MINIMAL POLYNOMIALS OF THE IMAGES OF THE UNIPOTENT ELEMENTS OF NON-PRIME ORDER IN THE IRREDUCIBLE REPRESENTATIONS OF AN ALGEBRAIC GROUP OF TYPE F 4

A bstract. The minimal polynomials of the images of the unipotent elements of non-prime order in the irreducible repre sentations of an algebraic group of type F 4 in characteristics 3 and 7 are found. This completes the solution of the minimal polynomial problem for unipotent elements in the irreducible representations of such a group in an odd characteristic.

Introduction. The investigation of the minimal polynomial problem for the images of unipotent elements in irreducible representations of the simple algebraic groups is continued. In this paper the problem is solved for unipotent elements of non-prime order and irreducible representations of an algebraic group of type F 4 in characteristics 3 and 7. Earlier this problem has been solved for unipotent elements of prime order and all simple algebraic groups [1], for unipotent elements of non-prime order and irreducible representations of the classical algebraic groups in an odd characteristic [2], and for such elements in exceptional groups in certain characteristics [3]. In [3] the following cases are settled: the groups of types E 6 in characteristic at least 5, the groups of type E 7 in characteristics 5, 7, and 17, the groups of type E 8 in characteristics 7 and 29, the groups of type F 4 in characteristics 5 and 11, and the groups of type G 2 in all characteristics. As the groups of type F 4 have unipotent elements of non-prime order only in characteristics at most 11, now for these groups the minimal polynomial problem is completely solved in all odd characteristics.
The minimal polynomials of the images of individual elements in representations yield important invariants of these representations useful for solving problems on recognizing representations and linear groups by the presence of particular matrices. Results on these polynomials proven for irreducible representations of algebraic groups can be immediately transferred to absolutely irreducible repre sentations of finite Chevalley groups in the defining characteristic what increases the range of their potential applications. Therefore such results can be regarded as a contribution to the programme of extending the fundamental results of Hall and Higman [4] on the minimal polynomials of p-elements in finite irreducible p-solvable linear groups in characteristic p to groups that are not p-solvable. In [2] one can find a short discussion of some results on the minimal polynomial problem for irreducible representations of finite groups close to simple.
The main part. In what follows  is the complex field, K is an algebraically closed field of an odd characteristic p,  and  + are the sets of integers and nonnegative integers, respectively, G = F 4 (K), G  = F 4 (), ω i , 1 ≤ i ≤ 4, are the fundamental weights of G, ω(ϕ) is the highest weight of a representation ϕ. For an element x and a representation ρ of some algebraic group, the symbol d ρ (x) denotes the degree of the minimal polynomial of ρ(x); |x| is the order of x; áµ, αñ is the value of a weight µ on a root α (the canonical pairing in the sense of [5, Section 1]). If ϕ is an irreducible representation of G, then ϕ  is the irreducible representation of G  with highest weight ω(ϕ). There exists a canonical bijection f from the set of unipotent conjugacy classes of G onto the analogical set for G  determined with the help of the distinguished parabolic subgroups in the Levi subgroups of G (see, for instance, comments in the Introduction of [6]). In what follows if x Î G is a unipotent element from a class C, then Recall that an irreducible representation of a semisimple algebraic group over K is p-restricted if all coefficients of its highest weight are less than p.
It is clear that the minimal polynomial of the image of a unipotent element in a rational representation of an algebraic group has the form (t -1) d and hence is completely determined by its degree. It is well known that the maximal order of a unipotent element in G is equal to 27 for p = 3 and to 49 for p = 7; if p = 3, only regular unipotent elements have order 27, other unipotent elements have smaller orders and are conjugate to elements from proper subsystem subgroups of G whose simple components are classical groups; for p = 7, the group G has two conjugacy classes of elements of order 49: regular unipotent elements and the class containing regular unipotent elements of a subsystem subgroup of type B 4 (see, for instance, [6]).
In what follows ϕ is a nontrivial irreducible representation of G with highest weight ω and M is a module affording ϕ.
T h e o r e m 1. Let p = 3, x Î G, and |x| = 9. Then d ϕ (x) = 9 or one of the following holds: 1) ω = 3 j ω 4 , x is conjugate to a regular unipotent element from a subsystem subgroup with a simple component of type C 2 , and d ϕ (x) = 5; 2) ω = 3 j ω 4 , x is conjugate to a regular unipotent element from a subsystem subgroup of type B 3 , and d ϕ (x) = 7; 3) ω = 3 j ω 1 , x is such as in Item 1), and d ϕ (x) = 7; 4) ω = 3 j ω 3 , x is such as in Item 2), and d ϕ (x) = 8 (here j is a nonnegative integer).
There are 3 conjugacy classes in G that satisfy the assumptions of Item 1) of Theorem 1. Regular unipotent elements from subsystem subgroups of types C 2 , C 2 × C 1 , and C 2 × C 1 × C 1 are their representatives.
T h e o r e m 2. Let p = 3 and x Î G be a regular unipotent element.
T h e o r e m 3. Let p = 7, x Î G be a regular unipotent element, and z Î G be such element of a subsystem subgroup of type B 4 . Assume that ϕ is p-restricted. Then 4 , ω 1 + ω 4 , ω 3 + ω 4 }, and d ϕ (g) = 49 otherwise for g = x or z.
According to [6, Tables A and D], for p = 7, the element z 7 is a long root element and x 7 is a product of commuting long and short root elements. Let 4 1 .
where all ρ k are p-restricted and Fr is the Frobenius morphism determined by raising the elements of K to the power p. Set The weight ω'(ρ) is uniquely determined. We call an irreducible representation ρ of a simple algebraic group G over K p-large if áϕ'(ρ), βñ ≥ p for a maximal root β of G.
As for other simple algebraic groups and characteristics, when the minimal polynomial problem is solved for p-restricted representations, we can apply the Steinberg tensor product theorem and the formulas for a tensor product of unipotent Jordan blocks from [8] to pass to arbitrary irreducible representations. In particular, if ϕ = ϕ 1 Äϕ 2 and 1 We use the notation G  , ω i , and ω(ϕ) in the same manner as for G. Throughout the text dim V is the dimension of a subspace V. If λ Î L + (G), then M(λ) and V(λ) are the irreducible module and the Weyl module of G with highest weight λ; ω(m) is the weight of a weight vector m from some module. If H is a subgroup of G, then M | H is the restriction of a G-module M to H. We assume that the weights and the roots of G are considered with respect to a fixed maximal torus T. If T Ç H is a maximal torus in H, then ω | H is the restriction of a weight ω to T Ç H. In this case for a weight vector m from some G-module, we set ω H (m) = ω(m) | H. If M is an irreducible G-module, then v Î M is a nonzero highest weight vector. For G = A 1 (K), the set L(G) is canonically identified with : The following facts are used intensively in the proofs of the main results.    Table 4], Theorem 5, Proposition 2, and Lemmas 3 and 1.
One easily observes that G(α 2 + 2α 3 + 2α 4 , 1, 2, 3) @ B 4 (K). We use this group when analyzing the restrictions of representations to a subsystem subgroup of type B 4 . Now assume that x Î G is a regular unipotent element. Set y = x 9 for p = 3 and y = x 7 for p = 7. Define M y such as in Proposition 1. Assume that ϕ is p-restricted. First suppose that p = 3. Now we state two lemmas that play an important role in the proof of Theorem 2. Let α be the maximal root of G. It is well known that α = 2α 1 + 3α 2 + 4α 3 + 2α 4 . L e m m a 7. Let x Î U + . Then . y α ∈X On the proof of Lemma 7. Set R 9 = {α 1 + 2α 2 + 4α 3 + 2α 4 , α 1 + 3α 2 + 4α 3 + 2α 4 , α}. One easily concludes that R 9 is the set of roots β of G with h(β) ≥ 9. By Lemma 4, 9 . y R β ∈ | β∈ X Using conjugation by elements of U + , one can assume that y = x δ (t) where δ Î R 9 . Analyzing the action of the elements x and y on the irreducible module N with highest weight ω 4 , we deduce that δ = α, otherwise we would get that d N (x) > 15 and obtain a contradiction due to [6, Table 3]. L e m m a 8. Let G = C 3 (K), u Î G be a regular unipotent element, and ρ be an irreducible representation of G. Assume that d ρ (u) < 9. Then ω(ρ) Î {0, 3 j ω 1 , 3 j ω 3 } ( j Î  + ).
The proof of Lemma 8 follows from [2, Table VII] Tables 3 and 4], and Formula (1). Now let p = 7. Lemma 1 implies that for proving Theorem 3 for the element x, it suffices to consider only irreducible representations ϕ with d ϕ (z) < 49 for a regular unipotent element z of a subsystem subgroup of type B 4 , in particular, Proposition 2 and Theorem 5 allow us to exclude p-large representations. Now we state three results that are used for computing the minimal polynomial of ϕ(x) in the remaining cases.
One easily checks that R 7 coincides with the set of roots δ with h(δ) ≥ 7. By Lemma 4, 7 y R δ | δ∈ ∈ X if x Î U + . Taking these facts into account and applying the conjugation in U + , we obtain the required equality for y.
L e m m a 9. The group C G (y) contains a subgroup A @ A 1 (K) such that u Î A Ì G(3, 4) and A contains a nontrivial element from . L e m m a 10. Let G = A 2 (K), g Î U + (G) be a regular unipotent element, N be an irreducible p-restricted G-module with highest weight ω = a 1 ω 1 +a 2 ω 2 , and w Î N be a nonzero lowest weight vector. Assume that the Weyl module V(ω) is irreducible. Put l = min{2(a 1 + a 2 ), 6}. Then the vector (z -1) l w has a nontrivial weight component of weight ω(w) + b 1 α 1 + b 2 α 2 with b 1 + b 2 = l.
To prove Theorem 3 for x, we show that in all cases under consideration Conclusion. Theorems 1 and 2 can be used for computing the minimal polynomials of the images of unipotent elements in irreducible representations of the group E 6 (K) for p = 3. This would complete the solution of the minimal polynomial problem for the latter group in an odd characteristic.