Abstract
Humphreys’ conjecture on blocks parametrises the blocks of a reduced enveloping algebra of the Lie algebras of a reductive algebraic group over an algebraically closed field of positive characteristic. It is well-known to hold under Jantzen’s standard assumptions. We note here that it holds under slightly weaker assumptions, by utilising the full generality of certain results in the literature. We also provide a new approach to prove the result in type G2 in characteristic 3, a case in which the previously mentioned weaker assumptions do not hold. This approach requires some dimensional calculations for certain centralisers, which we conduct in the Appendix for all the exceptional Lie algebras in bad characteristic.
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Acknowledgements
The author would like to thank Ami Braun and Dmitriy Rumynin for suggesting this question, and Simon Goodwin for engaging in many useful discussions regarding this subject and comments on earlier versions of this paper.
Funding
The author was supported during this research by a PhD studentship from the Engineering and Physical Sciences Research Council, award reference 1789943, by the Engineering and Physical Sciences Research Council grant EP/R018952/1, and later by a research fellowship from the Royal Commission for the Exhibition of 1851.
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Appendix A: Divisibility bounds
Appendix A: Divisibility bounds
By Proposition 4.2, Humphreys’ conjecture on blocks holds whenever, for each λ ∈Λ0, there exists \(\mu _{\lambda }\in {\mathfrak b}^{\perp }\) such that \(Z_{\mu _{\lambda }}(\lambda )\) is an irreducible \(U_{\mu _{\lambda }}({\mathfrak g})\)-module. The natural choice for such μλ is the \(\chi \in {\mathfrak g}^{*}\) which is regular nilpotent in standard Levi form. For such χ, one way to try to show that each Zχ(λ) is irreducible is to show that each \(\dim (Z_{\chi }(\lambda ))\) is divisible by pN, where \(N=\left \vert {\Phi }^{+}\right \vert \). This appendix contains some computations to determine some k ≤ N such that all \(U_{\chi }({\mathfrak g})\)-modules have dimension divisible by pk. Unfortunately, except for the case of G2 in characteristic 3, we do not find k to be equal to N when assumptions (B) or (C) fail. In a few cases, we are even able to show that pN does not divide \(\dim (Z_{\chi }(\lambda ))\) for some λ ∈Λ0.
In this appendix, we assume G is an almost-simple simply-connected algebraic group over an algebraically closed field \({\mathbb K}\) of positive characteristic p > 0, and we write Φ for its (indecomposable) root system. Specifically, let \(G_{{\mathbb Z}}\) be a split reductive group scheme over \({\mathbb Z}\) with root data (X(T),Φ,α↦α∨), let \(T_{{\mathbb Z}}\) be a split maximal torus of \(G_{{\mathbb Z}}\), and let \({\mathfrak g}_{{\mathbb Z}}\) be the Lie ring of \(G_{{\mathbb Z}}\). Throughout this appendix, we think of G as being obtained from \(G_{{\mathbb Z}}\) through base change, so \(G=(G_{{\mathbb Z}})_{{\mathbb K}}\), \(T=(T_{{\mathbb Z}})_{{\mathbb K}}\) and \({\mathfrak g}={\mathfrak g}_{{\mathbb Z}}\otimes _{{\mathbb Z}}{\mathbb K}\). In particular, the elements eβ (β ∈Φ) and hα (α ∈π) form a Chevalley basis of \({\mathfrak g}\).
Under these assumptions, \({\mathfrak g}\) is a simple Lie algebra unless Φ is of type An with p dividing n + 1; of type Bn, Cn, Dn, F4 or E7 with p = 2; or of type E6 or G2 with p = 3 (see, for example, [13, 6.4(b)]). If \({\mathfrak g}\) is simple then there exists a G-equivariant isomorphism \({\mathfrak g}\xrightarrow {\sim }{\mathfrak g}^{*}\) coming from the Killing form, so assumption (C) holds. We also note that assumption (A) holds for all such G, since G equals its derived subgroup.
We consider here both those G which satisfy assumption (C) and those which don’t (i.e. we also consider those G with \({\mathfrak g}\) non-simple). We focus our attention on the exceptional types E6, E7, E8, F4 and G2. We generally assume throughout this appendix that χ is in standard Levi form with I = π, although we don’t make that assumption in this preliminary discussion.
When G satisfies assumptions (A), (B) and (C), Premet’s theorem [18, 20] (proving the second Kac-Weisfeiler conjecture [16]) shows that the dimension of each \(U_{\chi }({\mathfrak g})\)-module is divisible by \(p^{\dim (G\cdot \chi )/2}\). We note also that when (A) and (B) hold but (C) does not - i.e. when Φ = An and p divides n + 1 - Premet’s theorem shows the same result for faithful irreducible \(U_{\chi }({\mathfrak g})\)-modules. When χ is regular nilpotent and assumption (C) holds, we know that \(\dim (G\cdot \chi ) /2=N\). Hence, in this situation we have that all irreducible \(U_{\chi }({\mathfrak g})\)-modules have dimension divisible by pN. This means that all baby Verma modules are irreducible, and so all irreducible \(U_{\chi }({\mathfrak g})\)-modules are baby Verma modules.
Outside of the setting of Premet’s theorem, there are other ways to determine powers of p which divide the dimensions of all \(U_{\chi }({\mathfrak g})\)-modules. Two particular results are relevant here. Both utilize the centraliser in \({\mathfrak g}\) of \(\chi \in {\mathfrak g}^{*}\), which the reader will recall is defined as \(c_{{\mathfrak g}}(\chi ):= \{x\in {\mathfrak g} \vert \chi ([x,{\mathfrak g}])=0\}.\)
The first result comes from Premet and Skryabin [21], and applies when the prime p is non-special for the root system Φ. This means that p≠ 2 when Φ is Bn, Cn or F4, and p≠ 3 when Φ = G2 (i.e. p does not divide any non-zero off-diagonal entry of the Cartan matrix).
Proposition A.1
Let \(\chi \in {\mathfrak b}^{\perp }\), and let \(d(\chi ):=\frac {1}{2}{{\text {codim}}}_{{\mathfrak g}}(c_{{\mathfrak g}}(\chi ))\). If p is non-special for Φ, then every \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by pd(χ).
The second proposition we use is also due to Premet [13, 18, 19]. To apply it, recall that a restricted Lie algebra is called unipotent if for all \(x\in {\mathfrak g}\) there exists r > 0 such that \(x^{[p^{r}]}=0\), where \(x^{[p^{r}]}\) denotes the image of x under r applications of [p]. In particular, this applies to \({\mathfrak n}^{-}\) and any restricted subalgebras of it.
Proposition A.2
Let \(\chi \in {\mathfrak g}^{*}\). If \({\mathfrak m}\) is a unipotent restricted subalgebra of \({\mathfrak g}\) with \(\chi ([{\mathfrak m},{\mathfrak m}])=0\), \(\chi ({\mathfrak m}^{[p]})=0\) and \({\mathfrak m}\cap c_{{\mathfrak g}}(\chi )=0\), then every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module is free over \(U_{\chi }({\mathfrak m})\).
In applying the second proposition when \(\chi \in {\mathfrak b}^{\perp }\), the reader should note the following. Suppose \({\mathfrak m}\), a \({\mathbb K}\)-subspace of \({\mathfrak g}\), has a basis consisting of elements e−α for α ∈Ψ, where Ψ is some subset of Φ+. The condition \(\chi ([{\mathfrak m},{\mathfrak m}])=0\) is clearly satisfied if χ(e−α−β) = 0 for all α,β ∈Ψ. Furthermore, we have in \(U({\mathfrak g})\) that
by the semilinearity of the map x↦xp − x[p], and we have
where we interpret \(e_{-\gamma _{1}-\gamma _{2}-\cdots -\gamma _{p}}=0\) if − γ1 − γ2 −⋯ − γp∉Φ. We hence conclude that
In particular, if \(\chi (e_{-\gamma _{1}-\gamma _{2}-\cdots -\gamma _{p}})=0\) for all γ1,…,γp ∈Ψ, we find that \(\chi ({\mathfrak m}^{[p]})=0\). Furthermore, if Ψ satisfies the condition that α,β ∈Ψ, α + β ∈Φ implies α + β ∈Ψ (we call this the condition of Ψ being closed), then it is enough to check that χ(e−α−β) = 0 for all α,β ∈Ψ. Finally, we observe that Ψ being closed is enough to show that \({\mathfrak m}\) is a subalgebra. So we may obtain a corollary to Proposition A.2:
Corollary A.3
Let \(\chi \in {\mathfrak b}^{\perp }\) and let Ψ be a closed subset of Φ+. Suppose that χ(e−α−β) = 0 for all α,β ∈Ψ. Furthermore, let \({\mathfrak m}\) be the subspace of \({\mathfrak g}\) with basis consisting of the e−α with α ∈Ψ, and suppose that \({\mathfrak m}\cap c_{{\mathfrak g}}(\chi )=0\). Then every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by p|Ψ|.
The above discussion actually shows that this corollary can be improved a bit. Given two roots α,β ∈Φ, write Cα,β := q + 1 where \(q\in {\mathbb N}\) is maximal for the condition that β − qα lies in Φ (so, in particular, [eα,eβ] = ±Cα,βeα+β if α + β ∈Φ). Let us say that Ψ is p-closed if, for all α,β ∈Ψ with α + β ∈Φ, either α + β ∈Ψ or p divides Cγ,δ for all γ,δ ∈Ψ with γ + δ = α + β. Then we easily obtain the following.
Corollary A.4
Let \(\chi \in {\mathfrak b}^{\perp }\), and let Ψ be a p-closed subset of Φ+. Suppose that χ(e−α−β) = 0 for all α,β ∈Ψ with α + β ∈Ψ. Furthermore, let \({\mathfrak m}\) be the subspace of \({\mathfrak g}\) with basis consisting of the e−α with α ∈Ψ, and suppose that \({\mathfrak m}\cap c_{{\mathfrak g}}(\chi )=0\). Then every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by p|Ψ|.
Let us consider a bit further the condition that \({\mathfrak m}\cap c_{{\mathfrak g}}(\chi )=0\). Let \(x\in {\mathfrak m}\cap c_{{\mathfrak g}}(\chi )\). We can then write
The fact that \(x\in c_{{\mathfrak g}}(\chi )\) means that \(\chi ([x,{\mathfrak g}])=0\). This is equivalent to the requirement that χ([x,eβ]) = 0 for all β ∈Φ and χ([x,h]) = 0 for all \(h\in {\mathfrak h}\). Let Δ be the subset of Φ− such that χ(eα)≠ 0 for α ∈Δ. We then have, for β ∈Φ, that
and, for \(h\in {\mathfrak h}\), that
Showing that \({\mathfrak m}\cap c_{{\mathfrak g}}(\chi )=0\) then involves showing that there is no non-zero solution to these equations in cγ.
We now turn to the application of these propositions. In each case, we take χ to be regular nilpotent in standard Levi form and we apply one of the propositions or its corollaries to determine a divisibility bound for the dimensions of \(U_{\chi }({\mathfrak g})\)-modules. We do this for Φ of exceptional type. Principally, we compute the centraliser \({\mathfrak c}_{{\mathfrak g}}(\chi )\) and use its description to determine the bound. For Φ = G2 we give the explicit computations, but for the larger rank examples the results were obtained using Sage [22]. Because of this, when there is a choice we take the structure coefficients to be as used in the Sage class LieAlgebraChevalleyBasis_with_category. However, we use the labelling of the simple roots as given in [23].
Remark 1
Our computations of \(\dim {\mathfrak c}_{{\mathfrak g}}(\chi )\) can be compared with the computations of \(\dim {\mathfrak c}_{{\mathfrak g}}(e)\) for \(e={\sum }_{\alpha \in {\Pi }}e_{\alpha }\) which can be deduced from [23, Cor. 2.5, Thm. 2.6]. The results are listed in Table 1.
When \({\mathfrak g}\) is simple, χ and e are identified through the G-equivariant isomorphism \({\mathfrak g}\xrightarrow {\sim }{\mathfrak g}^{*}\), and thus \({\mathfrak c}_{{\mathfrak g}}(\chi )={\mathfrak c}_{{\mathfrak g}}(e)\). In the subsections below, we nonetheless include calculations of \({\mathfrak c}_{{\mathfrak g}}(\chi )\) for the bad primes for which \({\mathfrak g}\) is simple, since we give explicit bases for the centralisers in these case and in some instances we use such bases to show the reducibility of the corresponding baby Verma modules.
In the other cases (which we label with an asterisk (*) in Table 1), however, we find that the dimensions of \({\mathfrak c}_{{\mathfrak g}}(e)\) and \({\mathfrak c}_{{\mathfrak g}}(\chi )\) differ from each other. Note also that we give in Table 1 the dimension of \({\mathfrak c}_{{\mathfrak g}}(\chi )\) for G2 in characteristic 3, even though we do not give it in Section A below, because it is easy to compute.
Remark 2
In our discussion of \({\mathfrak g}\) so far, the Lie algebra \({\mathfrak g}\) of G has been obtained as \({\mathfrak g}={\mathfrak g}_{{\mathbb Z}}\otimes _{{\mathbb Z}} {\mathbb K}\), where \({\mathfrak g}_{{\mathbb Z}}\) is a \({\mathbb Z}\)-form of the complex simple Lie algebra \({\mathfrak g}_{{\mathbb C}}\). In particular, \({\mathfrak g}_{{\mathbb Z}}\) is the \({\mathbb Z}\)-form coming from the chosen Chevalley basis of \({\mathfrak g}_{{\mathbb C}}\), which is what gives our Chevalley basis of \({\mathfrak g}\). We may then also define \({\mathfrak g}_{{\mathbb F}_{p}}={\mathfrak g}_{{\mathbb Z}}\otimes _{{\mathbb Z}} {\mathbb F}_{p}\), so that \({\mathfrak g}={\mathfrak g}_{{\mathbb F}_{p}}\otimes _{{\mathbb F}_{p}}{\mathbb K}\). Therefore, if \(\chi _{{\mathbb F}_{p}}:{\mathfrak g}_{{\mathbb F}_{p}}\to {\mathbb F}_{p}\) is a linear form, we may define \(\chi :{\mathfrak g}\to {\mathbb K}\) by linear extension. It is clear that any χ in standard Levi form may be obtained in this way. Our calculations in Sage are calculations with \({\mathfrak g}_{{\mathbb F}_{p}}\) and \(\chi _{{\mathbb F}_{p}}\) rather than \({\mathfrak g}\). However, when χ is obtained through scalar extension from an \({\mathbb F}_{p}\)-linear form, the above discussion shows that determining the elements of \({\mathfrak g}\) which lie in \({\mathfrak c}_{{\mathfrak g}}(\chi )\) comes down to finding solutions to certain linear equations with coefficients in \({\mathbb F}_{p}\). This in particular shows that \({\mathfrak c}_{{\mathfrak g}_{{\mathbb F}_{p}}}(\chi _{{\mathbb F}_{p}})\otimes _{{\mathbb F}_{p}}{\mathbb K}={\mathfrak c}_{{\mathfrak g}}(\chi )\), so our calculations over \({\mathbb F}_{p}\) also lead to the results over \({\mathbb K}\).
1.1 A.1 G 2 in characteristic 2
Suppose Φ = G2 and p = 2. Since p is non-special in this case, we may apply Proposition A.1. Let us therefore compute \(c_{{\mathfrak g}}(\chi )\). Set \(x\in {\mathfrak g}\) be written as \(x={\sum }_{\gamma \in {\Phi }}c_{\gamma } e_{\gamma } + {\sum }_{\gamma \in {\Pi }}d_{\gamma } h_{\gamma }\), with the cγ, dγ lying in \({\mathbb K}\). Then the relations required for \(x\in c_{{\mathfrak g}}(\chi )\) are as follows:
-
0 = χ([x,e3α+ 2β]) = 0,
-
0 = χ([x,e3α+β]) = c− 3α− 2βχ([e− 3α− 2β,e3α+β]) = c− 3α− 2βχ(e−β) = c− 3α− 2β,
-
0 = χ([x,e2α+β]) = c− 3α−βχ([e− 3α−β,e2α+β]) = c− 3α−βχ(e−α) = c− 3α−β,
-
0 = χ([x,eα+β]) = c− 2α−βχ([e− 2α−β,eα+β]) = c− 2α−βχ(2e−α) = 0,
-
0 = χ([x,eβ]) = c−α−βχ([e−α−β,eβ]) = c−α−βχ(e−α) = c−α−β,
-
0 = χ([x,eα]) = c−α−βχ([e−α−β,eα]) = c−α−βχ(− 3e−α) = − 3c−α−β = c−α−β,
-
0 = χ([x,hα]) = c−αχ(α(hα)e−α) + c−βχ(β(hα)e−β) = 2c−α − 3c−β = c−β,
-
0 = χ([x,hβ]) = c−αχ(α(hβ)e−α) + c−βχ(β(hβ)e−β) = −c−α + 2c−β = c−α,
-
0 = χ([x,e−α]) = dαχ([hα,e−α])+dβχ([hβ,e−α]) = dαχ(−α(hα)e−α)+dβχ(−α(hβ)e−α) = − 2dα+dβ = dβ,
-
0 = χ([x,e−β]) = dαχ([hα,e−β])+dβχ([hβ,e−β]) = dαχ(−β(hα)e−β)+dβχ(−β(hβ)e−β) = 3dα− 2dβ = dα,
-
0 = χ([x,e−α−β]) = cαχ([eα,e−α−β])+cβχ([eβ,e−α−β]) = 3cαχ(e−β)−cβχ(e−α) = cα+cβ,
-
0 = χ([x,e− 2α−β]) = cα+βχ([eα+β,e− 2α−β]) = − 2cα+βχ(e−α) = 0,
-
0 = χ([x,e− 3α−β]) = c2α+βχ([e2α+β,e− 3α−β]) = −c2α+βχ(e−α) = c2α+β,
-
0 = χ([x,e− 3α− 2β]) = c3α+βχ([e3α+β,e− 3α− 2β]) = −c3α+βχ(e−α) = c3α+β.
We therefore conclude that
and so is 4-dimensional. Hence, \(d(\chi )=\frac {1}{2}(14-4)=5\), and so by Proposition A.1 we conclude that every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 25.
We furthermore note that a \(U_{\chi }({\mathfrak g})\)-module of dimension 25 does indeed exist in this case. Let λ ∈Λ0 be such that λ(hβ) = 0, and let us write ω1 ∈Λ0 for the map with ω1(hα) = 1 and ω1(hβ) = 0. We may then define a \(U_{\chi }({\mathfrak g})\)-module homomorphism
This has a kernel of dimension 25 and so both the kernel and image of this homomorphism are \(U_{\chi }({\mathfrak g})\)-modules of dimension 25.
1.2 A.2 G 2 in characteristic 3
Suppose Φ = G2 and p = 3. Note that this Lie algebra is not simple, since it has an ideal generated by the short roots. In this case p is not non-special for Φ so we cannot apply Proposition A.1. Instead, we want to apply Proposition A.2, and so we need to find an appropriate \({\mathfrak m}\). Take \({\mathfrak m}={\mathfrak n}^{-}\). In this case, Ψ = Φ+ is closed and χ(e−γ−δ) = 0 for all γ,δ ∈Ψ. In the notation of the previous discussion, we have Δ = {−α,−β}.
Let \(x={\sum }_{\gamma \in {\Phi }^{+}} c_{\alpha } e_{-\alpha }\). Then the relations required for \(x\in c_{{\mathfrak g}}(\chi )\) are as follows:
-
0 = χ([x,e3α+ 2β]) = 0,
-
0 = χ([x,e3α+β]) = c3α+ 2βχ([e− 3α− 2β,e3α+β]) = c3α+ 2βχ(e−β) = c3α+ 2β,
-
0 = χ([x,e2α+β]) = c3α+βχ([e− 3α−β,e2α+β]) = c3α+βχ(e−α) = c3α+β,
-
0 = χ([x,eα+β]) = c2α+βχ([e− 2α−β,eα+β]) = c2α+βχ(2e−α) = 2c2α+β,
-
0 = χ([x,eβ]) = cα+βχ([e−α−β,eβ]) = cα+βχ(e−α) = cα+β,
-
0 = χ([x,eα]) = cα+βχ([e−α−β,eα]) = cα+βχ(− 3e−α) = 0,
-
0 = χ([x,hα]) = cαχ(α(hα)e−α) + cβχ(β(hα)e−β) = 2cα − 3cβ = 2cα,
-
0 = χ([x,hβ]) = cαχ(α(hβ)e−α) + cβχ(β(hβ)e−β) = −cα + 2cβ.
It is easy to see that these relations force x = 0, so \({\mathfrak m}\cap c_{{\mathfrak g}}(\chi )=0\). Hence, Proposition A.2 shows that every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 36, which is \(3^{\dim {\mathfrak n}^{-}}\). So in this case each baby Verma module Zχ(λ) is irreducible.
1.3 A.3 F 4 in characteristic 2
Set Φ = F4 and p = 2. Since p is not non-special in this case we need to use Proposition A.2; in fact, we use Corollary A.4. Set \({\mathfrak m}\) to be the subspace of \({\mathfrak n}^{-}\) with basis given by the elements e−α for α ∈Ψ := Φ+ ∖{α2 + 2α3}. It is straightforward to see that Ψ is 2-closed. We want to see that \({\mathfrak m}\cap {\mathfrak c}_{{\mathfrak g}}(\chi )=0\). We do this by giving a basis of \({\mathfrak c}_{{\mathfrak g}}(\chi )\) as follows:
-
(1)
\(e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}} + e_{-\alpha _{2}-2\alpha _{3}}\);
-
(2)
\(e_{\alpha _{3}} +e_{\alpha _{4}}\);
-
(3)
\(e_{\alpha _{3}+\alpha _{4}}\);
-
(4)
\(e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+\alpha _{4}} + e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}}\);
-
(5)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+\alpha _{4}}+e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}}\);
-
(6)
\(e_{2\alpha _{1}+3\alpha _{2}+4\alpha _{3}+2\alpha _{4}}\).
It is clear from this basis description that \({\mathfrak c}_{{\mathfrak g}}(\chi )\cap {\mathfrak m}=0\). Hence, Corollary A.4 applies and we get that every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by \(2^{\left \vert {\Psi }\right \vert }=2^{\left \vert {\Phi }^{+}\right \vert -1}=2^{23}\).
Now, set \({\mathfrak r}\) to be the \({\mathbb K}\)-subspace of \({\mathfrak g}\) generated by eβ for all β ∈Φ+ ∖{α3,α4}, by h1,h2 + h3,h3 + h4, by \(e_{\alpha _{3}}+e_{\alpha _{4}}\), and by \(e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}}+e_{-\alpha _{2}-2\alpha _{3}}, e_{-\alpha _{3}-\alpha _{4}}\) and \(e_{-\alpha _{3}}+e_{-\alpha _{4}}\). This has dimension 29. One may check that it is in fact a subalgebra of \({\mathfrak g}\) (using that the characteristic of \({\mathbb K}\) is 2). One may also check that \(\chi ([\mathfrak r,\mathfrak r])=0\), and that \(\chi (\mathfrak r^{[p]})=0\) (since the characteristic is 2, we have (x + y)[2] = [x,y] whenever x[2] = y[2] = 0).
Then \(U_{\chi }({\mathfrak r})\) has dimension 229 and has a 1-dimensional trivial module \({\mathbb K}_{\chi }\). Therefore, \(U_{\chi }({\mathfrak g})\otimes _{U_{\chi }({\mathfrak r})}{\mathbb K}_{\chi }\) is a \(U_{\chi }({\mathfrak g})\)-module of dimension 223, so the divisibility bound we found is strict.
1.4 A.4 F 4 in characteristic 3
Set Φ = F4 and p = 3. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{2}-2\alpha _{3}-\alpha _{4}} + 2e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4}} + e_{-\alpha _{1}-\alpha _{2}-2\alpha _{3}}\);
-
(2)
\(2e_{\alpha _{1}}+2e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}}\);
-
(3)
\(e_{\alpha _{2}+\alpha _{3}+\alpha _{4}}+2e_{\alpha _{2}+2\alpha _{3}} + 2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}}\),
-
(4)
\(e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}} + e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}} + e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+\alpha _{4}}\);
-
(5)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+\alpha _{4}}+2e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}}\);
-
(6)
\(e_{2\alpha _{1}+3\alpha _{2}+4\alpha _{3}+2\alpha _{4}}\).
Therefore, \(\dim c_{{\mathfrak g}}(\chi )=6\), and so \(d(\chi )=23=\left \vert {\Phi }^{+}\right \vert -1\). Hence, every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 323.
1.5 A.5 E 6 in characteristic 2
Suppose Φ = E6 and p = 2. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}}+e_{-\alpha _{2}-\alpha _{3}-\alpha _{6}} + e_{-\alpha _{3}-\alpha _{4}-\alpha _{6}}\);
-
(2)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}\);
-
(3)
\(e_{\alpha _{1}+\alpha _{2}}+e_{\alpha _{2}+\alpha _{3}} + e_{\alpha _{3}+\alpha _{4}} + e_{\alpha _{3}+\alpha _{6}} + e_{\alpha _{4}+\alpha _{5}}\);
-
(4)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{6}}+e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}}\);
-
(5)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{6}}+e_{\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{6}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}}\);
-
(6)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{6}}+e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} + e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}}\);
-
(7)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} + e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4} +\alpha _{5} +\alpha _{6}}\);
-
(8)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+2\alpha _{4}+\alpha _{5}+2\alpha _{6}}\).
In particular we see that \(\dim c_{{\mathfrak g}}(\chi )=8\), and so \(d(\chi )=35=\left \vert {\Phi }^{+}\right \vert -1\). Hence, every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 235.
1.6 A.6 E 6 in characteristic 3
Suppose Φ = E6 and p = 3. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}}+2e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}} + e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{6}} + 2e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4}}+e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{6}}\);
-
(2)
\(2e_{-\alpha _{1}}+e_{-\alpha _{2}}+2e_{-\alpha _{4}}+e_{-\alpha _{5}}\);
-
(3)
h1 + 2h2 + h4 + 2h5;
-
(4)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}\);
-
(5)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{6}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{6}}\);
-
(6)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}} + 2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{6}}+2e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}}\);
-
(7)
\(2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{6}}+2e_{\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{6}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}}+2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}}\);
-
(8)
\(2e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{6}}+e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} + e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}}\);
-
(9)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} + 2e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4} +\alpha _{5} +\alpha _{6}}\);
-
(10)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+2\alpha _{4}+\alpha _{5}+2\alpha _{6}}\).
Hence, \(\dim (c_{{\mathfrak g}}(\chi ))=10\) and so \(d(\chi )=34=\left \vert {\Phi }^{+}\right \vert -2\). Proposition A.1 then says that all \(U_{\chi }({\mathfrak g})\)-modules have dimension divisible by 334.
Now, set \({\mathfrak r}\) be the \({\mathbb K}\)-subspace of \({\mathfrak g}\) generated by eβ for all β ∈Φ+, by h1,h2,h4,h5 and h6, and by \(2e_{-\alpha _{1}}+e_{-\alpha _{2}}\) and \(2e_{-\alpha _{4}}+e_{-\alpha _{5}}\). This has dimension 43. One may check that it is in fact a subalgebra of \({\mathfrak g}\) (using that the characteristic is 3). One may also check that \(\chi ([\mathfrak r,\mathfrak r])=0\) and that \(\chi (\mathfrak r^{[p]})=0\).
Then \(U_{\chi }({\mathfrak r})\) has dimension 343 and has a 1-dimensional module \({\mathbb K}_{\chi }\). Therefore \(U_{\chi }({\mathfrak g})\otimes _{U_{\chi }({\mathfrak r})}{\mathbb K}_{\chi }\) is a \(U_{\chi }({\mathfrak g})\)-module of dimension 335. In particular, this shows that it is not true in this case that all baby Verma modules are irreducible \(U_{\chi }({\mathfrak g})\)-modules. It obviously, however, does not imply that our divisibility bound is strict.
1.7 A.7 E 7 in characteristic 2
Suppose Φ = E7 and p = 2. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{1}-2\alpha _{2}-2\alpha _{3}-2\alpha _{4}-\alpha _{5}-\alpha _{7}} + e_{-\alpha _{2}-2\alpha _{3}-2\alpha _{4}-2\alpha _{5}-\alpha _{6}-\alpha _{7}} + e_{-\alpha _{1}-\alpha _{2}-2\alpha _{3}-2\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{7}} \\ + e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-2\alpha _{4}-2\alpha _{5}-\alpha _{6}-\alpha _{7}}\);
-
(2)
\(e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}}+e_{-\alpha _{3}-2\alpha _{4}-\alpha _{5}-\alpha _{7}} + e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{7}} + e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{7}};\)
-
(3)
\(e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}} + e_{-\alpha _{4}-\alpha _{5}-\alpha _{7}}+e_{-\alpha _{3}-\alpha _{4}-\alpha _{7}}\);
-
(4)
\(e_{-\alpha _{1}}+e_{-\alpha _{3}}+e_{-\alpha _{7}}\);
-
(5)
h1 + h3 + h7;
-
(6)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}+e_{\alpha _{7}}\);
-
(7)
\(e_{\alpha _{1}+\alpha _{2}}+e_{\alpha _{2}+\alpha _{3}} + e_{\alpha _{3}+\alpha _{4}} + e_{\alpha _{4}+\alpha _{5}} + e_{\alpha _{4}+\alpha _{7}}+e_{\alpha _{5}+\alpha _{6}}\);
-
(8)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{7}} +e_{\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}+e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}}\);
-
(9)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{7}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{7}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}+e_{\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}\);
-
(10)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}+e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}} + e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}} + e_{\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(11)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}} + e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4} +\alpha _{5} +\alpha _{6}+\alpha _{7}} +e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(12)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}+e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}} + e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(13)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}+e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}} + e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}+2\alpha _{7}}\);
-
(14)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+3\alpha _{3}+2\alpha _{5}+\alpha _{6}+\alpha _{7}} + e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}+2\alpha _{7}}\);
-
(15)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+3\alpha _{5}+2\alpha _{6}+2\alpha _{7}}\).
We conclude that \(\dim ({\mathfrak c}_{{\mathfrak g}}(\chi ))=15\) and so \(d(\chi )=59=\left \vert {\Phi }^{+}\right \vert -4\). We then conclude from Proposition A.1 that every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 259.
1.8 A.8 E 7 in characteristic 3
Suppose Φ = E7 and p = 3. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}}+2e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{7}} + 2e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}} +e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{7}}+e_{-\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{7}}\);
-
(2)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}+e_{\alpha _{7}}\);
-
(3)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{7}} + e_{\alpha _{4}+\alpha _{5}+\alpha _{6}}+e_{\alpha _{4}+\alpha _{5}+\alpha _{7}}\);
-
(4)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} + 2e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{7}} +e_{\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}+e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(5)
\(2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}+e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}} + e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}+e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}\\+2e_{\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(6)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{7}}+e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}} + 2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(7)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}} + 2e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+3\alpha _{4} +2\alpha _{5} +\alpha _{6}+\alpha _{7}} +e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}+2\alpha _{7}}\);
-
(8)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}}+2e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+2\alpha _{5}+\alpha _{6}+2\alpha _{7}}\);
-
(9)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+3\alpha _{5}+2\alpha _{6}+2\alpha _{7}}\).
In particular we see that \(\dim c_{{\mathfrak g}}(\chi )=9\), and so \(d(\chi )=62=\left \vert {\Phi }^{+}\right \vert -1\). Every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module therefore has dimension divisible by 362.
1.9 A.9 E 8 in characteristic 2
Suppose Φ = E8 and p = 2. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{2}-2\alpha _{3}-3\alpha _{4}-4\alpha _{5} - 2\alpha _{6}-\alpha _{7}-2\alpha _{8}}{+}e_{-\alpha _{1}-\alpha _{2}-2\alpha _{3}-3\alpha _{4} -3\alpha _{5}-2\alpha _{6}-\alpha _{7}-2\alpha _{8}} + e_{-\alpha _{1}-2\alpha _{2}-2\alpha _{3}-3\alpha _{4}-3\alpha _{5}-2\alpha _{6}-\alpha _{7}-\alpha _{8}} \\ +e_{-\alpha _{1}-2\alpha _{2}-2\alpha _{3}-2\alpha _{4}-3\alpha _{5}-2\alpha _{6}-\alpha _{7}-2\alpha _{8}}\);
-
(2)
\(e_{-\alpha _{2}-2\alpha _{3}-2\alpha _{4}-2\alpha _{5}-\alpha _{6}-\alpha _{8}} + e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-2\alpha _{5}-2\alpha _{6}-\alpha _{7}-\alpha _{8}} + e_{-\alpha _{2}-\alpha _{3}-2\alpha _{4}-2\alpha _{5}-\alpha _{6}-\alpha _{7}-\alpha _{8}} \\ + e_{-\alpha _{3}-2\alpha _{4}-2\alpha _{5}-2\alpha _{6}-\alpha _{7}-\alpha _{8}}\);
-
(3)
\(e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}}+e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{8}} + e_{-\alpha _{4}-2\alpha _{5}-\alpha _{6}-\alpha _{8}} + e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{8}}\);
-
(4)
\(e_{-\alpha _{4}-\alpha _{5}-\alpha _{6}} + e_{-\alpha _{4}-\alpha _{5}-\alpha _{8}} + e_{-\alpha _{5}-\alpha _{6}-\alpha _{8}}\);
-
(5)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}+e_{\alpha _{7}}+e_{\alpha _{8}}\);
-
(6)
\(e_{\alpha _{1}+\alpha _{2}}+e_{\alpha _{2}+\alpha _{3}} + e_{\alpha _{3}+\alpha _{4}} + e_{\alpha _{4}+\alpha _{5}} + e_{\alpha _{5}+\alpha _{6}}+e_{\alpha _{5}+\alpha _{8}} + e_{\alpha _{6}+\alpha _{7}} \)
-
(7)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} +e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{8}}+e_{\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}} + e_{\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(8)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{8}}{+}e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}} {+}e_{\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}}{+}e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}}\\{{+}\alpha _{5}+\alpha _{6}+\alpha _{7}} +e_{\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}} + e_{\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(9)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}} + e_{\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}\\+e_{\alpha _{3}+\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(10)
\(e_{\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}} + e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4} +2\alpha _{5} +2\alpha _{6}+\alpha _{7}+\alpha _{8}}+e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\\+e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(11)
\(e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}}\\{+2\alpha _{6}+\alpha _{7}+\alpha _{8}}+e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(12)
\(e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+3\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}} {+} e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}}\\{+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(13)
\(e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+2\alpha _{7}+2\alpha _{8}} {+} e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+\alpha _{7}+2\alpha _{8}} {+} e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+3\alpha _{4}}\\{+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}} + e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}\);
-
(14)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+4\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}} {+} e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+2\alpha _{7}+2\alpha _{8}} {+} e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}}\\{+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+\alpha _{7}+2\alpha _{8}}\);
-
(15)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+5\alpha _{5}+4\alpha _{6}+2\alpha _{7}+2\alpha _{8}} + e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+5\alpha _{5}+3\alpha _{6}+2\alpha _{7}+3\alpha _{8}}\);
-
(16)
\(e_{2\alpha _{1}+3\alpha _{2}+4\alpha _{3}+5\alpha _{4}+6\alpha _{5}+4\alpha _{6}+2\alpha _{7}+3\alpha _{8}}\).
In particular, \(\dim ({\mathfrak c}_{{\mathfrak g}}(\chi ))=16\) and so \(d(\chi )=116=\left \vert {\Phi }^{+}\right \vert -4\). Proposition A.1 then says that all finite-dimensional \(U_{\chi }({\mathfrak g})\)-modules have dimension divisible by 2116.
1.10 A.10 E 8 in characteristic 3
Suppose Φ = E8 and p = 3. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{1}-\alpha _{2}-2\alpha _{3}-2\alpha _{4} - 2\alpha _{5}-\alpha _{6}-\alpha _{8}}{+}e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-2\alpha _{4} -2\alpha _{5}-\alpha _{6}-\alpha _{7}-\alpha _{8}} {+} e_{-\alpha _{2}-2\alpha _{3}-2\alpha _{4}-2\alpha _{5}}\\{-\alpha _{6}-\alpha _{7}-\alpha _{8}}+2e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4}-2\alpha _{5}-2\alpha _{6}-\alpha _{7}-\alpha _{8}}+2e_{-\alpha _{3}-2\alpha _{4}-3\alpha _{5}-2\alpha _{6}-\alpha _{7}-\alpha _{8}}+e_{-\alpha _{2}-\alpha _{3}-2\alpha _{4}-2\alpha _{5}-2\alpha _{6}-\alpha _{7}-\alpha _{8}}\);
-
(2)
\(e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}} + 2e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{8}} + e_{-\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{8}} + e_{-\alpha _{5}-\alpha _{6}-\alpha _{7}-\alpha _{8}} + 2e_{-\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{7}}\);
-
(3)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}+e_{\alpha _{7}}+e_{\alpha _{8}}\);
-
(4)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}} + e_{\alpha _{3}+\alpha _{4}+\alpha _{5}} + e_{\alpha _{4}+\alpha _{5}+\alpha _{6}} + e_{\alpha _{4}+\alpha _{5}+\alpha _{8}}+e_{\alpha _{5}+\alpha _{6}+\alpha _{8}} + e_{\alpha _{5}+\alpha _{6}+\alpha _{7}}\);
-
(5)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{8}} + 2e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}} + e_{\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}}\\+2e_{\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}} + e_{\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(6)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}} {+}2e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}}\\{+\alpha _{8}}+e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}+e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}}\);
-
(7)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}}{+}e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}} {+} e_{\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}}\\{+2\alpha _{8}}{+}e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}2e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}} {+} e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}}\\{+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(8)
\(e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}} {+} 2e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4} +3\alpha _{5} +2\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}}\\{+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}} {+} 2e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}}\\{+\alpha _{8}}\);
-
(9)
\(e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+2\alpha _{7}+2\alpha _{8}}{+}2e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}2e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+3\alpha _{4}}\\{+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}\);
-
(10)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+4\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}+e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+2\alpha _{7}+2\alpha _{8}} \\+ 2e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+\alpha _{7}+2\alpha _{8}}\)
-
(11)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+5\alpha _{5}+4\alpha _{6}+2\alpha _{7}+2\alpha _{8}} + 2e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+5\alpha _{5}+3\alpha _{6}+2\alpha _{7}+3\alpha _{8}}\);
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(12)
\(e_{2\alpha _{1}+3\alpha _{2}+4\alpha _{3}+5\alpha _{4}+6\alpha _{5}+4\alpha _{6}+2\alpha _{7}+3\alpha _{8}}\).
Therefore \(\dim ({\mathfrak c}_{{\mathfrak g}}(\chi ))=12\) and so \(d(\chi )=118=\left \vert {\Phi }^{+}\right \vert -2\). Proposition A.1 then says that each \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 3118.
1.11 A.11 E 8 in characteristic 5
Suppose Φ = E8 and p = 5. Since p is non-special in this case, we may apply Proposition A.1. We must therefore give \(c_{{\mathfrak g}}(\chi )\), and Sage computations show that \({\mathfrak c}_{{\mathfrak g}}(\chi )\) is the \({\mathbb K}\)-subspace of \({\mathfrak g}\) with the following basis:
-
(1)
\(e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4} - \alpha _{5}-\alpha _{6}}{+}4e_{-\alpha _{1}-\alpha _{2}-\alpha _{3}-\alpha _{4} -\alpha _{5}-\alpha _{8}} {+} e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{8}}{+}2e_{-\alpha _{3}-\alpha _{4}}\\{-2\alpha _{5}-\alpha _{6}-\alpha _{8}}{+}2e_{-\alpha _{2}-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}-\alpha _{7}}{+}2e_{-\alpha _{4}-2\alpha _{5}-\alpha _{6}-\alpha _{7}-\alpha _{8}} {+} 3e_{-\alpha _{3}-\alpha _{4}-\alpha _{5}-\alpha _{6}}\\{-\alpha _{7}-\alpha _{8}}\);
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(2)
\(e_{\alpha _{1}}+e_{\alpha _{2}}+e_{\alpha _{3}} +e_{\alpha _{4}} + e_{\alpha _{5}} +e_{\alpha _{6}}+e_{\alpha _{7}}+e_{\alpha _{8}}\);
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(3)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}}+e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}} + e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{8}} {+} 2e_{\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}} {+} 3e_{\alpha _{3}+\alpha _{4}}\\{+\alpha _{5}+\alpha _{6}+\alpha _{8}}+e_{\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}+2e_{\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
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(4)
\(e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{8}} + 4e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}} + e_{\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}}+3e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}}\\+4e_{\alpha _{4}+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}+3e_{\alpha _{2}+\alpha _{3}+\alpha _{4}+\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}} + e_{\alpha _{3}+\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(5)
\(e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{8}}{+}e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}+\alpha _{6}+\alpha _{7}+\alpha _{8}} {+} 2e_{\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}}\\{+2\alpha _{8}}{+}2e_{\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}3e_{\alpha _{2}+\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}}\\{+2\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}\);
-
(6)
\(e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}4e_{\alpha _{2}+2\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}} {+} e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+2\alpha _{4}+2\alpha _{5}}\\{+2\alpha _{6}+\alpha _{7}+\alpha _{8}}{+}2e_{\alpha _{1}+\alpha _{2}+\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}4e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+2\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}}\\{+\alpha _{8}}\);
-
(7)
\(e_{\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+2\alpha _{7}+2\alpha _{8}} {+} 4e_{\alpha _{1}+\alpha _{2}+2\alpha _{3}+3\alpha _{4} +4\alpha _{5} +3\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}4e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}}\\{+3\alpha _{4}+3\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}{+}e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}\);
-
(8)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+4\alpha _{5}+2\alpha _{6}+\alpha _{7}+2\alpha _{8}}+e_{\alpha _{1}+2\alpha _{2}+2\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+2\alpha _{7}+2\alpha _{8}}\\+4e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+3\alpha _{4}+4\alpha _{5}+3\alpha _{6}+\alpha _{7}+2\alpha _{8}}\);
-
(9)
\(e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+5\alpha _{5}+4\alpha _{6}+2\alpha _{7}+2\alpha _{8}}+4e_{\alpha _{1}+2\alpha _{2}+3\alpha _{3}+4\alpha _{4}+5\alpha _{5}+3\alpha _{6}+2\alpha _{7}+3\alpha _{8}}\);
-
(10)
\(e_{2\alpha _{1}+3\alpha _{2}+4\alpha _{3}+5\alpha _{4}+6\alpha _{5}+4\alpha _{6}+2\alpha _{7}+3\alpha _{8}}\).
In particular we see that \(\dim c_{{\mathfrak g}}(\chi )=10\), and so \(d(\chi )=119=\left \vert {\Phi }^{+}\right \vert -1\). Hence, every finite-dimensional \(U_{\chi }({\mathfrak g})\)-module has dimension divisible by 5119.
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Westaway, M. A Note on Humphreys’ Conjecture on Blocks. Algebr Represent Theor 26, 2585–2604 (2023). https://doi.org/10.1007/s10468-022-10190-x
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DOI: https://doi.org/10.1007/s10468-022-10190-x