 6 (4, 5, 7, 9, 5, 0) (4, 6, 8, 12, 10, 8)

  9

  2 ,

  0, 4,

  2 ,

  5

  2 ,

  1

  2 ) 3 (

  21

  2 , 21,

  63

  2 , 28, 42,

  49

  2 ) 2 (3, 0, 5, 6, 6, 5, 3) (14,

  17

  17,

  51

  In type E 7 , the special element is δ = (17,

  1 1 (0,

  49

  2 , 33, 48,

  75

  2 , 26,

  27

  2 ). The δ ’s and δ 1 ’s obtained by removing the j -th vertex from the Dynkin diagram are listed below. j δ δ

  15

  34,

  2 ,

  15

  2 ,

  14, 12, 9, 5) (17, 17,

  51

  2 ,

  65, 91, 130, 104, 78, 52, 26)

  4 (1,

2.6 F 4 . Here we have δ = (8, 15, 21, 11). Removing the j -th vertex from the Dynkin diagram will give the following result.

  2 ). When the 2nd vertex is removed, we get δ = (

  . Knowing the Cartan integers hα i

  1 , α j i to get λ j

  1 , we only need to divide it by hδ

  j Having found δ

  2 , 3).

  9

  0) and δ 1 = (

  2 ,

  1

  5

  1 , α j i.

  1 = (5,

  2 ) and δ

  1

  2.7 G 2 . The special element is δ = (5, 3). Removing either vertex from the Dynkin diagram will yield a diagram for A 1 . When the 1st vertex is removed, we get δ = (0,

  11)

  2 ,

  33

  11,

  2 ,

  , α j i makes it easy to calculate hδ

  Below are the values of hδ

  0) (

  (l ≥ 3) : hδ

  − 1, j = l − 1, l.

  2 (2l − j − 1), j = 1, . . . , l − 2; l

  1

  ½

  (l ≥ 4) : hδ 1 , α j i =

  = 1, . . . , l. D l

  2 (2l − j + 1), j

  1

  1 , α j i =

  = 1, . . . , l − 1; l, j = l. C l

  1 , α j i for root systems of type A – G.

  2 (2l − j ), j

  1

  ½

  (l ≥ 2) : hδ 1 , α j i =

  B l

  2 (l + 1), j = 1, . . . , l.

  1

  (l ≥ 1) : hδ 1 , α j i =

  For type A – D, we have the following result. A l

  11

  2 ,

  1

  55,

  2 , 33, 48,

  49

  2 ) 8 (17,

  57

  2 ) (38, 57, 76, 114, 95, 76, 57,

  1

  7 (8, 11, 15, 21, 15, 8, 0,

  2 ) 6 (4, 5, 7, 9, 5, 0, 1, 1) (42, 63, 84, 126, 105, 84, 56, 28)

  55

  2 ,

  2 , 26,

  165

  2 ) (44, 66, 88, 132, 110,

  3

  2,

  2 ,

  3

  2 , 90, 135, 108, 81, 54, 27) 5 (2, 2, 3, 3, 0,

  135

  2 , 1, 0, 2, 3, 3, 2) (45,

  75

  27

  9

  15

  4,

  2 ,

  5

  2 ) 4 (

  21

  2 ) (7, 14, 21,

  1

  15, 20, 10) 3 (1, 1, 0,

  2 ,

  0, 1, 1) (

  2 , 0) (29,

  2 ,

  1

  1 1 (0, 3, 5, 3) (8, 12, 16, 8) 2 (

  j δ δ

  2 , 29)

  87

  2 , 58,

  145

  2 , 58, 87,

  87

3. Computing λ

  For type E – G, we have the following result. (The values of hδ 1 , α i are listed as j j increases from 1 to l.)

  11

  9

  7

9 E

  , , , , 6 : 6, 6.

  2

  2

  2

  2

  17

  11

13 E

  , , , 7 : 7, 4, 5, 9.

  2

  2

  2

  23

  17

  13

  9

  11

  19

29 E 8 : , , , , , 7, , .

  2

  2

  2

  2

  2

  2

  2

  5

  7

11 F , , .

  4 : 4,

  2

  2

  2

  5

  3 G , . 2 :

  2

  2 The above results clearly verify the fundamental dominant weights of root systems of type A – G listed in [4, p. 69].

  References [1] J.-P. Antoine and D. Speiser, Characters of irreducible representations of the simple groups. II. Applications to classical groups , J. Mathematical Physics, 5(1964), 1560-1572.

  [2] M. Cowling and C. Meaney, On a maximal function on compact Lie groups, Trans. Amer. Math. Soc., 315(1989), 811-822. [3] H. Gunawan, A generalization of maximal functions on compact semisimple Lie groups, Pac. J. Math., 156(1992), 119-134.

  [4] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer- Verlag, New York, 1972.