Abstract
Let k be an algebraically closed field of characteristic , let G be a simple simply connected classical linear algebraic group of rank and let T be a maximal torus in G with rational character group . For a nonzero p-restricted dominant weight , let V be the associated irreducible kG-module. We define as the minimum codimension of any eigenspace on V for any non-central element of G. In this paper, we determine lower-bounds for for G of type and , and for G of type , or and . Moreover, we give the exact value of for G of type with ; for G of type or with ; and for G of type with .
2020 MATHEMATICS SUBJECT CLASSIFICATION:
1 Introduction
Let k be an algebraically closed field of characteristic , let V be a finite-dimensional k-vector space and let H be a group acting linearly on V. For denote by the eigenspace corresponding to the eigenvalue of h on V. Set
In [Citation2], one can find the classification of groups H acting linearly, irreducibly and primitively on a vector space V (over a field of characteristic zero) that contain an element h for which is small when compared to . The following year, Hall, Liebeck and Seitz, [Citation6], expanded on Gordeev’s result by working over algebraically closed fields of arbitrary characteristic, and they proved that, in the case of linear algebraic groups, if H is classical, we have , where is the rank of H and V is a faithful rational irreducible kH-module of dimension n; while, if H is not of classical type, then . Now, with the lower-bounds for known, the following natural step was to start the classification of pairs (H, V) with bounded from above, in particular the pairs (H, V) with or have been of great interest, see for example [Citation9, Citation10, Citation23]. In [Citation4], the irreducible subgroups H of , where V is a finite-dimensional k-vector space of dimension n > 1, which act primitively and tensor-indecomposably on V and have been classified.
Let G be a simple simply connected classical linear algebraic group of rank with over k and let V be a nontrivial rational irreducible tensor-indecomposable kG-module. In this paper, we determine in the following cases:
G is of type with and ;
G is of type with and ;
G is of type with and ;
G is of type with and .
Moreover, for the groups of smaller rank and their corresponding irreducible tensor-indecomposable modules with dimensions satisfying the above bounds, we improve the known lower-bounds for (see [Citation4, Theorem 8.4]). The origin of this paper is the PhD thesis of the author, in which the classification of pairs (G, V) with was established. We now state the main results of this paper. The notation used will be introduced in Section 2.
Theorem 1.1.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected classical linear algebraic group of rank over k. When G is of type , we assume that . Let T be a maximal torus in G and let , where λ is a nonzero p-restricted dominant weight. When G is of type with assume that . In all other cases (G of type with ; with ; with ) assume that . The value of is as given .
In the following section, we fix the notation and terminology used throughout the text. In Section 3 we go over preliminary results, we establish an algorithm for calculating , and, for each type of classical group G, we determine the complete list of kG-modules that are candidates for Theorem 1.1.
The proof of Theorem 1.1 is given in Sections 4–7, where each section is dedicated to one of the types of classical groups.
2 Notation
Throughout the text k is an algebraically closed field of characteristic . Note that when we write , for some prime p0, we allow p = 0. Let G be a simple simply connected classical linear algebraic group of rank and let T be a maximal torus in G with rational character group . Let be the group of rational cocharacters of T and let be the perfect natural pairing on . We denote by the root system of G corresponding to T and by a set of simple roots in , where we use the standard Bourbaki labeling, as given in [Citation7, 11.4, p.58]. Let be the set of positive roots of G. Following [Citation1, Section 2.1], we fix a total order on : for we have if and only if , or with , and ar > 0.
For each , let be its corresponding coroot, let be its corresponding root subgroup and let be an isomorphism of algebraic groups with the property that for all and all . Let Gs be the set of semisimple elements of G and let Gu be the set of unipotent elements of G. Any can be written , where , respectively any can be written as , where and the product respects . Lastly, let B be the positive Borel subgroup of G, be the Weyl group of G corresponding to T, and be the longest word.
The set of dominant weights of G with respect to Δ is denoted by , and the set of p-restricted dominant weights by . We adopt the usual convention that when p = 0, all weights are p-restricted. For , we denote by the irreducible kG-module of highest weight λ. Further, we denote by ωi, , the fundamental dominant weight of G with respect to αi.
All representations and all modules of a linear algebraic group are assumed to be rational and nonzero. For a kG-module V we will use the notation to express that V has a composition series with composition factors . Further, by Vm we denote the direct sum , in which V occurs m times. When p > 0, we denote by the kG-module obtained from V by twisting the action of G with the Frobenius endomorphism i times, see [Citation16, Section 16.2]. Lastly, we will denote the natural module of G by W.
For , we define by if and if .
3 Preliminary results
To begin, we prove the following result which gives us the strategy we will use for calculating .
Proposition 3.1.
Let G be a simple linear algebraic group and let V be an irreducible kG-module. Then:
Proof.
Let be the associated representation and let . We write the Jordan decomposition of , where and . By [Citation16, Theorem 2.5], is the Jordan decomposition of in . We choose a basis of V with the property that is written in its Jordan normal form. Then, with respect to this basis, is the diagonal matrix whose entries are just the diagonal entries of , while is the unipotent matrix obtained from the Jordan normal form of by dividing all entries of each Jordan block by the diagonal element. We distinguish the following two cases:
Case 1: Assume . First, we remark that , as . Secondly, as , it follows that for some . Thereby, c is the sole eigenvalue of on V and we have .
Case 2: Assume . Then, since is a diagonal matrix with entries the diagonal entries of , we determine that and have the same eigenvalues on V and, for any such eigenvalue we have , where the last equality follows by [Citation16, Corollary 4.5 and Theorem 4.4]. □
3.1 Group isogenies and irreducible modules
In this section, we will assume that the simple algebraic group G is not simply connected, and we let be its simply connected cover. Fix a central isogeny with and . Let be a maximal torus in with the property that and, similarly, let be the Borel subgroup of given by . Let . Since , we will denote by the weight λ when viewing it as an element of . By [Citation8, II.2.10], as , it follows that . Moreover, by the same result, we have that is a simple -module and as -modules.
Lemma 3.2.
We have and . In particular, we have .
Proof.
Let and let be an eigenvalue of on . Let and note that . Denote by the eigenvalue of on corresponding to under . We have that:
Lastly, let be such that . As the map is surjective, let be an arbitrary preimage of in and be the eigenvalue of on corresponding to under . Then:
□
The following result justifies why we only treat groups of type and their respective modules over fields of characteristic different to 2.
Lemma 3.3.
Let p = 2. Let B, respectively C, be a simple simply connected linear algebraic group of type , respectively of type . Let , respectively , be the fundamental dominant weights of B, respectively of C. Then, for any 2-restricted dominant weight of B we have that:
Proof.
As p = 2, there exists an exceptional isogeny between the two groups, see [Citation21, Theorem 28]. Consequently, we can induce irreducible kC-modules from irreducible kB-modules by twisting with the isogeny . Therefore, we have:
□
Remark 3.4.
In view of Lemma 3.3, for any 2-restricted dominant weight of B, we have , where . Similarly, for the weight , we have . Lastly, in the case of weights of the form , where and there exists such that di = 1, in order to determine it suffices to calculate .
3.2 Restriction to Levi subgroups
We return to the situation where G is simply connected. For each , let Pi be the maximal parabolic subgroup of G corresponding to , and let be a Levi subgroup of Pi. The root system of Li is , in which Δi is a set of simple roots. Now, we have that , where is a one-dimensional subtorus of G and is a semisimple simply connected linear algebraic group of rank , see [Citation16, Proposition 12.14]. Lastly, let be a maximal torus in , contained in the Borel subgroup . We will abuse notation and denote the fundamental dominant weights of Li corresponding to Δi by .
Let , let be the associated irreducible kG-module, and let be the set of weights of V. Fix some . We say that a weight has αi-level j if , where . The maximum αi-level of weights in V will be denoted by . By [Citation8, II, Proposition 2.4(b)], we have that is equal to the αi-level of . Now, consider the Levi subgroup Li of Pi. For each , define the subspace of V and note that Vj is invariant under Li. Then, as a -module, V admits the following decomposition: where, by [Citation20, Proposition], is the irreducible -module of highest weight .
Lemma 3.5.
Assume V is a self-dual kG-module. Then, for all , we have , as -modules.
Proof.
We note that, as V is self-dual, we have and V is equipped with a nondegenerate bilinear form . Let be such that . Let and . Then . Therefore , as , and so . Moreover, as is nondegenerate, it follows that for all .
Secondly, let be a weight of αi-level j, where . We will show that has αi-level . On one hand, we know that is equal to the αi-level of , thus , where . On the other hand, as , for , we have , where for all . Thus, has αi-level equal to . In particular, as for all , it follows that .
Lastly, as is self-dual, it follows that . Furthermore, as V is equipped with a nondegenerate bilinear form, we have that , for all . As , it follows that . By the same argument, this time applied to , we determine that . Therefore, , thus , and we conclude that . □
Remark 3.6.
Applying Lemma 3.5, let , where . As , it follows that V is self-dual if . Thus, for groups of type , V is self-dual if for all . For groups of type and , as , all irreducible kG-modules are self-dual. Lastly, for groups of type with even, all irreducible kG-modules are self-dual, while for groups of type with odd, V is self-dual if .
In what follows, we give a formula for , the maximum α1-level of weights in , for the classical linear algebraic groups. Further, for groups of type , we also give a formula for .
Lemma 3.7.
Let G be of type and let . Then .
Proof.
In order to determine we have to calculate the α1-level of . We have that (3.1) (3.1)
Using [Citation7, , p. 69], we write the ωi’s, , in terms of the simple roots αj, , and we see that for , we have ; and if is odd, we have . Substituting in (3.1), we determine that . □
Lemma 3.8.
Let G be of type and let . Then and .
Proof.
Note that we have , hence . We write the ωi’s, , in terms of the simple roots αj, , see [Citation7, , p. 69], and we get:
We remark that the coefficient of each αi is a nonnegative integer and the result follows. □
Lemma 3.9.
Let G be of type and let . Then .
Proof.
We have that , hence . Writing the fundamental dominant weights ωi in terms of the simple roots αj, we see that: therefore . □
Lemma 3.10.
Let G be of type and let . Then .
Proof.
We first assume that is even. Then , hence , and so
Thus . We now assume that is odd. We note that , for all , and . It follows that: and so . □
3.3 The algorithm for calculating
Let for some . Consider the restriction , where and , for all . In view of Proposition 3.1, in order to determine , one has to calculate and . In this section we will outline an algorithm for calculating and .
First, let . Then, in particular, and so , where and . As and is a one-dimensional torus, there exists and , such that . Moreover, we have for all , . On the other hand, as , we have , where for all . Now, as , z acts on each Vj, , as scalar multiplication by , where: (3.2) (3.2)
Lastly, let , be the distinct eigenvalues of h on Vj, , and let be their respective multiplicities. Then, as , it follows that the eigenvalues of s on Vj are and they are distinct, as the ’s are, with respective multiplicities . This proves the following:
Lemma 3.11.
Let with and . Let , be the distinct eigenvalues of h on Vj, , with respective multiplicities . Then:
z acts on Vj as scalar multiplication by , where is given in (3.2);
the distinct eigenvalues of s on Vj are , with multiplicities ;
the eigenvalues of s on V are , with respective multiplicities at least .
An algorithm for calculating
First, assume that s admits an eigenvalue μ on V with the property that for some . In this case , and so it acts on each Vj as scalar multiplication by . Therefore, the eigenvalues of s on V, not necessarily distinct, are . We also remark that the ’s are not all equal, as . We have that , where is such that for any we have . Therefore, , where the calculation of the latter maximum is straightforward. Secondly, assume that for all eigenvalues μ of s on V and all . This case is solved inductively. We write , where and . Recall that is a semisimple simply connected group of rank . We have that , where and for all . Therefore, , and we use induction to determine this upper-bound.
We now let . We will denote by the group algebra of over k.
Lemma 3.12.
Let be a unipotent element and let V be a finite-dimensional kG-module. Let , where , be a filtration of -submodules of V. Then:
Moreover, suppose that for each i, we have a u-invariant decomposition with as -modules. Then .
Proof.
For each , we fix a basis in Mi with the property that the matrix associated to the action of u on is upper-triangular. Then, the matrix of the action of u on V is the block upper-triangular matrix:
Using , we calculate the matrix of the action of on V: where is the matrix of the action of on , , with respect to the basis of Mi we have previously fixed. It follows that: and, consequently, we have . Now, as we determine that .
Lastly, for all , assume that there exists a -submodule of Mi such that . Then , and so there exists a basis of V with the property that:
thereby . Arguing as above, we establish that . □
An algorithm for calculating
Let be the Levi decomposition of the maximal parabolic subgroup Pi of G. Let , , where the product respects the total order on and . Now, as and , it follows that u admits a decomposition , where each of the products respects and , for all . We set and , and we note that and . Recall that for some and that . Let , with corresponding weight space , and let . As , see [Citation16, Lemma 15.4], we have and . Therefore, V admits a filtration of -submodules, where for all . We see that u acts on each , , as and so, by Lemma 3.12, we determine that . Therefore, if we identify the kLi-composition factors of each Vj, , then using already proven results and Lemma 3.12, we can establish an upper-bound for each . Now, assuming that , the upper-bound we obtain for , hence for , will be strictly smaller than . Lastly, we remark that if , i.e. , then , for all , and thus, by Lemma 3.12, it follows that .
We end this section with two lemmas concerning the behavior of unipotent elements. The first one is due to Guralnick and Lawther, [Citation5], and tells us which unipotent conjugacy classes in G afford the largest dimensional eigenspaces.
Lemma 3.13.
[Citation5, p.19 and Lemmas 1.4.1 and 1.4.4] We have , if belongs to a unipotent conjugacy class of root elements and belongs to any nontrivial unipotent class, unless and one of the following holds:
, p = 2, u1 belongs to the unipotent conjugacy class of and u2 belongs to the unipotent conjugacy class of .
, u1 belongs to the unipotent conjugacy class of and u2 belongs to the unipotent conjugacy class of .
, u1 belongs to the unipotent conjugacy class of and u2 belongs to the unipotent conjugacy class of .
The second lemma gives us , when , V is a finite-dimensional k-vector space and the unipotent element u acts as a single Jordan block on . For each , let Vi be the indecomposable -module with and on which u acts as the full Jordan block Ji of size i. Note that is a set of representatives of the isomorphisms classes of indecomposable -modules.
Lemma 3.14.
Let k be a field of characteristic p = 2 and let V be a vector space of dimension over k. Let u be a unipotent element acting as a single Jordan block in . Then .
Proof.
We will prove the result by induction on . First, we note that both cases i = 1 and i = 2 follow directly from the structure of . Hence, we assume that and that the result holds for all . Let m be the unique nonnegative integer for which and set . Now, up to isomorphism, there exist exactly q indecomposable -modules: , where and u acts on Vj as Jj. Therefore, as -modules, we have . Now, by [Citation3, Theorem 2], we have , and so (3.3) (3.3)
As and as u acts as a single Jordan block on Vq and , respectively, it follows that and . Furthermore, we note that, as , we have and, by applying induction, it follows that . Substituting in (3.3) we obtain . □
3.4 The list of modules
Lemma 3.15.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected classical linear algebraic group. Let .
Let G be of type with and assume that . If , then, up to duality of the corresponding kG-module, we have . If , the additional λ’s are given in .
Let G be of type with and assume that . If , we have that . If , the additional λ’s are given in .
Let and let G be of type with . Assume that . If , we have . If , the additional λ’s are given in .
Let G be of type with and assume that . If , we have . If , the additional λ’s are given in (up to duality or outer automorphisms of the corresponding kG-module).
Proof.
The result follows by [Citation17, Theorem 1.2] and [Citation14]. □
4 Proof of Theorem 1.1 for groups of type
Let G be a simple simply connected linear algebraic group of type with . In view of Proposition 3.1, it is sufficient to know and in order to determine , where V is any irreducible kG-module. In this section we prove Theorems 4.1 and 4.2 which provide the values of and for all kG-modules with , and . As a corollary, the part of Theorem 1.1 concerning simple simply connected linear algebraic groups of type will follow.
Theorem 4.1.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected linear algebraic group of type with . Let , where and , be such that . The value of is as given in .
Proof.
To begin, recall that we have denoted by W the natural module of G, see end of Section 2. Let be a triplet featured in . In view of Lemma 3.13, for any , we have . Thus, in what follows, we will focus on calculating . To ease notation, we reference each triplet featured in by its corresponding number in column ‘Ref’.
4.1.1: Follows because as kG-modules, and acts on W as .
4.1.2: Note that by [Citation18, Proposition 4.2.2]. We write , where and acts on W1 as J2; and and acts trivially on W2. Since , we get . As acts as a single Jordan block on , using [Citation15, Lemma 3.4], respectively Lemma 3.14 when p = 2, we get . Further, as acts trivially on W2, we have . Lastly, as acts on as , we have , by [Citation15, Lemma 3.4]. It follows that .
4.1.3: Note that by [Citation18, Proposition 4.2.2]. We write , where and acts on W1 as J2; and and acts trivially on W2. Then, since , we argue as above to show .
4.1.4: Follows from [Citation18, Proposition 4.6.10] and [Citation11, Theorem 6.1], arguing as in 4.1.2 and 4.1.3.
4.1.5, 4.1.6, 4.1.8, and 4.1.9: Follow from [Citation18, Proposition 4.2.2], arguing as in 4.1.2 and 4.1.3.
4.1.7: First, assume . By [Citation18, Proposition 4.6.10], we have that , and so . As acts on as and on as , see 4.1.2, one shows that . The result now follows by 4.1.1, 4.1.2, and 4.1.5.
Now, let p = 3. Set and . By Lemma 3.7, we have , therefore . First, by [Citation20, Proposition], we have . Secondly, the weight admits a maximal vector in V1, thus V1 has a composition factor isomorphic to . Lastly, the weight admits a maximal vector in V2, thus V2 has a composition factor isomorphic to . By dimensional considerations, we determine that
By 4.1.1 and 4.1.3, we have . Recursively and using 4.1.5, for the base case of , we get .
□
Theorem 4.2.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected linear algebraic group of type with . Let , where and , be such that . The value of is as given in .
In order to prove Theorem 4.2, in addition to the algorithm for calculating outlined in Section 3.3, we will sometimes use the following algorithm from [Citation5, Section 2.2], which gives lower bounds for . We give a brief description of it in what follows. Let be a standard subsystem of and let be its Weyl group. We define . For , set , and for , define . By [Citation5, Prop. 2.2.1], we have . Now, as , it will prove extremely useful to give a formula for , . To this end, we first note that , is of type , thus and . Moreover, as , we have , therefore (4.1) (4.1)
Let . To improve readability, we will use the expression “s is as in ” to mean that s satisfies the following: , where for all i < j, and . Note that any is as in .
Proposition 4.3.
Let . Then , where the maximum is attained if and only if s is conjugate to with and .
Proof.
As as kG-modules, for all we have . Now, equality holds if and only if s and μ are as in the statement of the result. □
Proposition 4.4.
Let and let . Then , where the maximum is attained if and only if
and s is conjugate to with , and .
and s is conjugate to with and , and .
and s is conjugate to with , and .
Proof.
Let be as in . As , see [Citation18, Proposition 4.2.2], we deduce that the eigenvalues of s on V, not necessarily distinct, have one of the following forms: (4.2) (4.2)
In order for s to have an eigenvalue of the form , there has to exist some with . Suppose there exists such i, and consider the eigenvalue of s on V. Now, since the μr’s are distinct, it follows that for all , hence:
Let and assume . Then , which does not hold as . Thus, we let and assume that . Then and, since , the inequality holds if and only if . Hence, m = 2, and , as and . This gives for all , where equality holds if and only if i = 1 and s is conjugate to with , as in (3). This completes the case of eigenvalues of the form of s on V.
Fix and consider the eigenvalue of s on V. Since the μr’s are distinct, we remark that:
By (4.2), these account for at least eigenvalues of s on V different to . It follows that:
As in the previous case, we begin with . Then, since , we have . Assume that . Then , which holds if and only if , i.e. m = 2, , as , and , as . Thus for all and all i < j, where equality holds if and only if s is conjugate to with , as in (1). We now let and assume . Then: (4.3) (4.3)
Since , we have and , therefore, by inequality (4.3), it follows that . If , then, for inequality (4.3) to hold, we must have , hence , contradicting . On the other hand, if , then m = 2 and, by (4.3), we determine that . Now, the inequality holds if and only if and , as . In this case, and with and . Therefore, for all and all i < j, where equality holds if and only if and s is conjugate to with and , as in (2). □
Proposition 4.5.
Assume and let . Then , where the maximum is attained if and only if s is conjugate to with , and .
Proof.
Let be as in . Since, , see [Citation18, Proposition 4.2.2], the eigenvalues of s on V, not necessarily distinct, have one of the following forms: (4.4) (4.4)
Fix some i and consider the eigenvalue of s on V. Since the μr’s are distinct, we deduce that:
Assume . Then , which holds if and only if . In both cases, we get , where equality holds if and only if for all , i.e. m = 2 and with .
Fix i < j and consider the eigenvalue of s on V. Since the μr’s are distinct, we have
By (4.4), these account for at least eigenvalues of s on V which are different to . Hence, we have . Assume . Then: (4.5) (4.5)
As , we have , therefore . If , then and inequality (4.5) does not hold. If , then m = 2, and inequality (4.5) does not hold, as . Therefore, for all and all i < j. We conclude that . □
Proposition 4.6.
Let and . Then , where the maximum is attained if and only if
and s is conjugate to with .
, μ = 1 and s is conjugate to with .
, μ = 1 and s is conjugate to with .
Proof.
First, we note that if , then , while, if , then , see [Citation18, Proposition 4.6.10]. Let be as in . We determine that the eigenvalues of s on V, not necessarily distinct, have one of the following forms:
We first consider the eigenvalue 1 of s on V. Since the μi’s are distinct, it follows that:
Assume . Since , we have that: (4.6) (4.6) which holds if and only if either , m = 2, and ; or , m = 2, and . In both cases, as and , we get and . Thus, for all and equality holds if and only if p, , s and μ are as in (2), or (3).
We now fix i < j and consider the eigenvalue of s on V. In the case when , one shows that . We thus assume that and . Since the μr’s are distinct, we remark that:
By (4.6), it follows that . Assume . Then: (4.7) (4.7) and, for it to hold, we must have , i.e. m = 2 and . Substituting in (4.7) gives , and we get and . One checks that the inequality holds only for . Thus, we have , where equality holds if and only if p, , s and μ are as in (1). □
Proposition 4.7.
Let and . Then .
Proof.
Set and . By Lemma 3.7, we have , therefore . By [Citation20, Proposition], we have and, since the weight admits a maximal vector in V1, by dimensional considerations, it follows that: (4.8) (4.8)
Let . If for some eigenvalue μ of s on V, where i = 0, 1, then and acts on Vi as scalar multiplication by . As , we have for all eigenvalues μ of s on V. We thus assume that for all eigenvalues μ of s on V and for both i = 0, 1. We write , where and . First, let . Using (4.8) and Proposition 4.4, we determine that for all μ. Therefore, . We now assume that . By (4.8) and Proposition 4.4, we have . Recursively and using the result for , we get for all eigenvalues μ of s on V. □
Proposition 4.8.
Assume and let . Then .
Proof.
We will apply the algorithm from [Citation5, Section 2.2] (described earlier in this section) to determine a lower bound for . Afterwards, we will show that this bound is attained. By [Citation14], the sub-dominant weights in V are and ω3. Therefore, . As is of type , we have and , hence . Lastly, using (4.1), we get , therefore .
By Lemma 3.7, we have , therefore . Now, we argue as we did in the proof of Proposition 4.7 to determine the composition factors of each Vi. It will follow that and . Let . We note that s acts on each Vi as scalar multiplication by . For , we have , therefore . Since , it follows that . □
Proof of Theorem 4.2.
To ease notation, we will reference each triplet featured in by its corresponding number in column “Ref”. Note that 4.2.1–4.2.6 have been solved in Propositions 4.3–4.8. The remaining cases are more straightforward: one has to first determine the structure of and then apply the algorithm of Section 3.3. In what follows, we will only indicate the results that are used in the inductive step of the algorithm and mention if there are small special cases to consider.
4.2.7: If p = 3, the proof is analogous to that of Proposition 4.8. Thus, we assume . The result follows recursively. The base case of follows by Propositions 4.3, 4.5, and 4.6. Note that one has to treat the case when , where and is conjugate to with , separately. For , by Propositions 4.3, 4.4, and 4.5, we have . Recursively, it follows that .
4.2.8: Follows by Proposition 4.7.
4.2.9: Follows by 4.2.8. □
4.1 Supplementary results
At this point, we have completed the proofs of Theorems 4.1 and 4.2. However, we will require additional result for the groups of type in the proofs of Theorems 5.1, 5.2, and 7.1, 7.2, respectively. We collect these in Theorems 4.9 and 4.10.
Theorem 4.9.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected linear algebraic group of type with . Let , where and . We have that
Proof.
Let be a triplet listed in . Let and let μ be an eigenvalue of s on V. To calculate upper-bounds for in cases 4.3.2–4.3.8, one will first determine the structure of , and then apply the algorithm of Section 3.3. Because the proofs are this straightforward, we only indicate the results that are used in the inductive step of the algorithm and mention if there are small special cases to consider. To ease notation, we will reference each triplet by its corresponding number in column “Ref”.
4.3.1: The eigenvalues of s on V, not necessarily distinct, are , where and, consequently, for all .
Assume m is even. If , then , as V is self-dual. Let μ = 1. If for some , then, as and , at most of the eigenvalues can equal 1, where . We deduce that . If , then at most of the eigenvalues can equal –1, where . We deduce that . The case of m odd is analogous.
To show that equality holds, consider , where . Then, since , we have .
4.3.2: When p = 2 and , where and , one shows that the eigenvalues of s on V have the form: , c2, and , where and . Therefore . When , the result follows by Propositions 4.3, 4.5, and 4.8.
4.3.3: When with and , one has to show that for all . The result then follows by Propositions 4.3, 4.5, and 4.8 and 4.3.1.
4.3.4: When p = 2 and with and , one has to show that for all μh of h. The result then follows by Propositions 4.3, 4.5, 4.6, and 4.3.2.
4.3.5: Follows by Propositions 4.5 and 4.6.
4.3.6: Follows by Proposition 4.8 and 4.3.2.
4.3.7: When with and , one has to show that the result holds in two special cases. First case happens when and h conjugate to with . This is solved by showing that the eigenvalues of h on V1, respectively on V2, are ±1 with and , respectively with and . The second case occurs when p = 2 and, by Proposition 4.3, 4.3.2 and the structure of V1, we have . One has to show that equality does not hold. If it did, then, by Proposition 4.3, h would be conjugate to with , and we would have , where . However, since p = 2, we get , a contradiction. Outside the two special cases, the result follows by Propositions 4.3, 4.5, 4.8, 4.6, and 4.3.2.
4.3.8: Follows by Propositions 4.3, 4.5, 4.6, 4.3.2, and 4.3.3. □
Theorem 4.10.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected linear algebraic group of type with . Let , where and . We have that
Proof.
Let be a triple listed in . To begin, note that, by Lemma 3.13, we have . Now, to calculate we use the structure of , Lemma 3.12 and the algorithm of Section 3.3. Since the proofs are very similar to the ones of Theorem 4.1, we will only mention the results used in the inductive step of the algorithm.
4.4.1: Follows from [Citation22, Theorem 1.9].
4.4.2: If p = 2, we have , and acts as on both and . If , Theorem 4.1[4.1.1, 4.1.3, 4.1.6] gives the result.
4.4.3: Follows by Theorem 4.1[4.1.1, 4.1.3, 4.1.6] and the fact that .
4.4.4: Follows by Theorem 4.1[4.1.1, 4.1.3, 4.1.7] and 4.4.2.
4.4.5: Follows by Theorem 4.1[4.1.3, 4.1.4].
4.4.6: Follows by Theorem 4.1[4.1.6] and 4.4.2.
4.4.7: When , it follows by Theorem 4.1[4.1.1, 4.1.3, 4.1.4, 4.1.6] and 4.4.2. When p = 2, one shows that acts on as , and the result follows by Theorem 4.1[4.1.1] and 4.4.2.
4.4.8: Follows by Theorem 4.1[4.1.1, 4.1.3, 4.1.4], 4.4.2 and 4.4.3. □
5 Proof of Theorem 1.1 for groups of type
Let G be a simple simply connected linear algebraic group of type with . This section is dedicated to Theorems 5.1 and 5.2, which give the values of and for all kG-modules with , and . As a corollary, the part of Theorem 1.1 concerning simple simply connected linear algebraic groups of type will follow.
Theorem 5.1.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected linear algebraic group of type with . Let , where and , be such that . The value of is as given in .
Proof.
Recall that we have denoted by W the natural module of G. To begin, let be a triplet listed in . Remark that, by Lemma 3.13, we have , where . For 5.1.7 - 5.1.31, to calculate , we use: the structure of either one, or both of and ; Lemma 3.12 and the algorithm of Section 3.3. When the proofs are very similar to ones of Theorem 4.1, we will only mention the results used in the inductive step.
5.1.1: Note that , respectively , acts on W as , respectively as .
5.1.2: Note that , if , while if , we have , see [Citation18, Lemma 4.8.2]. By the proof of Theorem 4.1[4.1.2] we get . To calculate , we write , where and acts as on W1; and and acts trivially on W2. Using [Citation15, Lemma 3.4] and Lemma 3.14, we determine that . Lastly, by [Citation18, Lemma 4.8.2] and [Citation11, Corollary 6.2], or [Citation12, Theorem B] if p = 2, we determine that and .
5.1.3: Follows by [Citation18, Proposition 4.2.2] and Theorem 4.1[4.1.3].
5.1.4: The result for follows by 5.1.1 and 5.1.2. Assume . We note that if , we have , while, if , then , as kG-modules, see [Citation18, Lemma 4.8.2]. For , we argue as in 5.1.2 to calculate , and we deduce . For , using Theorem 4.1[4.1.5], one shows that , where equality holds for . To show that equality also holds for , we determine the structure of and use 5.1.2.
5.1.5: Follows by [Citation18, Proposition 4.2.2], arguing as in 5.1.2.
5.1.6: Let . When , the result follows by Theorem 4.1[4.1.1, 4.1.3]. When p = 2, we have , as kG-modules, see [Citation19, (1.6)]. Using the Jordan form of , i = 1, 2, on and [Citation12, Theorem B], one shows that .
Let . If p = 3, the result follows by 5.1.1, 5.1.3 and the result for . Assume . First, we argue recursively, using 5.1.1, 5.1.2, 5.1.3 and the result for , to show that . Lastly, assume that . By [Citation18, Lemmas 4.9.1 and 4.9.2], we know that V is a composition factor of the kG-module , where . Note that by we understand the character of the Weyl kG-module of highest weight . Therefore, in view of [Citation18, Lemmas 4.8.2 and 4.9.2], we have for all . For , it follows that .
5.1.7: Follows by Theorem 4.1[4.1.1, 4.1.3, 4.1.6].
5.1.8: Follows by Theorems 4.1[4.1.1, 4.1.3, 4.1.6] and 4.10[4.4.1].
5.1.9: Because we will need to know also when , we will not limit ourselves to the case p = 7 and, instead, we will only assume that . In this case, by Theorems 4.1[4.1.3, 4.1.6] and 4.10[4.4.1], we get , where equality holds for .
5.1.10: As above, we will only assume that . Then, by Theorems 4.1[4.1.1, 4.1.3, 4.1.6] and 4.10[4.4.1], we deduce that .
5.1.11: Follows by 5.1.1, 5.1.2, 5.1.3, and 5.1.6.
5.1.12: Follows by 5.2.6 and 5.2.7.
5.1.13: As in 5.1.9, we only assume . Then, by 5.1.3, 5.1.6, and 5.1.7, we determine that , where equality holds for .
5.1.14: Again, we only assume . Then, by 5.1.1, 5.1.2, 5.1.3, 5.1.6, and 5.1.7, we determine that .
5.1.15: Follows by 5.1.2 and 5.1.4.
5.1.16: Follows recursively, using 5.1.11 and 5.1.30.
5.1.17: Follows by 5.1.4, 5.1.6 and 5.1.11.
5.1.18: Follows by 5.1.1, 5.1.2, 5.1.6, and 5.1.11.
5.1.19: Follows by 5.1.2, 5.1.4, 5.1.15, and 5.1.30.
5.1.20: Follows by 5.1.4 and 5.1.15.
5.1.21: Follows by 5.1.2, 5.1.4 and 5.1.19.
5.1.22: Follows by 5.1.4, 5.1.19, 5.1.20, and 5.1.30.
5.1.23: Follows by 5.1.19 and 5.1.20.
5.1.24: Follows by 5.1.2, 5.1.4 and 5.1.21.
5.1.25 and 5.1.26: Follow by 5.1.22 and 5.1.23.
5.1.27: Follows by 5.1.21 and 5.1.22.
5.1.28: Follows by 5.1.2, 5.1.4, and 5.1.24.
5.1.29: Follows by 5.1.2, 5.1.4, and .
5.1.30: Follows recursively, using 5.1.4.
5.1.31: Note that and , by [Citation19, (1.6)]. The result follows by 5.1.11 and 5.1.16. □
Theorem 5.2.
Let k be an algebraically closed field of characteristic and let G be a simple simply connected linear algebraic group of type with . Let , where and , be such that . The value of is as given in .
Let . As in Section 4, to improve readability, we will say “s is as in ” to express the fact that s satisfies the following: , where for all i < j, and ; and if m = 1, then . Note that any is as in .
Proposition 5.3.
Let . Then , where the maximum is attained if and only if and s is conjugate to with .
Proof.
The proof is analogous to that of Proposition 4.3. □
Proposition 5.4.
Let . Then
and , where the maximum is attained if and only if
(1.1) and s is conjugate to .
(1.2) p = 2, μ = 1 and s is conjugate to with .
(1.3) p = 2, and s is conjugate to with .
and , where the maximum is attained if and only if
(2.1) , μ = 1 and s is conjugate to with .
(2.2) and s is conjugate to .
and , where the maximum is attained if and only if
(3.1) and s is conjugate to .
(3.2) p = 2, μ = 1 and s is conjugate to with .
(3.3) p = 2, μ = 1 and s is conjugate to with .
and , where the maximum is attained if and only if μ = 1 and s is conjugate to with .
Proof.
To ease notation, define , in the following way: ; ; ; and for all . Let be as in . Using the structure of as a kG-module, see [Citation18, Lemma 4.8.2], we deduce that the eigenvalues of s on V, not necessarily distinct, have one of the following forms: (5.1) (5.1)
Let be an eigenvalue of s on V. If , then, as V is self-dual, we have , where equality holds if and only if , p, s and μ are as in (1.3). Thus, for the remainder of the proof, we assume that .
Let m = 1, i.e. and . Using (5.1), we show that , where equality holds if and only if , p, s and μ are as in (1.2), (2.1), and (3.2). Thus, for the remainder of the proof, we assume that .
If , then m = 2, i.e. . Using (5.1), we determine that , where equality holds if and only if p, s and μ are as in (1.1). If , then . If m = 2, i.e. and , then where equality holds if and only if p, s and μ are as in (2.2). If m = 3, by (5.1), we get ; and . Further, as –1 can equal at most one eigenvalue of the form and at most one of the form , we deduce that . For the remainder of the proof, we assume that .
For the eigenvalue μ = 1, as for all , we have for all i < j, thus: (5.2) (5.2)
Let and assume that . It follows that, and, as and for all r < q, we get . Thus, holds if and only if p = 2, m = 2, and . Substituting in (5.2), gives , where equality holds if and only if s is as in (3.3). We now let and assume that . By (5.2), we get: (5.3) (5.3)
As , it follows that . But and , as for all , therefore inequality (5.3) holds if and only if m = 2, and . Substituting in (5.2), gives , where equality holds if and only if all eigenvalues of s on V different to are equal to 1, i.e. if and only if s is as in (4).
Lastly, let . We remark that , see (5.1). If for all i < j, we have . We thus assume that there exist i < j such that . Then and, since the μi’s are distinct, we have that: (5.4) (5.4)
By (5.1), all of the above account for at least additional eigenvalues of s on V different to –1. This gives: (5.5) (5.5)
Let and assume . Then and we have . For the inequality does not hold. For we get , see (5.5), where equality holds if and only if s is as in . For we have , see (5.5). Since we are assuming , we must have nr = 1 for all , therefore m = 4 and ni = 1 for all . Substituting in (5.5) gives , where equality holds if and only if all eigenvalues of s on V different to 1 and the ones listed in (5.4) are equal to –1. However, as at most one eigenvalue of the form can equal –1, we see that the condition for equality cannot be satisfied. This completes the case of . Thus, let and assume . It follows that: (5.6) (5.6)
If , then, as , we have m = 2 and so . Substituting in (5.6) gives , which does not hold as . If , then, for (5.6) to hold, we must have , hence . If , inequality (5.6) does not hold. If , substituting in (5.6) gives . One checks that this inequality holds if and only if , nr = 1 for all , and when . In both cases, we can assume without loss of generality that . As the μr’s are distinct, at most one eigenvalue of each of the forms , and , can equal –1. This gives an additional eigenvalues of s on V that are different to –1. Consequently, we have . This completes the proof. □
Proposition 5.5.
Assume and let . Then , where the maximum is attained if and only if
and s is conjugate to with .
, μ = 1 and s is conjugate to .
Proof.
Let be as in . We note that , see [Citation18, Proposition 4.2.2], therefore the eigenvalues of s on V, not necessarily distinct, have one of the following forms: (5.7) (5.7)
The cases of ; with m = 1; and with and μ = 1 are handled as in the proof of Proposition 5.4. The only case we have left to consider is with and . We note that . If for all i < j, then, by (5.7), there are at least additional eigenvalues of s on V different to –1. This gives: (5.8) (5.8)
Therefore , where equality holds if and only if and all eigenvalues of s on V different to 1, and are equal to –1. But then, by (5.7), we must have , a contradiction. We thus assume that there exist i < j such that . Then , and:
By (5.7), these amount to at least additional eigenvalues of s on V different to –1. This gives: (5.9) (5.9)
If , it follows that
which does not hold. This completes the proof of the proposition. □
Proposition 5.6.
Let and . Then .
Proof.
First, one determines the structure of and then applies the algorithm from Section 3.3, using Theorem 5.2[5.2.4 for ] and Propositions 5.4 and 5.5. For p = 3, it follows that , while, for , we get and that there exist with . In what follows we will show that .
Assume there exist with . Then , and we write with and . If , by Proposition 5.4, h must be conjugate to one of . In all cases, one shows that for all μ. Let p = 2. In view of Proposition 5.4, assume h is conjugate to one of . Using the weight structure of , one shows that for all μ. On the other hand, if h belongs to a different conjugacy class, then by Propositions 5.4, 5.5 and the weight structure of V0, it follows that for all μ. □
Proposition 5.7.
Let and . Then .
Proof.
To prove the result, we determine the structure of and apply the algorithm of Section 3.3. Note that the case when is handled recursively. We write , where and . For , we use Propositions 5.3, 5.4, and 5.6, to show that . For , using the result for , one shows that:
□
Proof of Theorem 5.2.
The proofs of 5.2.1, 5.2.2, and 5.2.3 are covered in Propositions 5.3, 5.4, and 5.5. We have made this choice as they require more in-depth analysis. The proofs of the remaining results are much more straightforward: for each triplet we first determine the structure of , i = 1 or i = 2, and then apply the algorithm of Section 3.3.
5.2.4: Propositions 5.3 and 5.4 give the result for , while Proposition 5.6, respectively 5.7, gives the result for , respectively .
5.2.5: The proof is analogous to that of Proposition 5.7, using Propositions 5.3 and 5.5.
5.2.6: Let . If p = 5, the result follows from Propositions 4.3, 4.5, and 4.8. If p = 2, one identifies the eigenvalues of s on V and determines that . If , one uses the weight structure of V1 and Proposition 4.3 to prove the result. Assume . If p = 3, the result follows recursively, using Propositions 5.3, 5.5, and 5.2.6. If , the proof is analogous to that of Proposition 5.7 and uses 5.2.4 and 5.2.30.
5.2.7: Follows by Propositions 4.3, 4.5, and 4.8.
5.2.8: Follows by Propositions 4.5 and 4.8. Note that when , one has to show that there do not exist μ with . If this were the case, then, as V is self-dual, we would have and for all . We write with and . Then for all , where . In particular, by Propositions 4.5 and 4.8, we have , thereby , and with , thereby , contradicting .
5.2.9: Because we will need to know also when , we will not limit ourselves to the case p = 7 and, instead, only assume that . Using Propositions 4.5, 4.8, and Theorem 4.9[4.3.1], we deduce that .
5.2.10: As above, we only assume . Using Proposition 4.5 we deduce that . Note that when (, where and ), one has to show that and for all , i = 1, 2. Lastly, equality is shown to hold for and .
5.2.11: Follows by Propositions 5.3, 5.4, 5.5, and 5.2.6. Note that when , one has to show that for all , where with and .
5.2.12: Follows by 5.2.6 and 5.2.7.
5.2.13: As in 5.2.9, we only assume . Using Proposition 5.5, 5.2.6, and 5.2.7, we deduce that .
5.2.14: We only assume . Using Propositions 5.3, 5.4, 5.5, 5.2.6, and 5.2.7, we deduce that . Note that one has to show that there do not exist with . If this were the case, then ( with and ) and h would be conjugate to one of . In both cases, one shows that .
5.2.15: Follows by Proposition 5.4 and 5.2.4.
5.2.16: Follows recursively, using 5.2.11 and 5.2.30.
5.2.17: Follows by 5.2.4, 5.2.6, and 5.2.11.
5.2.18: Follows by Propositions 5.3, 5.4, 5.2.6, and 5.2.11.
5.2.19: Follows by Propositions 5.4, 5.6, 5.2.30, and 5.2.15.
5.2.20: Follows by Proposition 5.6 and 5.2.15.
5.2.21: Follows by Propositions 5.4, 5.7 and 5.2.19.
5.2.22: Follows by Proposition 5.7, 5.2.19, 5.2.20, and 5.2.30.
5.2.23: Follows by 5.2.19 and 5.2.20.
5.2.24: Follows by Propositions 5.4, 5.7, and 5.2.21.
5.2.25 and 5.2.26: Follow by 5.2.22 and 5.2.23.
5.2.27: Follows by 5.2.21 and 5.2.22.
5.2.28: Follows by Propositions 5.4, 5.7, and 5.2.24.
5.2.29: Follows by Propositions 5.4, 5.7, and .
5.2.30: Follows by 5.2.4.
5.2.31: Proof is analogous to that of Proposition 5.7, and uses 5.2.4 and 5.2.30. □
6 Proof of Theorem 1.1 for groups of type
Let k be an algebraically closed field of characteristic and let G, respectively , be a simple adjoint, respectively simply connected, linear algebraic group of type with . We fix a central isogeny with and , and let , respectively , be a maximal torus, respectively a Borel subgroup, in with the property that , respectively . We denote by , and the character group of , the center of , the set of unipotent elements in , the set of simple roots in corresponding to , and the fundamental dominant weights of corresponding to . We denote by a Levi subgroup of the maximal parabolic subgroup of corresponding to , , and we let . Now, for a p-restricted dominant weight, we let and we have , where , .
This section is dedicated to Theorems 6.1 and 6.2 which give the values of and for all -modules , with a nonzero p-restricted dominant weight, and . As a corollary, the part of Theorem 1.1 concerning simple simply connected linear algebraic groups of type will follow.
Theorem 6.1.
Let k be an algebraically closed field of characteristic and let be a simple simply connected linear algebraic group of type with . Let , where is a nonzero p-restricted dominant weight, be such that . The value of is as given in .
Proof.
For 6.1.1–6.1.5 we will deduce the result for by calculating , see Section 3.1 and Lemmas 3.2 and 3.13. The proofs for 6.1.6 - 6.1.17 are much more straightforward: first one determines the structure of , and then applies the algorithm of Section 3.3. Once more, by Lemma 3.13, we have .
6.1.1: Note that as -modules and as kG-modules. We have that , respectively , acts on W as , respectively as .
6.1.2: Note that as -modules and as kG-modules, see [Citation18, Proposition 4.2.2]. For , we write , where and acts as J3 on W1; and and acts trivially on W2. Using [Citation15, Lemma 3.4], we show that . Similarly, for , we write , where and acts as on ; and and acts trivially on . One shows that .
6.1.3: Note that as -modules. Moreover, by [Citation18, Proposition 4.7.3], we have if , and if . We argue as in 6.1.2 to show that . Then, , by [Citation11, Corollary 6.3].
6.1.4: For , we use Theorem 5.1[5.1.1]. Assume . Note that as -modules and that as kG-modules, see [Citation18, Proposition 4.2.2]. We now argue as in 6.1.2 to obtain the result.
6.1.5: Note that as -modules and that if , respectively if , see [Citation18, Propositions 4.7.4]. We argue in 6.1.2 to show that . Now, one has to establish the structure of and apply the algorithm of Section 3.3. Equality will follow recursively, using 6.1.1 and 6.1.3.
6.1.6: For , it follows by Theorem 5.1[5.1.2, 5.1.3, 5.1.7, 5.1.10]. For , it follows recursively by 6.1.1, 6.1.2, 6.1.3 and the result for .
6.1.7: Follows by Theorem 5.1[5.1.2, 5.1.3].
6.1.8, 6.1.9, and 6.1.10: Follow by Theorem 5.1[5.1.5, 5.1.10].
6.1.11: Follows by 6.1.2, 6.1.4, and 6.1.7.
6.1.12 and 6.1.13: Follow by 6.1.2, 6.1.4, and 6.1.11.
6.1.14: Follows by 6.1.2, 6.1.4, and 6.1.12.
6.1.15: Follows by 6.1.2, 6.1.4, and 6.1.14.
6.1.16: Follows recursively, by 6.1.17 and Theorem 5.1[5.1.1, 5.1.6] for the base case.
6.1.17: Follows recursively, using 6.1.4 to prove the base case. □
Theorem 6.2.
Let k be an algebraically closed field of characteristic and let be a simple simply connected linear algebraic group of type with . Let , where is a nonzero p-restricted dominant weight, be such that . The value of is as given in .
Before we begin, we recall that G is a simple adjoint linear algebraic group of type , and that we have denoted by W the natural module of G. Let . As in Section 4, we will say“s is as in ” to mean that s satisfies the following: with and , where and . Note that every is as in .
Proposition 6.3.
Let . Then .
Proof.
Set . We note that as -modules and that as kG-modules. The proof now follows that of Proposition 4.3. □
Proposition 6.4.
Let . Then .
Proof.
Set and note that as -modules. Further, by [Citation18, Proposition 4.2.2], we have as kG-modules. Let be as in . The eigenvalues of s on V, not necessarily distinct, have one of the following forms: (6.1) (6.1)
Let be an eigenvalue of s on V. If , one shows that . We thus assume for the remainder of the proof that .
Assume m = 1. Then , as . By (6.1), the eigenvalues of s on V, not necessarily distinct, are each with multiplicity at least ; each with multiplicity at least nn1; and 1 with multiplicity at least . Let μ = 1. Since , we have , where equality holds if and only if and . Now, let . If , then , while, if , then , therefore for all s with m = 1. We thus assume that .
Let μ = 1. Since for all , by (6.1), we determine that . Assume . Then, as , it follows that (6.2) (6.2) which does not hold. Therefore for all with .
Lastly, let . If for all i, then . Suppose that . Then . Since , we must have , which does not hold. We thus assume there exist i such that . Then, since the μi’s are distinct and different to 1, by (6.1), we determine that . Suppose . Then: (6.3) (6.3)
We have that , as , and that . Substituting in (6.3) gives , which does not hold. Therefore for all with . □
Proposition 6.5.
Let . Then .
Proof.
Set . Note that as -modules and that if ; while if , see [Citation18, Proposition 4.7.3]. Let be as in . The eigenvalues of s on V, not necessarily distinct, have one of the following forms: (6.4) (6.4)
The cases and with m = 1 are handled as in the proof of Proposition 6.4. The only case we have left is that of with . For μ = 1, as for all , by (6.4), we determine that , see (6.2). Thus, let . Suppose that for all i. Then, by (6.4), we have . One shows, as in the corresponding case in the proof of Proposition 6.4, that . Lastly assume that there exists i such that . Then for all and, by (6.4), we get . Once more, as in the proof of Proposition 6.4, we show that . □
Proof of Theorem 6.2.
The results 6.2.1, 6.2.2, and 6.2.3 are covered in Propositions 6.3, 6.4, and 6.5, respectively. The proofs for 6.2.4–6.1.17 are much more straightforward: first one determines the structure of and then applies the algorithm of Section 3.3.
6.2.4: The case follows by Proposition 5.3, while the case follows by Propositions 6.3, 6.4 and 6.2.7. Note that in the latter one has to show that for with , we have . The case follows recursively, by Propositions 6.3, 6.4 and the result for . Moreover, one shows that for even and with , respectively odd and with , we have .
6.2.5: For , it follows by Proposition 5.4 and Theorem 5.2[5.2.7, 5.2.9]. Also, one shows that equality holds for with and . For , the result is obtained recursively, using Propositions 6.3, 6.5 and the result for . Moreover, one shows that equality holds for even and with , respectively odd and with .
6.2.6: For , it follows by Propositions 5.4, 5.5 and Theorem 5.2[5.2.7, 5.2.10]. For , it follows recursively from 6.3, 6.4, 6.5 and the result for .
6.2.7: Follows by Propositions 5.4 and 5.5. Note that when , where and , one has to first show that for all .
6.2.8: Follows by Proposition 5.3 and Theorem 5.2 [5.2.10].
6.2.9 and 6.2.10: Follow by Theorem 5.2[5.2.5, 5.2.10].
6.2.11: Follows by Proposition 6.4 and 6.2.7. Moreover, one shows that there do not exist with .
6.2.12 and 6.2.13: Follow by Proposition 6.4, 6.2.4, and 6.2.11.
6.2.14: Follows by Proposition 6.4, 6.2.4, and 6.2.12.
6.2.15: Follows by Proposition 6.4, 6.2.4, and 6.2.14.
6.2.16: Follows recursively, by 6.2.17, Proposition 5.3 and Theorem 5.2[5.2.6] for the base case.
6.2.17: Follows recursively, using 6.2.4 to prove the base case. □
7 Proof of Theorem 1.1 for groups of type
Let k be an algebraically closed field of characteristic , let W be a -dimensional k-vector space equipped with a nondegenerate quadratic form Q and let . Note that G is a simple algebraic group of type . We let be a simply connected cover of G, and we fix a central isogeny with and . As in Section 6, we will denote by the object in corresponding to the object X in G under . For example, is a maximal torus in with .
Let a be a nondegenerate alternating bilinear form on W and let . Note that H is a simple simply connected linear algebraic group of type . Let TH be a maximal torus in H. Note that we can choose a symplectic basis and an orthogonal basis in W such that . Lastly, we denote by the fundamental dominant weights of H.
This section is dedicated to Theorems 7.1 and 7.2 which give the values of and for all -modules , with a nonzero p-restricted dominant weight, and . As a corollary, the part of Theorem 1.1 concerning simple simply connected linear algebraic groups of type will follow.
Theorem 7.1.
Let k be an algebraically closed field of characteristic and let be a simple simply connected linear algebraic group of type with . Let , where is a nonzero p-restricted dominant weight, be such that . The value of is as given in
Proof.
For 7.1.1–7.1.5, 7.1.7, 7.1.10, 7.1.19, and 7.1.24 we will deduce the result for by calculating , see Section 3.1 and Lemmas 3.2 and 3.13. The proofs for the remaining results are more straightforward: first one determines the structure of , and then applies the algorithm of Section 3.3. Once more, we will use Lemma 3.13, by which we have .
7.1.1: Note that as -modules and as kG-modules. Moreover, acts on W as .
7.1.2: Note that as -modules. The result follows by [Citation18, Proposition 4.2.2], respectively by Lemma 7.3 if p = 2, the proof of Theorem 5.1[5.1.2] and, when , by [Citation12, Theorem B].
7.1.3: Note that as -modules. The result follows by [Citation18, Propositions 4.7.3], the proof of Theorem 5.1[5.1.3] and [Citation11, Corollary 6.2].
7.1.4: Note that as -modules. If , the result follows by [Citation18, Proposition 4.2.2] and Theorem 5.1[5.1.4]. Assume p = 2. Then as kG-modules, see [Citation19, ]. Moreover, if and if , see [Citation18, Lemma 4.8.2]. Then . Equality is shown recursively, using the structure of , 7.1.1 and 7.1.2.
7.1.5: Note that as -modules. If , then , while, if , then as kG-modules, see [Citation18, Proposition 4.7.4]. Then . Equality is shown recursively, using the structure of , 7.1.1, 7.1.3 and Theorem 5.1[5.1.2, 5.1.9, 5.1.14].
7.1.6: The result follows recursively, using 7.1.1 and 7.1.3 (and 7.1.2 when ). To prove the base case of , we need Theorems 4.1[4.1.2] (and [4.1.4] when ) and 4.10[4.3.4, 4.3.5].
7.1.7: Note that as -modules. If p = 2, then as kG-modules, see [Citation19, ] and [Citation18, Lemma 4.8.2]. The result now follows by Theorem 5.1[5.1.4]. If , it follows by the structure of and Theorem 4.1[4.1.2, 4.1.3, 4.1.4].
7.1.8: Follows by Theorem 4.10[4.4.4, 4.4.5, 4.4.8].
7.1.9: Follows by Theorems 4.1[4.1.1, 4.1.7] and 4.10[4.4.7].
7.1.10: Follows by [Citation19, ] and Theorem 5.1[5.1.18].
7.1.11: Follows by 7.1.3 and 7.1.7.
7.1.12: Follows by 7.1.2, 7.1.3, and 7.1.7.
7.1.13: Follows by 7.1.1 and 7.1.7.
7.1.14: Follows by 7.1.7.
7.1.15: Follows by 7.1.11 and 7.1.12.
7.1.16: Follows by 7.1.4, 7.1.11, 7.1.12, and 7.1.25.
7.1.17: Follows by 7.1.13 and 7.1.25.
7.1.18: Follows by 7.1.2, 7.1.4, and 7.1.12.
7.1.19: Follows by [Citation19, ] and Theorem 5.1[5.1.19].
7.1.20: Follows by 7.1.17 and 7.1.25.
7.1.21: Follows by 7.1.2, 7.1.4, and 7.1.18.
7.1.22: Follows by 7.1.2, 7.1.4, and 7.1.21.
7.1.23: Follows by 7.1.20 and 7.1.25.
7.1.24: Follows by [Citation19, ] and Theorem 5.1[5.1.29].
7.1.25: Follows recursively, using 7.1.1. □
Theorem 7.2.
Let k be an algebraically closed field of characteristic and let be a simple simply connected linear algebraic group of type with . Let , where is a nonzero p-restricted dominant weight, be such that . The value of is as given in .
Before we begin, recall that is a simple simply connected linear algebraic group of type with maximal torus TH with the property that . Thus, any is conjugate in H to an element of the form with for all i < j, and .
Lemma 7.3.
Let p = 2. If , then , as kG-modules. If , then the kG-module has three composition factors one isomorphic to and two to .
Proof.
By [Citation19, 1.15], the kG-module admits a unique nontrivial composition factor of highest weight ω2. Since , see [Citation13, ], we determine that, if , then has two composition factors: one isomorphic to and one to , hence , by [Citation8, II.2.14]. If , then has three composition factors: one isomorphic to and two to . □
Proposition 7.4.
Let . Then .
Proof.
Let and note that, in particular, we have . Let μ be an eigenvalue of s on V. Now, when , as , we have . Further, by [Citation18, Lemma 4.8.2] which gives the structure of as a kH-module, we have for , and . We now use Proposition 5.4 to get the result. Similarly, when p = 2, by the structure of as a kG-module, we determine that for , and . Arguing as in the previous case, we determine that for all μ, and the result follows by Proposition 5.4. □
Proposition 7.5.
Let and . Then .
Proof.
Set and note that as -modules. Further, if , then , while, if , then , see [Citation18, Propositions 4.7.3].
Let and note that, in particular, . Let μ be an eigenvalue of s on V. Arguing as in the proof of Proposition 7.4, we show that for ; and . The result for μ = 1 is given by Proposition 5.5. For μ such that we have , as V is self-dual. Thus, for the remainder of the proof, we let .
As , we write , where for all i < j, and . If for all , then , see inequality (5.8). If there exists i < j such that , then , see inequality (5.9). Assume that . Then:
As , we must have . For , we get , which does not hold. Similarly, if , then , which does not hold. Thus, for all . □
Proposition 7.6.
Let and let . If p = 3, then ; while if , then .
Proof.
The result follows recursively, using the algorithm of Section 3.3, the structure of , Proposition 7.5, Theorem 7.2[7.2.1] and, additionally, Proposition 7.4 when . In what follows we prove that the base case holds.
Let . If p = 3, we use Propositions 4.4 and Theorem 4.9[4.3.4, 4.3.5] to prove the result. Note that when with and , one has to treat the case when is conjugate to with separately. Thus, assume . The composition factors of are as follows: one isomorphic to , to , two to to and to .
When , i.e. with and , one has to eliminate the cases when there exists with and . If , the result then follows from Proposition 4.6 and Theorem 4.9[4.3.4, 4.3.5]. Thus, assume p = 2 and let be a Levi subgroup of the maximal parabolic subgroup of corresponding to . We once again abuse notation and denote by and the fundamental dominant weights of corresponding to the simple roots and . One shows that if , then for all . On the other hand, if , then , where with and . One treats the case when there exists such that separately, and afterwards concludes that . □
Proof of Theorem 7.2.
The proofs of 7.2.2, 7.2.3, and 7.2.6 are given in Propositions 7.4–7.6, respectively.
7.2.1: We argue as in the proofs of Theorem 7.1[7.1.1] and Proposition 4.3.
7.2.4: If p = 2, we argue as we did in the proof of Theorem 7.1[7.1.4] and use Theorem 5.2[5.2.4]. If , the result follows recursively, using [Citation18, Proposition 4.2.2], Theorem 5.2[5.2.4], Proposition 7.4, 7.2.1, and 7.2.7 for the base case of .
7.2.5: It follows recursively, using Proposition 7.5, 7.2.1, and, for the base case of , Proposition 4.4 and Theorem 4.9[4.3.5, 4.3.6].
7.2.7: If p = 2, we argue as in the proof of Theorem 7.1[7.1.7] and use Theorem 5.2[5.2.4]. When , we use Propositions 4.4–4.6, to prove the result. Note that when with and , one has to treat the case when is conjugate to with separately.
7.2.8: Follows by Theorem 4.9[4.3.4, 4.3.5, 4.3.8].
7.2.9: Follows by Proposition 4.3 and Theorems 4.2[4.2.7] and 4.9[4.3.7].
7.2.10: Follows by [Citation19, ] and Theorem 5.2[5.2.18].
7.2.11: Follows by Proposition 7.5 and 7.2.7.
7.2.12: Follows by Propositions 7.4, 7.5, and 7.2.7.
7.2.13: Follows by 7.2.1 and 7.2.7.
7.2.14: Follows by 7.2.7.
7.2.15: Follows by 7.2.11 and 7.2.12.
7.2.16: Follows by 7.2.4, 7.2.11, 7.2.12, and 7.2.25.
7.2.17: Follows by 7.2.13 and 7.2.25.
7.2.18: Follows by Proposition 7.4, 7.2.4, and 7.2.12.
7.2.19: Follows by [Citation19, ] and Theorem 5.2[5.2.19].
7.2.20: Follows by 7.2.17 and 7.2.25.
7.2.21: Follows by Proposition 7.4, 7.2.4, and 7.2.18.
7.2.22: Follows by Proposition 7.4, 7.2.4, and 7.2.21.
7.2.23: Follows by 7.2.20 and 7.2.25.
7.2.24: Follows by [Citation19, ] and Theorem 5.2[5.2.29].
7.2.25: Follows recursively, using 7.2.1. Note that when and , one has to prove that: there do not exist such that ; and that for with we have . □
Acknowledgments
The author is immensely grateful to Donna Testerman for her guidance. The author would also like to thank Simon Goodwin, Martin Liebeck and Adam Thomas for many helpful discussions and comments.
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References
- Carter, R. W. (1989). Simple Groups of Lie Type. Wiley Classics Library. London: Wiley.
- Gordeev, N. L. (1991). Coranks of elements of linear groups and the complexity of algebras of invariants. Leningrad Math. J. 2:245–267.
- Gow, R., Laffey, T. J. (2006). On the decomposition of the exterior square of an indecomposable module of a cyclic p-group. J. Group Theory 9(5):659–672.
- Guralnick, R. M., Saxl, J. (2003). Generation of finite almost simple groups by conjugates. J. Algebra 268(2):519–571. DOI: 10.1016/S0021-8693(03)00182-0.
- Guralnick, R. M., Lawther, R. (2019). Generic stabilizers in actions of simple algebraic groups i: modules and the first grassmanian varieties. arXiv: Group Theory.
- Hall, J. I., Liebeck, M. W., Seitz, G. M. (1992). Generators for finite simple groups, with applications to linear groups. Q. J. Math. 43(4):441–458. DOI: 10.1093/qmathj/43.4.441.
- Humphreys, J. E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. New York: Springer.
- Jantzen, J. C. (2007). Representations of Algebraic Groups. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society.
- Kac, V. G., Watanabe, K. (1982). Finite linear groups whose ring of invariants is a complete intersection. Bull. Amer. Math. Soc. 6:221–223. DOI: 10.1090/S0273-0979-1982-14989-8.
- Kemper, G., Malle, G. (1997). The finite irreducible linear groups with polynomial ring of invariants. Transformation Groups 2:57–89. DOI: 10.1007/BF01234631.
- Korhonen, M. (2019). Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic. Proc. Amer. Math. Soc. 147(10):4205–4219. DOI: 10.1090/proc/14570.
- Korhonen, M. (2020). Hesselink normal forms of unipotent elements in some representations of classical groups in characteristic two. J. Algebra 559:268–319. DOI: 10.1016/j.jalgebra.2020.03.035.
- Lübeck, F. (2001). Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math. 4:135–169. http://www.math.rwth-aachen.de/Frank.Luebeck/chev/WMSmall/index.html. DOI: 10.1112/S1461157000000838.
- Lübeck, F. (2001). Tables of weight multiplicities.
- Liebeck, M. W., Seitz, G. M. (2012). Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras. Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society.
- Malle, G., Testerman, D. (2011). Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics. Cambridge, UK: Cambridge University Press.
- Martínez, A. L. (2018). Low-dimensional irreducible rational representations of classical algebraic groups. DOI: 10.48550/ARXIV.1811.07019.
- McNinch, G. J. (1998). Dimensional criteria for semisimplicity of representations. Proc. London Math. Soc. 76(1):95–149. DOI: 10.1112/S0024611598000045.
- Seitz, G. M. (1987). The Maximal Subgroups of Classical Algebraic Groups, 365. Providence, RI: American Mathematical Society.
- Smith, S. D. (1982). Irreducible modules and parabolic subgroups. J. Algebra 75(1):286–289. DOI: 10.1016/0021-8693(82)90076-X.
- Steinberg, R. (2016). Lectures on Chevalley Groups. University Lecture Series, Vol. 66. Providence, RI: American Mathematical Society.
- Suprunenko, I. D. (1983). Preservation of systems of weights of irreducible representations of an algebraic group and a Lie algebra of type A with bounded higher weights in reduction modulo p. Vestsi Akad. Na uk BSSR, Ser. Fiz.-Mat. Na uk (2):18–22.
- Verbitsky, M. (1999). Holomorphic symplectic geometry and orbifold singularities. Asian J. Math. 4:553–564. DOI: 10.4310/AJM.2000.v4.n3.a4.