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Mod 3 Chern classes and generators.

1. Introduction. Let p be a prime number. In the study of mod p cohomology of the classifying space of a simply-connected, simple, compact connected Lie group G, Stiefel-Whitney classes and Chern classes play an important role. For example, the mod 2 cohomology of the classifying space of the exceptional Lie group [E.sub.6] is generated by two generators of degree 4 and of degree 32 as an algebra over the mod 2 Steenrod algebra, and Toda pointed out that the generator of degree 32 could be given as the Chern class of an irreducible representation [[rho].sub.6] : [E.sub.6] [right arrow] SU(27) in [12]. Mimura and Nishimoto [8], Kono [7] and the author [5] proved that Stiefel-Whitney classes [w.sub.16]([[rho].sub.4]), [w.sub.128]([[rho].sub.8]) and Chern classes [c.sub.16]([[rho].sub.6]), [c.sub.32]([[rho].sub.7]) are algebra generators of the mod 2 cohomology of the classifying space BG for G = [F.sub.4]; [E.sub.6]; [E.sub.7]; [E.sub.7], where [[rho].sub.4], [[rho].sub.8] are real irreducible representations of dimension 26, 248, and [[rho].sub.6], [[rho].sub.7] are complex irreducible representations of dimension 27, 56, respectively. For G = [F.sub.4]; [E.sub.6]; [E.sub.7], the mod 2 cohomology of the classifying space is generated by two elements, that is, one is the element of degree 4 and the other is [w.sub.16]([[rho].sub.4]), [c.sub.16] ([[rho].sub.6]), [c.sub.32]([[rho].sub.7]), respectively. In the case G = [E.sub.6] and p = 2; 3, the mod p cohomology of the classifying space is not yet computed. Since the non-triviality of the Stiefel-Whitney class [w.sub.128]([[rho].sub.8]) tells us that the differentials in the spectral sequence vanish on the corresponding element, we expect that it not only gives us a nice description for the generator but also helps us in the computation of the mod 2 cohomology of B[E.sub.8].

This paper is the sequel of [5] in the sense that we consider the mod 3 analogue of the above results.

In particular, we prove the non-triviality of the mod 3 Chern class [c.sub.162]([[rho].sub.8]) of degree 324. For an odd prime number p and for a simply-connected, simple, compact connected Lie group, the Rothenberg-Steenrod spectral sequence collapses at the [E.sub.2]-level and so at least additively the mod p cohomology is isomorphic to the cotorsion product of the mod p cohomology of G except for the case p = 3, G = [E.sub.8]. In [6], we proved that there exists an algebra generator of degree greater than or equal to 324 in the mod 3 cohomology ring of B[E.sub.8]. On the other hand, in [9,10], Mimura and Sambe proved that the [E.sub.2]-term of the Rothenberg-Steenrod spectral sequence is generated as an algebra by elements of degree less than or equal to 168. Hence the spectral sequence must not collapse at the [E.sub.2]-level. We expect that, in the mod 3 cohomology, the mod 3 Chern class [c.sub.162]([[rho].sub.8]) plays an important role similar to that of the Stiefel-Whitney class [w.sub.128]([[rho].sub.8]) in the mod 2 cohomology.

Now, we state our main theorem. Let T be a fixed maximal torus of the exceptional Lie group [F.sub.4]. We choose a maximal non-toral elementary abelian 3-subgroup A of [F.sub.4] so that T [intersection] A is nontrivial. We refer the reader to the paper of Andersen, Grodal, M0ller and Viruel [2, Section 8] for the details of non-toral elementary abelian p-subgroups of exceptional Lie groups and their Weyl groups. Let [mu] be a subgroup of T [intersection] A of order 3. The group p is the cyclic group of order 3. We consider the following diagram of inclusion maps.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We denote by [iota] : [mu] [right arrow] G the inclusion map of [mu] to G = [F.sub.4], [E.sub.6], [E.sub.7], [E.sub.8]. The mod 3 cohomology [H.sup.*](B[mu]; Z/3) of the classifying space B[mu] is isomorphic to

Z/3[[u.sub.2]] [cross product]] [LAMBDA]([u.sub.1]),

where [u.sub.2] is the image of the mod 3 Bockstein homomorphism of a generator [u.sub.1] of [H.sup.1]B[mu]; Z/3) = Z/3. From now on, we consider complex representations only and we denote complexifications of real representations [[rho].sub.4], [[rho].sub.8] by the same symbols [[rho].sub.4], [[rho].sub.8], respectively.

Theorem 1.1. The total Chern classes c([[iota].sup.*]([[rho].sub.i])) of the above induced representations [[iota].sup.*]([[rho].sub.i]), where i = 4, 6, 7, 8, are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As a corollary of this theorem, using Lemma 3.1, we have the following

Corollary 1.2. The Chern classes [c.sub.18]([[rho].sub.4]), [c.sub.18]([[rho].sub.6]), [c.sub.18]([[rho].sub.7]), [c.sub.162]([[rho].sub.8]) are nontrivial in [H.sup.*](B[F.sub.4]; Z/3), [H.sup.*](B[E.sub.6]; Z/3), [H.sup.*](B[E.sub.7]; Z/3), [H.sup.*](B[E.sub.8]; Z/3), respectively. Moreover, the Chern classes [c.sub.18]([[rho].sub.4]), [c.sub.18]([[rho].sub.6]), [c.sub.18]([[rho].sub.7]) are indecomposable, so that they are algebra generators.

This paper is organized as follows: In Section 2, we recall complex representations [[rho].sub.4],[[rho].sub.6], [[rho].sub.7], [[rho].sub.8] and their restrictions to Spin(8). In Section 3, we prove Theorem 1.1. We end this paper by showing the non-triviality of the mod 5 Chern class [c.sub.100]([[rho].sub.8]) of B[E.sub.8] in the appendix.

2. Complex representations. In this section, we consider complex representations [[rho].sub.4],[[rho].sub.6], [[rho].sub.7],[[rho].sub.8] in Theorem 1.1 and the complexification [[rho].sub.4] of the adjoint representation of [F.sub.4] and their restrictions to Spin(8). For the details of representation rings of Spin groups and cyclic groups, we refer the reader to standard textbooks on representation theory, e.g. Husemoller's book [4] and/or the book of Brocker and tom Dieck [3].

First, we recall the complex representation ring of Spin(2n). Let us consider the following pull-back diagram.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where SO(2n) is the special orthogonal group, [pi] : Spin(2n) [right arrow] SO(2n) is the universal covering, [T.sup.n] is the maximal torus of SO(2n) consisting of matrices of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[k.sub.n] is the inclusion map and [[??].sup.n] is a maximal torus of Spin(2n). The complex representation ring of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is R([S.sup.1]) = Z[z, [z.sup.-1]] where z is represented by the canonical complex line bundle. Considering [T.sup.n] as the product of n copies of [S.sup.1]'s, let [p.sub.i] : [T.sup.n] [right arrow] [S.sup.1] be the projection to the i-th factor. We denote by [z.sub.i] the element [p.sup.*.sub.i](z), [[pi].sup.*]([p.sup.*.sub.i](z)) in R[([T.sup.n])], R[([[??].sup.n])], respectively, so that [[pi].sup.*]([z.sub.i]) = [z.sub.i]. Then, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the complex representation ring of Spin(2n) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and [[epsilon].sub.i] [member of] {[+ or -]1}. For the sake of notational simplicity, from now on, we write [DELTA] for [[DELTA].sup.+] + [[DELTA].sup.-]. Let i : [mu] [right arrow] [S.sup.1] be the inclusion map. We denote by z the generator [i.sup.*](z) of R([mu]). Then, it is also known that R([mu]) = Z[z]/([z.sup.3]).

Next, we recall complex representations [[rho].sub.4]; [[rho].sub.6]; [[rho].sub.7]; [[rho].sub.8] of dimension 26; 27; 56; 24g in Section 1 and the complexification [[rho].sub.4] of the adjoint representation of [F.sub.4]. We consider the following commutative diagram.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is the obvious inclusion map. For [[rho].sub.4]; [[rho].sub.4], we refer the reader to Yokota's paper [14]. For [[rho].sub.6];[[rho].sub.7], we refer the reader to Adams' book [1, Corollaries 8.3, 8.2]. For [E.sub.8], from the construction of [E.sub.6] in Adams [1, Section 7] and the fact that the adjoint representation of Spin(2n) is the second exterior power of the standard representation, we have the following proposition.

Proposition 2.1. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

in R(Spin(g)), R(Spin(g)), R(Spin(10)), R(Spin(12)), R(Spin(16)), respectively.

Since the induced homomorphism [I.sup.*.sub.2n-2] maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively, we have the following proposition.

Proposition 2.2. For G = [F.sub.4]; [E.sub.6]; [E.sub.7]; [E.sub.6], let j : Spin(g) [right arrow] G be the inclusion map. In R(Spin(g)), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Mod 3 Chern classes. In this section, we prove Theorem 1.1. We consider the following diagram of inclusion maps.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The maximal torus [[??].sup.4] of Spin(g) is the maximal torus T of [F.sub.4] we mentioned in Section 1. By abuse of notation, we denote both the inclusion map of p to [[??].sup.4] and its composition with [[??].sup.4] by the same symbol [[iota].sub.0]. Let [square root of 0] be the nilradical of [H.sup.*](BA; Z/3) and [H.sup.*](B[mu]; Z/3), so that we have the induced homomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 3.1. The image of the induced homomorphism

[[iota].sup.*] : [H.sup.*](B[F.sub.4]; Z/3) [right arrow] [H.sup.*](B[mu]; Z/3)/[square root of 0]

is in Z/3[[u.sup.18.sub.2]], i.e. Im [[iota].sup.*] [subset] Z/3[[u.sup.18.sub.2]] [subset] Z/3[[u.sub.2]].

Proof. It is well-known that the Weyl group W(A) = N(A) / C(A) of A in [F.sub.4] is isomorphic to the special linear group SL3(Z /3). See the paper of Andersen, Grodal, Moller and Viruel [2, Section 8]. Moreover, [H.sup.*](BA; Z /3)/[square root of 0] is a polynomial algebra with 3 variables of degree 2 and S[L.sub.3] (Z/3) acts in the usual manner. The ring of invariants is also a polynomial algebra

[([H.sup.*](BA; Z /3)/[square root of 0]).sup.W(A)] = Z/3[[e.sub.3], [c.sub.3,1], [c.sub.3,2]].

The invariants [e.sup.2.sub.3] = [c.sub.3,0]; [c.sub.3,1]; [c.sub.3,2] are known as Dickson invariants and their degrees are 52; 4g; 36, respectively. Moreover, the induced homomorphism [[iota].sup.*.sub.1] maps [c.sub.3,0]; [c.sub.3,1]; [c.sub.3,2] to 0; 0,[u.sup.18.sub.2], respectively. See Wilkerson's paper [13, Corollary 1.4] for the details. Since the induced homomorphism [[iota].sup.*] factors through

[([H.sup.*](BA; Z /3)/[square root of 0]).sup.W(A)] [right arrow] [H.sup.*](B[mu]; Z /3)/[square root of 0],

the lemma follows.

Next, we compute the total Chern class c([[iota].sup.*.sub.0([[lambda].sub.1] + [DELTA])).

Proposition 3.2. The total Chern class c([[iota].sup.*.sub.0([[lambda].sub.1] + [DELTA])) is equal to 1 - [u.sup.18.sub.2].

Proof. Since dim([[lambda].sub.1] + [DELTA]) = 24, and since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by Lemma 3.1, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is equal to 1 + [alpha][u.sup.18.sub.2] for some [alpha] [member of] Z /3. On the other hand, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some [[alpha].sub.i] [member of] Z/ 3 and, since [[iota].sub.0] is the inclusion map, ([[alpha].sub.1]; [[alpha].sub.2]; [[alpha].sub.3]; [[alpha].sub.4]) [not equal to] (0; 0; 0; 0). So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and [[alpha].sub.1] [not equal to] 0 for some i. Hence, c([[iota].sup.*.sub.0]([[lambda].sub.1])) is divisible by 1 - [u.sup.2.sub.2]. Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is also divisible by 1 - [u.sup.2.sub.2] and so [alpha] = -1 in Z/3.

Next, we compute the total Chern class c([[iota].sup.*.sub.0]([[lambda].sub.2])).

Proposition 3.3. The total Chern class c([[iota].sup.*.sub.0]([[lambda].sub.2])) is equal to 1 - [u.sub.1],8.

Proof. As in the proof of the previous proposition, assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For ([[alpha].sup.2.sub.i], [[alpha].sup.2.sub.j]) = (1,1), we have

[f.sub.ij] = 1 - [u.sup.2.sub.2].

For ([[alpha].sup.2.sub.i], [[alpha].sup.2.sub.j]) = (1, 0) or (0,1), we have

[f.sub.ij] = 1 - 2[u.sup.2.sub.2] + [u.sup.4.sub.2] = [(1 - [u.sup.2.sub.2]).sup.2].

Since ([[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3], [[alpha].sub.4]) [not equal to] (0, 0, 0, 0), there exists (i,j) such that ([[alpha].sub.i], [[alpha].sub.j]) [not equal to] (0, 0). Hence the total Chern class c([[iota].sup.*.sub.0]([[lambda].sub.2])) is not trivial and it is divisible by 1 - [u.sup.2.sub.2].

Let us consider the total Chern class c([[iota].sup.*]([[rho].sub.4])). By Lemma 3.1, it is in Z/3[[u.sup.18.sub.2]] and by Proposition 3.2, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

So, c([[iota].sup.*.sub.0]([[lambda].sub.2])) is also in Z/3[[u.sup.18.sub.2]. Since dim [[lambda].sub.2] = 24, c([[iota].sup.*.sub.0]([[lambda].sub.2])) = 1 + [alpha][u.sup.18.sub.2] for some [alpha] [member of] Z/3. Since c([[iota].sup.*.sub.0]([[lambda].sub.2])) is divisible by 1 - [u.sup.2.sub.2], [alpha] = -1 as in the proof of the previous proposition.

Finally, we prove Theorem 1.1.

Proof of Theorem 1.1. Using Propositions 2.1, 2.2 and using Propositions 3.2, 3.3 above, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

A. Mod 5 Chern classes. Let p be an odd prime number. Let G be a simply-connected, simple, compact connected Lie group. If the integral homology of G has no p-torsion, then the mod p cohomology ring of its classifying space is a polynomial algebra and it is well-known. See, for example, the book of Mimura and Toda [11]. The integral homology of G has p-torsion if and only if (G,p) is one of ([F.sub.4], 3), ([E.sub.6], 3), ([E.sub.7], 3), ([E.sub.8], 3) and ([E.sub.8], 5). We dealt with the cases for p = 3 in this paper. For completeness, in this appendix, we deal with the remaining case, p = 5, G = [E.sub.8], that is, we prove the non-triviality of the mod 5 Chern class [c.sub.100]([[rho].sub.8]) of the complexification of the adjoint representation [[rho].sub.8] of the exceptional Lie group [E.sub.8].

The mod 5 analogue of Corollary 1.2 is as follows:

Theorem A.1. The mod 5 Chern class [c.sub.100]([[rho].sub.8]) is non-trivial. Moreover, the mod 5 Chern class [c.sub.100]([[rho].sub.8]) is indecomposable in [H.sup.*](B[E.sub.8]; Z/5).

To prove this theorem, we need the mod 5 analogue of Lemma 3.1. As in the case p = 3, G = [F.sub.4], there exists a non-toral maximal elementary abelian 5-subgroup of rank 3 in the exceptional Lie group [E.sub.8]. We choose the maximal torus T of [E.sub.8]. If necessary, by replacing A by its conjugate, we may assume that A [intersection] T is non-trivial. We choose a subgroup [mu] of A [intersection] T of order 5. Indeed, it is the cyclic group of order 5. We denote by [iota] : [mu] [right arrow] [E.sub.8] the inclusion map. The mod 5 cohomology of B[mu] is

[H.sup.*](B[mu]; Z/5) = Z/5[[u.sub.2] [cross product] [LAMBDA]([u.sub.1]),

where [u.sub.1] is a generator of [H.sup.1](B[mu]; Z/5) = Z/5 and [u.sub.2] is its image by the mod 5 Bockstein homomorphism. As in the previous section, we denote the nilradical by [square root of 0] and we denote the inclusion map of [mu] to A by [[iota].sub.1] : [mu] [right arrow] A.

Lemma A.2. The image of the induced homomorphism

[[iota].sup.*] : [H.sup.*](B[E.sub.8]; Z/5) [right arrow] [H.sup.*](B[mu]; Z/5)/[square root of 0]

is in Z/5[[u.sup.100.sub.2] [subset] [H.sup.*](B[mu]; Z/5)/[square root of 0].

Proof. Since the induced homomorphism [[iota].sup.*] factors through

[[iota].sup.*.sub.1] : ([H.sup.*][(BA; Z/5)/[square root of 0]).sup.W(A)] [right arrow] [H.sup.*](B[mu]; Z /5)/[square root of 0];

all we need to do is to recall the fact that the Weyl group W(A) of A in [E.sub.8] is S[L.sub.3](Z/5), that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and that the above induced homomorphism [[iota].sup.*.sub.1] maps [e.sub.3]; [c.sub.3,1]; [c.sub.3,2] to 0; 0, [u.sup.100.sub.2], respectively. We find these facts in [2, Section 8] and in [13, Corollary 1.4].

To compute [[iota].sup.*]([[rho].sub.8]), we need the following commutative diagram similar to the diagram in Section 3. However, in this case, the map [j.sub.16]: Spin(16) [right arrow] [E.sub.6] is not injective.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We choose the maximal torus T of [E.sub.8] so that [j.sub.16]([[??].sup.8]) = T. Then, since [[??].sup.8] [right arrow] T is a double cover and since [mu] is of order 5, there exists a map [[iota].sub.0] : [mu] [right arrow] [[??].sup.8] such that the above diagram commutes.

We use the following propositions to prove Theorem A.1.

Proposition A.3. The total mod 5 Chern class of [[iota].sup.*.sub.0] ([[lambda].sub.2]) is a product of copies of 1 - [u.sup.2.sub.2] and 1 + [u.sup.2.sub.2]. Moreover, it is non-trivial.

Proof. Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In Z/5, [[alpha].sup.2.sub.i] = 0 or [+ or -]1. So,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [mu] is a non-trivial subgroup of [[??].sup.8], [[alpha].sub.i] is nonzero for some i. So, the total Chern class is not equal to 1 and so we have the proposition.

Proposition A.4. The total mod 5 Chern class of [[iota].sup.*.sub.0]([[DELTA].sup.+]) is also a product of copies of 1 - [u.sup.2.sub.2] and 1 + [u.sup.2.sub.2].

Proof. Suppose that [i.sup.*.sub.0] : R(Spin(16)) [right arrow] R([mu]) maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then, it maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have the desired result.

Now we complete the proof of Theorem A.1.

Proof of Theorem A.1. By Propositions A.3, A.4, the total Chern class c([[iota].sup.*] ([[rho].sub.8])) is a product of copies of 1 - [u.sup.2.sub.2] and 1 + [u.sup.2.sub.2] and it is non-trivial. On the other hand, by Lemma A.2, since dim([[lambda].sub.2] + [[DELTA].sup.+]) = 240,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

for some [alpha], [beta] [member of] Z/5 and ([alpha], [beta]) [not equal to] (0; 0). Since it is divisible by 1 - [u.sub.2] or 1 + [u.sup.2.sub.2], we have 1 + [alpha] + [beta] = 0 in Z/5 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since it is a product of copies of 1 - [u.sup.2.sub.2] and 1 + [u.sup.2.sub.2], 1 + [beta][u.sup.100.sub.2] is also divisible by 1 - [u.sup.2.sub.2] or 1 + [u.sup.2.sub.2] if [beta] [not equal to] 0. So, [beta] = 0 or -1 and we have that c([[iota].sup.*]([[rho].sub.8])) is equal to 1 - [u.sup.100.sub.2] or [(1 - [u.sup.100.sub.2]).sup.2]. In particular, [c.sub.100]([[rho].sub.8]) = -[u.sup.100.sub.2] or -2[u.sup.100.sub.2] and by Lemma A.2, it is indecomposable in [H.sup.*](B[E.sub.6]; Z/5).

Acknowledgement. This work was supported by JSPS KAKENHI Grant Number JP25400097.

References

[1] J. F. Adams, Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.

[2] K. K. S. Andersen, J. Grodal, J. M. Moller and A. Viruel, The classification of p-compact groups for p odd, Ann. of Math. (2) 167 (2008), no. 1, 95-210.

[3] T. Brocker and T. tom Dieck, Representations of compact Lie groups, translated from the German manuscript, corrected reprint of the 1985 translation, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1995.

[4] D. Husemoller, Fibre bundles, 3rd ed., Graduate Texts in Mathematics, 20, Springer-Verlag, New York, 1994.

[5] M. Kameko, Chern classes and generators, Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 1, 21 23.

[6] M. Kameko and M. Mimura, On the Rothenberg-Steenrod spectral sequence for the mod 3 cohomology of the classifying space of the exceptional Lie group [E.sub.8], in Proceedings of the Nishida Fest (Kinosaki, 2003), 213 226, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, 2007.

[7] A. Kono, A note on the Stiefel-Whitney classes of representations of exceptional Lie groups, J. Math. Kyoto Univ. 45 (2005), no. 1, 217 219.

[8] M. Mimura and T. Nishimoto, On the Stiefel-Whitney classes of the representations associated with Spin(15), in Proceedings of the School and Conference in Algebraic Topology (Hanoi, 2004), 141 176, Geom. Topol. Monogr., 11, Geom. Topol. Publ., Coventry, 2007.

[9] M. Mimura and Y. Sambe, On the cohomology mod p of the classifying spaces of the exceptional Lie groups. II, J. Math. Kyoto Univ. 20 (1980), no. 2, 327 349.

[10] M. Mimura and Y. Sambe, On the cohomology mod p of the classifying spaces of the exceptional Lie groups. III, J. Math. Kyoto Univ. 20 (1980), no. 2, 351 379.

[11] M. Mimura and H. Toda, Topology of Lie groups. I, II, translated from the 1978 Japanese edition by the authors, Translations of Mathematical Monographs, 91, American Mathematical Society, Providence, RI, 1991.

[12] H. Toda, Cohomology of the classifying space of exceptional Lie groups, in Manifolds Tokyo 1973 (Proc. Internat. Conf, Tokyo, 1973), 265 271, Univ. Tokyo Press, Tokyo, 1975.

[13] C. Wilkerson, A primer on the Dickson invariants, in Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982), 421 434, Contemp. Math., 19, Amer. Math. Soc., Providence, RI, 1983.

[14] I. Yokota, Representation ring of Lie group [F.sub.4], Proc. Japan Acad. 43 (1967), 858 860.

By Masaki KAMEKO

Department of Mathematical Sciences, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitama 337-8570, Japan

(Communicated by Kenji FUKAYA, M.J.A., June 13, 2017)

2010 Mathematics Subject Classification. Primary 55R40, 55R35.

doi: 10.3792/pjaa.93.55
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