p-large representations and asymptotics: a survey and conjectures

Irina D. Suprunenko Institute of Mathematics Academy of Sciences of Belarus Minsk, 220072, Belarus e-mail: suprunenko@im.bas-net.by

To Alexei Ivanovich Kostrikin with gratitude and best wishes

Introduction

The goal of this survey is to discuss some asymptotic properties of irreducible representations of the classical algebraic groups in positive characteristic with highest weights large enough with respect to the characteristic. These properties are specific for positive characteristic, but almost do not depend upon the choice of the characteristic. Two topics will be considered: the behaviour of unipotent elements in the representations mentioned above and the restrictions of such representations to naturally embedded subgroups. Several conjectures will be formulated. Our experience shows that it may be useful to distinguish some classes of large enough representations when investigating properties specific for the positive characteristic case. In this connection the notion of a p-large representation was introduced in [24] where asymptotic estimates for numbers of Jordan blocks of size p in images of elements of order p in irreducible representations of the classical algebraic groups in characteristic p > 0 with large highest weights were obtained.

Asymptotic group theory is in a stage of revival now. Several profound surveys (Kantor [8] and [9], Pyber [16], etc.) have appeared recently. The majority of results in this direction is concerned with finite permutation groups, but some of them deal with finite classical and other finite groups of Lie type as well. Asymptotic results on groups of Lie type are concerned mainly with the probability of generating a group by elements of some kind, see Kantor and Lubotzky [10], Guralnick, Kantor, and Saxl [7], Malle, Saxl, and Weigel [14], and Liebeck and Shalev [11]-[13]. Proofs of asymptotic results on permutation groups often use the classification of finite simple groups and require highly nontrivial facts on finite groups of Lie type, their subgroup structure and representations. For instance, such are Saxl’s results on primitive permutation groups of special degrees discussed in [17]. So it seems worth to try to strengthen asymptotic trends in modular representation theory.

Due to famous Steinberg’s results [20] many problems on irreducible representations of finite groups of Lie type in describing characteristic can be better understood when considered for relevant algebraic groups over algebraically closed fields. The results and conjectures of this article are stated for the latter groups, but can be easily transferred to the former ones.

Before giving formal definitions and statements of results we would like to explain the approach informally.

The following notation is used throughout the article: C is the complex field, K is an algebraically closed field of characteristic p > 0, G is an algebraic group of a classical type over K, r is the rank of G, φ is an irreducible rational representation of G, and ω(φ) is its highest weight. Recall that there are four types of the classical algebraic groups: A, B, C, and D. Throughout the text simply connected algebraic groups are considered only. So the groups of type A and C are the special linear and the symplectic groups and those of types B and D are the spinor ones (of odd and even degrees, respectively). One can write the ”p-adic expansion”: ω(φ) = ∑_{j=0}^{m} p^{j}λ_{j} with p-restricted λ_{j}. A dominant weight λ is p-restricted if λ = ∑_{i=1}^{r} c_{i}ω_{i} where 0 ≤ c_{i} < p and ω_{i} are the fundamental weights. Put ω̄(φ) = ∑_{j=0}^{m} λ_{j}. Write ω̄(φ) = ∑_{i=1}^{r} a_{i}ω_{i}. In what follows the notation a_{i} is preserved for the parameters connected with an irreducible representation φ in this way. Denote by G_{C} the complex algebraic group of the same type as G, and let φ_{C} be the irreducible representation of G_{C} with highest weight ω̄(φ). The simple roots of G and G_{C} are denoted by the symbols α_{i}, 1 ≤ i ≤ r, their labelling corresponds to [1]. If μ is a weight and β is a root of G, we write ⟨μ, β⟩ for the value of μ on β. The symbol k_{ρ}(x) denotes the degree of the minimal polynomial of an element x ∈ G or G_{C} in a representation ρ.

The coefficients a_{i} are important for finding many linear invariants of representations. It is clear that the standard Frobenius morphism associated with raising of the elements of K to the p-th power does not influence such items as the behaviour of unipotent elements in representations, the complete reducibility of restrictions to subgroups, etc. On the other hand, it multiplies highest weights by p. So we prefer to look at ω̄(φ) rather than ω(φ) and to choose classes of p-large representations (and highest weights) excluding the growth of weights caused by the Frobenius morphism.

The common feature of the majority of results to be discussed here is that some regularities occur provided the highest weight is in a certain sense large enough with respect to p. Results of [24], [23] and [26] give some evidence in favour of the following approach to a number of representation-theoretic problems in positive characteristic, especially, to ones concerning properties of unipotent elements. We have a question and try to distinguish a class of p-large representations for this question which has some good properties. First a function s = s(a_{1}, …, a_{r}) on the set of irreducible representations of G which does not depend upon p and a function g on the set of primes are defined. An irreducible representation φ is called p-large (for our question) if s(φ) ≥ g(p). In this situation we wish the following conditions to hold.

a) Some regularities of asymptotic nature which do not depend or almost do not depend upon a fixed positive p hold for p-large representations; b) almost all (in some sense) irreducible representations are p-large;

c) not p-large representations have some properties similar to characteristic 0 case, more exactly, similar to those of the representation φ_{C};

d) s is a linear function of the coefficients a_{i};

e) the function s is defined in a similar manner for groups of arbitrary ranks of the same type;

f) the equality s(φ) ≥ g(p) is threshold for properties in item a), i.e. for arbitrarily large r and p there exist representations with s(φ) = g(p) − 1 for which these properties do not hold.

Conditions a) and b) are most important, and f) is in many cases too difficult.

1 Results on unipotent elements

These results are concerned with the number of Jordan blocks of size p in images of elements of order p and the minimal polynomials of arbitrary p-elements. In this section the following definition of a p-large representation introduced in [24] is used.

Definition. Let β be the maximal root of G. Put

l(φ) = ⟨ω̄(φ), β⟩.

A representation φ is called p-large if l(φ) ≥ p.

In a similar manner define the value l(ρ) for an irreducible representation ρ of G_{C} as well putting l(ρ) = ⟨ω(ρ), β⟩. Using the formulae for the maximal roots from [1, Tables 1-4], one easily deduces that

l(φ) = ∑_{i=1}^{r} a_{i} for G = A_{r}(K),

l(φ) = a_{1} + 2(∑_{i=2}^{r−1} a_{i}) + a_{r} for G = B_{r}(K),

l(φ) = ∑_{i=1}^{r} a_{i} for G = C_{r}(K),

and l(φ) = a_{1} + 2(∑_{i=2}^{r−2} a_{i}) + a_{r−1} + a_{r} for G = D_{r}(K).

The following theorem yields lower estimates for the number of Jordan blocks of size p in images of elements of order p in p-large representations.

Theorem 1.1 ([24, Theorem 1.1]) Assume that r > 2 and p > 2 for G = B_{r}(K), r > 1 for G = C_{r}(K) and r > 3 for G = D_{r}(K). Define the function f of the rank of G as follows: f(r) = 2r−2 for G = A_{r}(K); f(r) = 6r−7 for G = B_{r}(K), p = 3; f(r) = 8r−10 for G = B_{r}(K), p > 3; f(r) = 4r − 4 for G = C_{r}(K); f(r) = 4r − 8 for G = D_{r}(K), p = 2; f(r) = 6r − 10 for G = D_{r}(K), p = 3; and f(r) = 8r − 16 for G = D_{r}(K), p > 3. Let φ be p-large. Then the image in φ of any element of order p of G has at least f(r) Jordan blocks of size p.

Theorem 1.2 below shows that for groups of types A, B and D and p > 2 the estimates in Theorem 1.1 are asymptotically exact; in particular, we cannot take a linear function with a larger first coefficient for f. For groups of type C there may be some reasons to study the possibility of quadratic estimates.

Theorem 1.2 ([24, Theorem 1.4]) Let G = A_{r}(K), ω = a_{1}ω_{1} + a_{r}ω_{r}, a_{1} + a_{r} = p, a_{1}, a_{r} ≠ 0, or p > 2, G ∈ \{B_{r}(K), D_{r}(K)\}, ω = ω_{1} + \frac{1}{2}(p − 1)ω_{2}. Let ω(φ) = ω, and let d_{φ} be the number of Jordan blocks of size p in the image of a nonunity long root element in φ. Then l(φ) = p; d_{φ} ≤ 2r for G = A_{r}(K); d_{φ} ≤ 8r − 8 for G = B_{r}(K), p > 3; d_{φ} ≤ 6r − 5 for G = B_{r}(K), p = 3; d_{φ} ≤ 8r − 12 for G = D_{r}(K), p > 3; and d_{φ} ≤ 6r − 8 for G = D_{r}(K), p = 3.

For all four types of the classical groups and all r and p > 2 there exist p-restricted representations φ with l(φ) = p − 1 where the long root elements have just one block of size p. Hence the condition l(φ) = p is threshold for the number of Jordan blocks of size p in images of elements of order p.

Example 1.3 ([24, Theorem 1.4]) Let ω = a_{1}ω_{1} + a_{r}ω_{r} with a_{1} + a_{r} = p − 1 for G = A_{r}(K), ω = \frac{p−1}{2}ω_{2} for G = B_{r}(K) and D_{r}(K), and ω = (p − 1)ω_{1} for G = C_{r}(K). Let ω(φ) = ω. Then l(φ) = p − 1, and the image of a long root element in φ has only one Jordan block of size p.

Theorem 1.1 is a purely modular phenomenon.

Lemma 1.4 For each unipotent element y ∈ G_{C} there exist irreducible representations ψ of the group G_{C} with l(ψ) arbitrarily large such that ψ(y) has just one Jordan block of the maximal size. The image of a regular unipotent element in every irreducible representation of G_{C} has just one Jordan block of the maximal size.

Proof. Let X and X_{1} be the sets of weights of the groups G_{C} and A_{1}(C), respectively, and λ be the fundamental weight of A_{1}(C). For 1 ≤ i ≤ r denote by d_{i} the i-th label on the labelled (weighted) Dynkin diagram of y. Using the standard facts on labelled Dynkin diagrams (see [5, Section 5.6]), one can choose a subgroup G_{1} ≅ A_{1}(C) and maximal tori T ⊂ G_{C} and T_{1} ⊂ G_{1} such that y ∈ G_{1} and the restriction of weights from T to T_{1} induces a homomorphism τ : X → X_{1} with τ(α_{i}) = d_{i}λ. Set I = {i|1 ≤ i ≤ r, d_{i} ≠ 0} and denote by ρ the homomorphism X → Z which is a composition of τ and the homomorphism X_{1} → Z : aλ → a. Let ψ be an irreducible representation of G_{C} with highest weight ω = ∑_{i=1}^{r} a_{i}ω_{i} and assume that a_{i} = 0 for all i ∉ I. Let μ ≠ ω be a weight of ψ. It is well known that μ = ω − ∑_{i=1}^{r} b_{i}α_{i} and b_{i} ≠ 0 for some i ∈ I. We have ρ(ω) = c with c ∈ Z^{+}. Now it is clear that ρ(μ) = d with d ∈ Z and d < c. Standard facts of the representation theory of the group A_{1}(C) imply that ψ|G_{1} has a unique irreducible component with highest weight cλ and components with highest weights of form dλ with d < c. It is well known that a unipotent element of G_{1} has a single Jordan block of size a + 1 in the irreducible representation with highest weight aλ. This yields the first assertion of the lemma. To obtain the second one, recall that all d_{i} = 2 for regular y (this is well known and can be easily deduced, for instance, from the results of [5, Sections 5.6 and 5.7]. □

One easily observes that the quotient of the number of p-large p-restricted representations by that of all p-restricted representations → 1 when r → ∞. The same holds for q-restricted representations with q = p^{r}.

Now let us look at the p-large representations bearing in mind Conditions a)—f) of the Introduction. One can conclude that all of them hold. Indeed, by [23, Theorem 1.1 and Algorithm 1.4], the minimal polynomial of a long root element in a not p-large representation φ is such as in φ_{C}. Hence Condition c) holds. Theorem 1.1, Example 1.3 and the discussion above yield other conditions.

There are some regularities connected with the minimal polynomials of arbitrary unipotent elements in p-large representations. It is clear that the minimal polynomial of a unipotent element has the form (t − 1)^{k} where t is an indeterminate.

Theorem 1.5 ([25, Theorem 1.1] Let p > 2 and let φ be p-large. Then k_{φ}(x) = |x| for each unipotent element x ∈ G.

Here the condition l(φ) = p is also threshold.

Example 1.6 ([25, Theorem 1.2]) Let n be the dimension of the standard representation of G. Suppose that n = p^{l} + 1 for G = A_{r}(K) and C_{r}(K), n = p^{l} + 2 for G = B_{r}(K) and n = p^{l} + 3 for G = D_{r}(K) with l > 0 in all cases. Let ω(φ) = (p − 1)ω_{1} for G = A_{r}(K) and C_{r}(K) and ω(φ) = \frac{1}{2}(p − 1)ω_{2} for G = B_{r}(K) and D_{r}(K). Then l(φ) = p − 1 and

k_{φ}(x) = (p − 1)p^{l} + 1 < |x| = p^{l+1}.

for a regular unipotent element x ∈ G.

The following results show that if we are interested in the minimal polynomials of certain unipotent elements only and do not want to consider all unipotent elements, then it may be worth to replace l(φ) by other similar function.

Theorem 1.7 ([25, Theorem 1.3]) Let p > 2, G = C_{r}(K), φ be p-restricted, and let a_{1} + 2 ∑_{i=2}^{r} a_{i} ≥ p. Let x ∈ G be a unipotent element which lies in the subgroup A_{r−1}(K) naturally embedded into G. Then k_{φ}(x) = |x|.

Corollary 1.8 ([25, Corollary 1.4]) Let G and φ be as in Theorem 1.7. Assume that r ≥ p^{l} + 1, x ∈ G is a unipotent element of order p^{l+1} and in the standard realization of G the element x has either a Jordan block of size ≥ 2(p^{l} + 1), or two Jordan blocks of equal size t ≥ p^{l} + 1. Then k_{φ}(x) = |x|.

Set l_{1}(φ) = a_{1} + 2 ∑_{i=2}^{r} a_{i}. It is clear that the function l_{1} would satisfy Conditions a), b), d), and e) of the Introduction if one takes l_{1} for a function s which determines the set of p-large representations.

2 Questions and conjectures on unipotent elements

In this section some questions and conjectures connected with the behaviour of unipotent elements in p-large representations are discussed.

I. Further asymptotic estimates in the class of p-large representations. The goal is to generalize Theorem 1.1 and to obtain estimates of the form: if l(φ) ≥ g(p) where g is some polynomial, then the image φ(x) has at least r^{k} Jordan blocks of size p for each element x ∈ G of order p. Proposition 2.1 yields some very rough estimates for the groups of types A and C.

Proposition 2.1 Let G = A_{r}(K) or C_{r}(K), p > 2 for G = C_{r}(K), and let k ∈ Z^{+}. Assume that r > 5 if k = 2. Let φ be a p-restricted representation of G with l(φ) > 2k(p − 1). Then φ(x) has at least r^{k} Jordan blocks of size p for each element x ∈ G of order p.

Proof. We shall need the following claim.

(*) Let ρ be an irreducible representation of the group A_{s}(K) whose highest weight μ is a linear combination of the fundamental ones with at least t nonzero coefficients (t > 0). Then dim ρ ≥ (s + 1)s … (s + 2 − t).

To prove the claim, we shall show that the length of the orbit of μ under the action of the Weyl group W is at least (s + 1)s … (s + 2 − t). Indeed, let μ = ∑_{i=1}^{s} b_{i}ω_{i} and I = {i | b_{i} = 0}. Denote by S the stabilizer of μ in the Weyl group. The group S is generated by the reflections associated with the simple roots α_{i}, i ∈ I (see, for instance, [18, 1.10]). Hence S is isomorphic to a product of the symmetric groups S_{u_{1}}, … S_{u_{l}} with (u_{1} − 1) + … + (u_{l} − 1) = |I|. One easily observes that (a + 1)!(b + 1)! < (a + b + 1)! for positive integers a and b. This implies that |S| ≤ (|I| + 1)! ≤ (s − t + 1)!. Since |W | = (s + 1)!, we obtain the required estimate for the orbit of μ and prove our claim.

It follows from [24, Corollary 2.14] that for every element x ∈ G of order p the number of Jordan blocks of size p in φ(x) is not less than the similar number for a long root element (a root element if G = A_{r}(K)). Here arguments connected with the closures of unipotent conjugacy classes are used, and just here the assumption on p for G = C_{r}(K) is needed. So we proceed to prove our claim for a long root element. Let M be a module affording the representation φ and v ∈ M be a nonzero highest weight vector. Recall that l(φ) = ∑_{i=1}^{r} a_{i}. First assume that G = A_{r}(K). For k = 1 our proposition follows from Theorem 1.1, so we suppose that k > 1. Denote by H and H_{1} the subgroups of G generated by the root subgroups associated with the roots ±α_{i}, i ≠ 2, and by those associated with the roots ±α_{i}, i > 2, respectively. Then H_{1} ≅ A_{r−2}(K) and H ≅ A_{1}(K)A_{r−2}(K). Standard facts on representations of semisimple groups show that every irreducible representation of H can be realized as the tensor product of irreducible representations of A_{1}(K) and H_{1}. Let π be the irreducible representation of A_{1}(K) with highest weight (p−1)ω_{1}. We shall construct a composition factor ψ of the restriction φ|H isomorphic to π ⊗ ψ_{1} where ψ_{1} is an irreducible representation of H_{1} with dim ψ_{1} ≥ r^{k}. It is well known that dim π = p and that an element of order p has a single Jordan block in π. Now [6, ch. VIII, Lemma 2.2] implies that the image ψ(x_{1}(u)) has dim ψ_{1} Jordan blocks of size p if u ∈ K^{∗}. Hence it suffices to find the factor ψ.

Let Y_{i}, 1 ≤ i ≤ r, be the root element in the Lie algebra of G associated with the root −α_{i}. Construct a vector m ∈ M as follows. First assume that a_{1} < p − 1. Choose j such that a_{1} + … + a_{j−1} < p − 1 and a_{1} + … + a_{j} ≥ p − 1. Then p − 1 = d + ∑_{i=1}^{j−1} a_{i} with d ≤ a_{j}. Set d_{j} = d and d_{i} = a_{i} + d_{i+1} for 2 ≤ i ≤ j − 1. Now put m = Y_{2}^{d_{2}} … Y_{i}^{d_{i}} … Y_{j}^{d_{j}}v. If a_{1} = p − 1, set m = v. Denote by λ and ψ the weight of m with respect to H and the irreducible representation of H with highest weight λ. Let the weight λ_{1} and the representation ψ_{1} play the same role for H_{1}. Considering the weight system of φ, one easily deduces that the vector m is fixed by the root subgroups associated with the roots α_{i} with i ≠ 2. Hence m generates an indecomposable H-module with highest weight λ which has a composition factor isomorphic to ψ. One may regard ω_{1}, ω_{3}, …, ω_{r} as the set of the fundamental weights for H and write λ = (p − 1)ω_{1} + λ_{1}. Hence ψ ≅ π ⊗ ψ_{1}. It remains to prove that dim ψ_{1} ≥ r^{k}.

We shall show that λ_{1} is a linear combination of the fundamental weights with at least 2k − 1 nonzero coefficients. Then (*) will be applied. It is not difficult to verify the following formulae:

λ_{1} = a_{2}ω_{3} + … + a_{j−2}ω_{j−1} + (a_{j−1} + a_{j} − d)ω_{j} + (a_{j+1} + d)ω_{j+1} + a_{j+2}ω_{j+2} + … + a_{r}ω_{r}

if j > 3,

λ_{1} = (a_{2} + a_{3} − d)ω_{3} + (a_{4} + d)ω_{4} + a_{5}ω_{5} + … + a_{r}ω_{r}

for j = 3,

λ_{1} = (a_{3} + d)ω_{3} + a_{4}ω_{4} + … + a_{r}ω_{r}

for j = 2, and λ_{1} = a_{3}ω_{3} + … + a_{r}ω_{r} if a_{1} = p − 1. If a_{1} < p − 1, it is clear that ∑_{i=1}^{j} a_{i} < 2(p−1) and hence ∑_{i=j+1}^{r} a_{i} > (2k−2)(p−1). Since φ is p-restricted, at least 2k−1 of the integers a_{j+1}, …, a_{r} are nonzero. It is clear that ∑_{i=3}^{r} a_{i} > (2k−2)(p−1) and hence at least 2k−1 of the integers a_{3}, …, a_{r} are nonzero as a_{1} +a_{2} ≤ 2(p−1). In particular, r ≥ 2k+1. This yields that in all cases λ_{1} has ≥ 2k−1 nonzero coefficients. Now assertion (*) implies that dim ψ_{1} ≥ (r−1)(r−2) … (r−2k+1). Set D = (r−1)(r−2) … (r−2k+1). We claim that D ≥ r^{k} in our assumptions on k and r. Let k > 3. Then (r − k − 1) … (r − 2k + 1) is a product of k − 1 ≥ 3 consecutive integers the smallest of which is at least 2. Hence (r − k − 1) … (r − 2k + 1) > 2^{k} and D > (2r − 2)(2r − 4) … (2r − 2k) > r^{k} since each of the multipliers (2r − 2), …, (2r − 2k) is greater than r. Now let k = 3. Then r ≥ 7. It is not difficult to verify that D = (r − 1) … (r − 5) ≥ 6(r − 1)(r − 2)(r − 3) > r^{3}. If k = 2, one gets D = (r − 1)(r − 2)(r − 3) ≥ 3(r − 1)(r − 2) > r^{2} as r ≥ 6. This completes the proof for G = A_{r}(K).

The arguments for G = C_{r}(K) are quite similar. Here we take for H the subgroup generated by the root subgroups associated with the roots α_{i}, i ≠ r − 1, and α_{r} plays the same role as α_{1} in the proof for G = A_{r}(K). □

We can also prove a similar fact for the groups of type B and D with p > 2, but in those cases the estimates are still more rough. However, it seems very difficult to find threshold functions g for arbitrary k, i.e. such g that φ(x) has at least r^{k} Jordan blocks of size p provided |x| = p and l(φ) ≥ g(p), but for arbitrarily large r and p there exist irreducible representations φ with l(φ) = g(p) − 1 where some element x of order p has less than r^{k} blocks of size p. Another version of this problem is to replace r^{k} by p^{m}r^{k} or a similar function.

II. An analogue of Theorem 1.1 for elements of order > p. We would like to extend Theorem 1.1 to unipotent elements of arbitrary order replacing blocks of size p by those of size equal to the order of an element. However, the following lemma shows that this cannot be done without some extra assumptions.

Lemma 2.2 Let G = A_{r}(K), r = p^{s}, s > 0, φ be the irreducible representation of G with highest weight aω_{1} +bω_{r}, a, b > 0, a+b = p, and let x ∈ G be a regular unipotent element. Then φ(x) has at most 2 Jordan blocks of size p^{s+1}.

Proof. Set ω = aω_{1} + bω_{r}. Let ω′ be the lowest weight of φ. Denote by V and V_{C} the Weyl module and the irreducible G_{C}-module with highest weight ω, respectively. Then φ can be realized in a quotient M of V. It is clear that |x| = p^{s+1}. Let y ∈ G_{C} be a regular unipotent element. Put l = p^{s+1} − 1, n_{1} = dim(x − 1)^{l}M, n_{2} = dim(x − 1)^{l}V, and n_{3} = dim(y − 1)^{l}V_{C}. One easily observes that φ(x) has n_{1} Jordan blocks of size p^{s+1} and that n_{1} ≤ n_{2}. The construction of the Chevalley groups in [21, §3] and the arguments in [21, Theorem 39] show that there exist such bases B in V and B_{C} in V_{C} that the matrix of x in B can be obtained from that of y in B_{C} (an integer one) by reduction mod p. This forces n_{2} ≤ n_{3}. Hence it suffices to prove that n_{3} ≤ 2.

Now consider the subgroup G_{1} ⊂ G_{C} and the homomorphism ρ associated with the element y in Lemma 1.4. We can assume that y is a positive root element in G_{1} ⊂ G_{C} and lies in a maximal unipotent subgroup of G_{1} ⊂ G_{C} which is normalized by T_{1}. For k ∈ Z denote by V_{k} the sum of all the weight subspaces in V_{C} of weights μ with ρ(μ) = k. It is well known that ρ(ω_{1}) = ρ(ω_{r}) = r = p^{s}. This forces ρ(ω) = p^{s+1} = l + 1 and ρ(ω′) = −l−1. Since ρ(α_{i}) = 2 for all i, one gets (y−1)^{l}V_{k} ⊂ ∑_{j≥k+2l} V_{j} and |ρ(μ)| ≤ l−1 for ω′ < μ < ω. Now we can see that dim V_{k} = 1 for k = ±(l + 1), V_{k} = 0 for |k| > l + 1, and (y − 1)^{l}V_{C} ⊂ (y − 1)^{l}V_{−l−1} + V_{l+1}. This implies that n_{3} ≤ 2 and completes the proof. □

But it is possible that some asymptotic estimates can be found if r is large enough with respect to the order of an element.

III. Blocks of size p and the dimension of a representation. This question is concerned with the relations between the number of blocks of size p and the dimension of a representation. For an irreducible representation φ of G let N_{φ} be the minimal number of Jordan blocks of size p in the image φ(x) of an element x ∈ G of order p. It is well known that pN_{φ} = dim φ if ω(φ) = (p^{j_{1}} + … + p^{j_{s}})(p − 1)(ω_{1} + … + ω_{r}) with j_{1} < … < j_{s}. If φ is p-restricted and a_{i} < p − 1 for at least one i, it is not difficult to deduce that pN_{φ} < dim φ. Indeed, let H ⊂ G be the subgroup generated by the root subgroups associated with the roots α_{i} and −α_{i}. Then H ≅ A_{1}(K). Smith’s theorem [19] implies that the restriction φ|H has the irreducible representation of H with highest weight a_{i}ω_{1} as a direct summand. Then it follows from well-known facts of the representation theory of H that a root element associated with α_{i} has at least one Jordan block of size a_{i} + 1 < p in φ and hence pN_{φ} < dim φ. Is it true that pN_{φ}/ dim φ > δ for some fixed positive δ if l(φ) is large enough? A weaker version of this question: can one find a reasonable lower estimate for pN_{φ}/ dim φ provided l(φ) is large enough? Observe that there are no such estimates for the whole class of p-large representations. Indeed, take G = A_{r}(K) and consider an irreducible representation φ with ω_{(}φ) = a_{1}ω_{1} + a_{r}ω_{r}, a_{1} + a_{r} = p, and a_{1}, a_{r} ≠ 0. By Theorem 1.2, N_{φ} ≤ 2r. On the other hand, dim φ can be greater than r^{l} for any fixed l if p and r are large enough. Observe that in this case one can take a_{1} large enough with respect to l. The sets of weights of the representations φ and φ_{C} coincide ([15, Theorem 1] and [22]). Now it is clear that φ has no less weights than the irreducible representation of G with highest weight a_{1}ω_{1}. It is well known that the latter representation has (r + 1) … (r + a_{1})/a_{1}! weights which yields a very rough estimate dim φ ≥ (r + 1) … (r + a_{1})/a_{1}! sufficient for our claim if r is large with respect to a_{1}. It is quite probable that here one must consider assumptions of the form l(φ) > f(r, p) in order to obtain the required estimates.

IV. p-large representations for a given element. The objective is to obtain estimates of the following kind. Let x ∈ G be an element of order p^{l} and φ be an irreducible representation of G. Prove that the image φ(x) has at least F (r) Jordan blocks of size p^{l} if s(φ) is large enough with respect to p. Here F and s are some functions.

a) Elements of order p. First assume that |x| = p. Results of [23] on the minimal polynomials of elements of order p in irreducible representations of the semisimple algebraic groups in characteristic p prompt the choice of the function s. Assume that p > 2 if G ≠ A_{r}(K). By [23, Theorem 1.1],

k_{φ}(x) = min{p, 1 + ∑_{i=1}^{r} a_{i}b_{i}}

where b_{i} are constants depending upon x and 1 + ∑_{i=1}^{r} a_{i}b_{i} = k_{φ_{C}}(y) for a unipotent element y ∈ G_{C} with the same labelled Dynkin diagram as x (in the sense of [2] and [3]). Explicit algorithms for computing b_{i} with the use of the Jordan form of x (and y) in the standard realization of G (respectively, G_{C}) are given in [23, Algorithm 1.4]. In particular, for G = A_{r}(K) and a regular element

b_{i} = i(r + 1 − i).  (1)

Set s(φ) = ∑_{i=1}^{r} a_{i}b_{i} and call a representation φ p-large for x if s(φ) ≥ p. One easily observes that with this new definition almost all irreducible representations are p-large (in the same sense as before). It follows from the above that k_{φ}(x) = k_{φ_{C}}(y) for all not p-large φ. So Conditions b), c) and d) of the Introduction hold. However, one cannot obtain asymptotic estimates for the number of blocks of size p in the image φ(x) without additional assumptions as the following lemma holds.

Lemma 2.3 Let p ≡ 1(mod 3), G = A_{r}(K), r = (p+2)/3, ω_{(}φ) = ω_{1} +ω_{2}, and let x ∈ G be a regular unipotent element. Then |x| = s(φ) = p and φ(x) has at most 2 Jordan blocks of size p.

Proof. Since r + 1 < p, we have |x| = p. The claim on s(φ) follows from Formula (1). Next, the arguments from the proof of Lemma 2.2 are used. A regular unipotent element y ∈ G_{C} and the irreducible G_{C}-module V_{C} with highest weight ω_{(}φ) are considered. Set d = dim(y − 1)^{p−1}V_{C}. First one shows that the number of the blocks of size p in φ(x) is at most d. Then we prove that d ≤ 2. □

However, one can expect to obtain some asymptotic estimates for r large enough with respect to an element in the following sense. Assume that a unipotent element x ∈ G is conjugate to an element of a naturally embedded subgroup of rank i.

Conjecture 1. If r is large enough with respect to i and φ is p-large for x, then φ(x) has at least F (r) blocks of size p where F is an increasing function.

Observe that the p-large representations defined in Section 1 can be regarded as plarge representations for long root elements and turn out to be p-large for all elements of order p.

Using the notion of p-large representations for given elements, one can think about some kind of a hierarchy of representations. One can distinguish representations which are p-large for no elements of order p, then representations p-large for some ”big” elements only, then some intermediate classes, then p-large representations. For instance, the standard representations of the classical groups are p-large for no elements of order p; if p is large enough with respect to r, some other irreducible representations with this property exist. However, if p > 7 and an element x has a Jordan block of size p in the natural realization of G, then all p-restricted representations of G distinct from the standard one and its dual are p-large for x. These facts can be deduced from [23, Theorem 1.1 and Algorithm 1.4].

b) Elements of order > p. Here the situation is much more complicated. At present the minimal polynomials of unipotent elements of arbitrary orders are found for the special linear group and p > 2 only [27, Theorems 1-3]. If p > 2, G = A_{r}(K), φ is an irreducible representation of G, and x ∈ G is an element of order p^{s}, then

k_{φ}(x) = min{p^{s}, p^{l}(1 + ∑_{i=1}^{r} a_{i}b_{il}) | 0 ≤ l < s},  (2)

except some explicitly determined cases. Here b_{il} are some constants depending upon x. All the exceptions are concerned with representations with ω̄(φ) = ω_{p} or ω_{r+1−p} and elements having the minimal polynomial of degree p^{s−1} + p in the standard realization of G. For regular x the value 1 + ∑_{i=1}^{r} a_{i}b_{il} is equal to k_{φ_{C}}(y) where y ∈ G_{C} is an element with the same canonical Jordan form as x^{l}. For arbitrary x this does not hold. It may be worth to consider representations φ with the minimal value in Formula (2) equal to p^{s} as p-large for x. We would like to extend Conjecture 1 to elements of order greater than p, excluding the exceptional representations and the elements mentioned above.

3 Restrictions to semisimple subgroups

In this section the semisimplicity of restrictions of irreducible representations of the classical groups to their natural subgroups or the absence of this semisimplicity are discussed. Here l(φ) is such as in Section 1. Brundan, Kleshchev, and the author [4] have obtained a semisimplicity criterion for the restrictions of irreducible representations of the groups GL_{r+1}(K) and A_{r}(K) to GL_{r}(K) and A_{r−1}(K), respectively. This criterion is a rather complicated combinatorial condition stated in terms of partitions. In particular, results of [4] (Theorem 1.1 and the comments in the Introduction) yield that for a p-restricted representation φ of A_{r}(K) with all the coefficients a_{i} ≠ 0 the restriction φ|A_{r−1}(K) is semisimple (i.e. completely reducible) if r − 1 + ∑_{i=1}^{r} a_{i} ≤ p or a_{i} + a_{i+1} = p − 1 for all i, 1 ≤ i ≤ r − 1. This implies that for r large enough there exist p-restricted representations φ with arbitrarily large l(φ) with respect to p and semisimple restrictions to SL_{n}(K). However, this is not the case for restrictions to naturally embedded subgroups of relatively small ranks.

Theorem 3.1 ([26]). Let G = A_{r}(K), H = A_{n}(K) be a naturally embedded subgroup, and let φ be a p-restricted representation of G. Assume that l(φ) ≥ 2p − 1 and r ≥ 2n. Then the restriction φ|H is not semisimple.

It is quite possible that the estimates for l(φ) in Theorem 3.1 can be relaxed. For instance, by [26, Lemma 3], the restriction of φ to a naturally embedded A_{1}-subgroup is semisimple if and only if φ is not p-large.

Conjecture 2. Let G be an algebraic group of a classical type and rank r and G_{1} ⊂ G be a naturally embedded subgroup of the same type and rank n. If φ is an irreducible representation of G with l(φ) ≥ p and r is large enough with respect to n, then φ|G_{1} is not semisimple.

Question. Is this true for an arbitrary simple algebraic group of rank n embedded into G if r is large enough with respect to n?

Observe that the condition l(φ) ≥ p in these conjecture and question cannot be relaxed even if one assumes that n is large enough. Indeed, take G = A_{r}(K) and ω(φ) = aω_{i} +(p−1−a)ω_{i+1}, 0 < i < r. Representations φ are truncated symmetric powers of the standard representation of G. We have l(φ) = p − 1. A.E. Zalesskii and the author [28, Proposition 1.4] have proved that the restrictions of the representations φ to all naturally embedded subgroups of type A are semisimple.

Another topic we want to discuss here is concerned with completely splittable representations. An irreducible representation of a semisimple algebraic group is called completely splittable if its restrictions to all Levi subgroups are semisimple. For instance, such are the representations mentioned in the previous paragraph. Completely splittable representations of the groups A_{r}(K) and GL_{r+1}(K) are determined in [4].

Theorem 3.2 ([4, Theorem 6.2] Let φ be a p-restricted representation of the group A_{r}(K). Assume that a_{j} = 0 for 1 ≤ j < k and for m < j ≤ r, k ≤ m and a_{k}, a_{m} ≠ 0. Put t(φ) = m − k + ∑_{i=k}^{m} a_{i} = m − k + ∑_{i=1}^{r} a_{i}. The representation φ is completely splittable if and only if t(φ) ≤ p.

Let Fr be the Frobenius morphism of A_{r}(K) associated with raising elements of K to the p-th power. It follows from [4, Theorem 3.17] that a representation φ = ⊗_{j=0}^{s}φ_{j} Fr^{j} of A_{r}(K) with p-restricted φ_{j} is completely splittable if and only if all φ_{j} are such. One easily observes that t(φ) ≥ l(φ) for p-restricted φ and that t(φ) > l(φ) if l(φ) ≥ p. So in some cases it could be useful to divide the set of p-restricted representations into three classes: p-small (t(φ) ≤ p), ”middle” (l(φ) < p < t(φ)), and p-large (l(φ) ≥ p).

Acknowledgements

This research has been supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the Belarus State basic research programme ”Mathematical structures” and partially supported by INTAS, project INTAS 93-183 Ext. The conjectures stated here have been discussed in the author’s talk at the NATO Advanced Study Institute ”Modular representations of algebraic and related groups” (Cambridge, Isaac Newton Mathematical Sciences Institute, 1997). The author wishes to thank the organizers of the special programme at the Newton Institute M. Broue, R.W. Carter, and J. Saxl for the possibility to participate at the ASI and the staff of the Newton Institute for their hospitality.

References

[1] N .Bourbaki, ”Groupes et algebres de Lie”, Ch. IV-VI, Hermann, Paris, 1968.

[2] P. Bala and R.W. Carter, Classes of unipotent elements in algebraic groups. I, Math. Proc. Cambridge Phil. Soc., 79 (1976), 401-425.

[3] P. Bala and R.W. Carter, Classes of unipotent elements in algebraic groups. II, Math. Proc. Cambridge Phil. Soc., 80 (1976), 1-17.

[4] J. Brundan, A.S. Kleshchev and I.D. Suprunenko, Semisimple restrictions from GL(n) to GL(n − 1), to appear in J. fur de reine und angew. Math., 1998

[5] R. Carter, Finite groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, Chichester, 1985.

[6] W. Feit, The representation theory of finite groups, North-Holland Publishing Company, Amsterdam/ New York/ Oxford, 1982.

[7] R.M. Guralnick, W.M. Kantor, and J. Saxl, The probability of generating a classical group, Communications in Algebra, 22 (1994), 1395-1402.

[8] W.M. Kantor, Some topics in asymptotic group theory, in: ”Groups, Combinatorics and Geometry. Durham, 1990. London Math. Soc. Lecture Note Ser. 165”, Cambridge Univ. Press, 1992, 403-421.

[9] W.M. Kantor, Random remarks on permutation group algorithms, in:”Groups and Computation. Workshop in Groups and Computation. October 7-10. 1991. DIMACS Ser. in Discrete Math. and Theor. Comput. Sci.”, Providence, Amer. Math. Soc., 1993, Vol. 11, 127-131.

[10] W.M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Dedicata 36 (1990), 67-87.

[11] M.W. Liebeck and A. Shalev, The probability of generating a finite simple group, Geom. Dedicata 56 (1995), 103-113.

[12] M.W. Liebeck and A. Shalev, Simple groups, probabilistic methods, and a conjecture of Kantor and Lubotzky, J. Algebra, 184 (1996), 31-57.

[13] M.W. Liebeck and A. Shalev, Classical groups, probabilistic methods, and the (2,3)- generation problem, to appear in Annals of Mathematics.

[14] G. Malle, J. Saxl, and T. Weigel, Generation of classical groups, Geom. Dedicata 49 (1994), 85-116.

[15] A.A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Math. USSR-Sb. 61(1988), No 1, 167-183, Translated from Mat. Sb., 133 (1987), No 2, 167-183).

[16] L. Pyber, Asymptotic results for permutation groups, in: ”Groups and Computation. Workshop in Groups and Computation. October 7-10. 1991. DIMACS Ser. in Discrete Math. and Theor. Comput. Sci.”, Providence, Amer. Math. Soc., 1993, Vol. 11, 197-219.

[17] J.Saxl, Finite simple groups and finite permutation groups, in: ”Finite and locally finite groups. Proceedings of the NATO ASI on finite and locally finite groups, Istanbul, Turkey, 14-27 August, 1994”, Dordrecht/ Boston/ London, Kluwer Academic Publishers, NATO ASI Series C: Mathematical and Physical Sciences, 1995, Vol. 471, 97-110.

[18] G.M. Seitz, The maximal subgroups of classical algebraic groups, Mem. Amer. Math. Soc., no 365, AMS, Providence, 1987.

[19] S. Smith, Irreducible modules and parabolic subgroups, J. Algebra 75 (1982), 286-289.

[20] R. Steinberg, Representations of algebraic groups, Nagoya Math. J. 22 (1963), 33-56.

[21] R. Steinberg, ”Lectures on Chevalley groups”, Yale University, New Haven, 1968.

[22] I.D. Suprunenko, The invariance of the set of weights of irreducible representations of algebraic groups and Lie algebras of type A_{l} with restricted highest weights under reduction modulo p (in Russian), Vesti Acad Navuk BSSR. Ser. fiz.-mat. navuk (1983), No 2, 18-22.

[23] I.D. Suprunenko, The minimal polynomials of elements of order p in irreducible representations of Chevalley groups over fields of characteristic p, Siberian Advances in Math. 6 (1996), no 4, 97-150, Translated from Problems of Algebra and Logic, Trudy Inst. Math. Sib. Otdel. RAN, Novosibirsk, 30 (1996), 126-163.

[24] I.D. Suprunenko, On Jordan blocks of elements of order p in irreducible representations of classical groups with p-large highest weights, J. Algebra 191 (1997), no 2, 589-627.

[25] I.D. Suprunenko, The minimal polynomials of unipotent elements in irreducible representations of the classical groups in characteristic p with p-large highest weights, Preprint Inst. Exper. Math. Univ. Essen, No 6, 1997, 36 p.

[26] I.D. Suprunenko, Restrictions of modular irreducible representations of the special linear group with large highest weights with respect to characteristic to small natural subgroups are not completely reducible (in Russian, English summary), to appear in Doklady Akad. Nauk Belarusi, 1998, No 3.

[27] I.D. Suprunenko, Minimal polynomials of unipotent elements in irreducible representations of the special linear group, To appear in Acta Applicandae Math., 1998.

[28] I.D. Suprunenko and A.E. Zalesskii, Truncated symmetric powers of the natural realizations of the groups SL_{m}(P) and Sp_{m}(P) and their restrictions to subgroups, Siberian Math. J. 31 (1990), 555-566, Translated from Sibirsk. Mat. Zh. 31 (1990), No 4, 33-46.