858 Proc. Japan Acad., 43 (1967) [Vol. 43,

188. Representation Ring o f Lie Group F4 By Ichiro YOKOTA Departmentof Mathematics,Shinshu University, Matsumoto, Japan (Comm. by KinjiroKUNUGI, M.J.A., Nov. 13, 1967) 1. Introduction. The aim of this paper is to determine the representation ring R(F4) of group F4, which is a simply connected compact simple Lie group of exceptional type F. Let denote the Jordan algebra consisting of all 3-hermitian matrices over the division ring of Cayley numbers. The group F4 is obtained as the automorphism group of ~. Let o be the set of all elements of with zero trace. Then ~o is invariant by the operation of F4. Thus we have an F4-C-module ~o®RC.1' On the other hand, we know another F4-C-moduleF40RC, where F4 is the Lie algebra of F4. The result is as follows: R(F4) is a polynomial ring Z[21,22, 23, ,u] with 4 variables A1,22, 23, and p, where 2tiis the class of the exterior F4-C-module A2(jOORC)in R(F4) for i =1, 2, 3, and i is the class of 40RC in R(F4). In this paper, we shall describe the outline of our methods; these may be analogous to those as in the cases of classical groups [1[ and of group G2 [2[. The details will appear in the Journal of the Faculty of Science, Shinshu University, vol. 3, 1968. 2. Representation ring. Let G be a topological group. Let M(G) denote the set of all G-C-isomorphismclasses of G-C-modules. The direct sum V i3W and the tensor product V0 W of two G-C- modules V, W define a semiring structure on M(G). The represen- tation ring R(G) = (R(G), ~5)(where ~: M(G)->R(G) is a semiring homomorphism) is the universal ring associated with the semiring M(G). 3. Jordan algebra , group F4 and Lie algebra 40 RC. Let 6 denote the division ring of Cayley numbers and be the set of all 3-hermitian matrices X over . In ~, we define a Jordan multiplication by X oY= 1(X Y-[ YX ).

Then , is a 27-dimensional commutative distributive (non-asso- ciative) algebra over R. Let F4 denote the group of all automorphisms of ~. As is well known, F4 is a simply connected compact simple Lie group of exceptional type F. Obviously, is an F4-R-module.

1) R and C are the fields of real and complex numbers, respectively.

2 No. 9] Representation Ring of Lie Group F4 859

Let ~o be the set of all elements of with zero trace. ~o is a 26-dimensional R-submodule of ~. Since each x e F4 invaries the trace of every X e ~, ~o is also an F4-R-module and is decomposable into the direct sum of R (with trivial group action) and 3 =RO So. Thus we have an F4-C-module So®RC. Let 4 denote the Lie algebra of F4, which consists of all R-homomorphism A:--~~ satisfying A(X o Y) = A(X) o Y+ X o A(Y) for X, Y c ~. 4 is a 52-dimensional F4-R-module by the group operation (xA)(X) = x(A(x-1(X )) for x e F4, A E 4f X e ~. Thus we have an F4-C-module 40RC. 4. Maximal torus T and Weyl group W of F4, F4 has three subgroups of type Spin (9): Spin~l'(9), Spin~2'(9), Spin~3'(9), and has a subgroup Spin (8). That is, Spin(9)={x e F4 I x(E)=E}iifor z =1, 2, 3, where 1 0 0 0 0 0 0 0 0 E1=(o 0 0, E2= o 1 0, and E3=(0 0 0. 0 0 0 0 0 0 0 0 1 And Spin(8) =Spin~1'(9) n Spin~2'(9) (1Spin~3'(9). Since the ranks of F4 and Spin (8) are both 4, we choose a maximal torus T of F4 in Spin (8). The Weyl group W = W(F4) of F4 is NT(F4)/T, where NT(F4) is the normalizer of T in F4. Each element x c NT(F4) induces a permutation of E1, Elf E3. It follows that W(F4) is a semidirect product of W(Spin (8)) (the Weyl group of Spin (8)) and @?53 (the symmetric group of 3 factors). 5. Decompositions of ~O RC and %4®RC. Let j: T-->F4 denote the inclusion. Then j induces the inclusion R(F4) C R(T )W, where R(T )W is the subring of R(T) which is invariant by the Weyl group W. The representation ring R(T) of T is

R(T) = [a1f a-11 1 a2f a2llf a31 arl,3 a4f s-1f4 a11'2a21'2a3'2a44 and we have R(Spin (8))=Z[6, z, 4-, 4+] (cf. [1]), where a = (ai T ai 1) i=1 z = ~' (ai -I-ai 1)(a3+ as 1), 1Si

4 860 I. YOKOTA [Vol. 43,

(5.1) 21=c( So®RC)=2+a+4-+4+, (5.2) 1~=~(40Rc)=4+a+4-+4++z. Further by (5.1) we have (5.3) 22-Y'(112("soORc))=13+2(a+4 +4+)+(o4--[d-4+-[d+a)+3?, (5.4) 23= O(A3(So 0 RC)) = 24 + 8(a + 4- + 4)+ 2v(a + 4- + 4+) + 3(a4- + 4-4+ + 4+a) + a4-4+ + 6. 6. Ring structure of R(T)W. From (5.1)-(5.4) we have a+4-+4+=21-2

(6 ' 1) a4-+4-4++4+a=22+21-3p-3a4 4+=2 3-322-52 1 +7p +222-22p1 1 +5 z=p-21-2. Note that the left side formulae in (6.1) are polynomials in 21, 22, 23, and p. Now, let f be a W-invariant polynomial, We know that any W(Spin (8))-invariant polynomial is representable as a polynomial in a, v, 4-, 4+, and Weyl group W is the semidirect product of W(Spin (8)) and C~3(which is the permutation group of 3 factors a, 4-, 4+). Hence, f is a polynomial in a + 4- + 4+, a4- + 4-4+ + 4+a, a4-4+, and z. Thus, by (6.1) f is representable as a polynomial in 21, 22, 23, and p. Next, we shall show that 21, 22, 23, and p are algebraically independent. In fact, a, 4-, 4+, and v are algebraically independent, hence so are also a + 4- + 4+, a4- + 4-4+ + 4+a, a4-4+, and z-, and, therefore, by (5.1)-(5.4) 21, 22, 23, and p are algebraically independent. Thus we have proved the following Theorem. The representation ring R(F4) of F4 is a polynomial ring [21, 22, 23, ,~] with 4 variables 21, 22, 23, and p.

References [1] J, Milnor: The representation rings of some classical groups. Notes for Mathematics, 402 (1963). [2] I. Yokota: Representation ring of group C2. Jour. Fac. Sci., Shinshu Univ., 2 (1967).

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