Bowman, Anne Henke, and Stuart Martin. References [AK] H.H. Andersen and M. Kaneda, The rigidity of tilting modules, Moscow Math. J. 11 (2011). [DH] S.R.D. and A.E. Henke, Quart. J. Math. 56 (2005). [BDM1] C. Bowman, S.R.D., and S. Martin, Int. Elec. J. of Algebra 9 (2011). [Ringel] C.M. Ringel, Appendix to [BDM1]. [BDM2] C. Bowman, S.R.D., and S. Martin, preprint, arXiv:1111.5811. The references will repeat at the end.
semisimple, simply-connected algebraic group over an algebraically closed ﬁeld K. We consider only rational representations of G; i.e., rational G-modules. A module is with no proper submodules is called irreducible, or simple. A module with no non-trivial direct summands is called indecomposable. Let L,L be simple rational G-modules. Problem Understand the indecomposable direct summands of a tensor product L⊗L of two simple modules. (Tensor product over K.) This problem has been widely studied in case K has characteristic zero. Then the representation theory is completely reducible, so the indecomposable direct summands are again simple modules. In Type A you have the Littlewood–Richardson rule, etc.
K has characteristic p > 0. Then the rep theory of G is no longer semisimple, so the indecomposable direct summands of L⊗L are not necessarily simple anymore. In general we still lack a character formula for the simple modules, but in low rank cases they are known.
important role in the rep theory. (Fix a maximal torus T contained in a Borel subgroup B, and let B correspond to the negative roots.) Let λ ∈ X(T)+ = set of dominant weights. L(λ) = simple module of highest weight λ; ∇(λ) = indG B K λ = induced module of highest weight λ; ∆(λ) = Weyl module of highest weight λ; T(λ) = indecomposable tilting module of highest weight λ; Pπ (λ) = projective indecomposable module of highest weight λ for some generalized Schur algebra S(π), where λ ∈ π. We have L(λ) = socle ∇(λ) = head ∆(λ). Furthermore, ∇(λ) ∼ = ∆(λ)◦, the contravariant dual. So: If ∇(λ) is simple then it must be tilting, and L(λ) = ∇(λ) = ∆(λ) = T(λ).
a ∇-ﬁltration (resp., ∆-ﬁltration). Objects in F(∇)∩F(∆) are called tilting. Then T(λ) is the unique indecomposable module of highest weight λ in F(∇)∩F(∆) [Ringel, Donkin]. Can always arrange the ﬁltrations so that ∇(λ) appears as a quotient and ∆(λ) appears as a submodule of T(λ). T(λ) is contravariantly self dual, i.e., T(λ)◦ = T(λ). The category of tilting modules is closed under tensor product. The projective Pπ (λ) belongs to F(∆) and satisﬁes the reciprocity property: [Pπ (λ) : ∆(µ)] = [∇(µ) : L(λ)], (µ ∈ π). Strong linkage principle: If [∇(λ) : L(µ)] = 0 then µ ↑ λ.
all simple roots α} (restricted weights). Any λ ∈ X(T)+ has a unique p-adic expansion in the form λ = λ0 +λ1p +λ2p2 +···λmpm such that each λj ∈ X1(T). Theorem (Steinberg’s tensor product theorem) L(λ) ∼ = L(λ0)⊗L(λ1)(1) ⊗···⊗L(λm)(m). Here M(j) means the module structure is twisted by composing the action with the jth iterate Fj of the Frobenius pth power map F : G → G.
tilting modules, due to S. Donkin [Math Zeit. (1993)]. Another fact used repeatedly in calculations is that the second socle layer of ∇(λ) (equivalently, the second radical layer of ∆(λ)) determines extensions between simple modules. This is expressed by the isomorphism Ext1 G (L(λ),L(µ)) ∼ = HomG (L(λ),∇(µ)/L(µ)), (λ > µ). I think this goes back to Cline, Parshall, and Scott. For small rank groups the structure of Weyl modules can be computed by various techniques, so we have such extension information.
∼ = j L(λj )(j), L(µ) ∼ = j L(µj )(j) we have L(λ)⊗L(µ) ∼ = j L(λj )⊗L(µj ) (j) . This suggests a restricted question: Study L⊗L where L,L are restricted simples. Suppose that has been solved, so L⊗L ∼ = Q∈F(G) [L⊗L : Q]Q where F(G) is the ﬁnite set of isomorphism classes of indecomposable direct summands that occur in a tensor product of two restricted simples. Then (∗) L(λ)⊗L(µ) ∼ = S ∏j [L(λj )⊗L(µj ) : Qj ] j Q(j) j . where the direct sum is over all ﬁnite sequences S = (Q0,Q1,Q2,...) of members of F(G).
indecomposable for any sequence S = (Q0,Q1,Q2,...) then we have solved the problem, in some sense. For example, if each member Q of F(G) has a restricted simple G1T-socle then (by Steinberg’s tensor product theorem) it follows that any twisted tensor product Q(S) = j Q(j) j also has simple G-socle, and hence must be indecomposable as G-module. This is exactly what happens in case G = SL2 (for any characteristic p) and in case G = SL3 (in characteristic 2) but not in general.
= s β is a ﬁnite Coxeter group, generated by reﬂections (for β a positive root) s β (v) = v − v,β∨ β (v ∈ X(T)⊗ Z R) in the hyperplane H⊥ β = {v : v,β∨ = 0}. This action restricts to an action of W on X(T). Let Wp be the aﬃne Weyl group, which is generated by all aﬃne reﬂections s β,mp (v) = s β (v)+mpβ. Wp also acts on X(T) as well as the Euclidean space X(T)⊗ Z R. Usually we shift the actions of W and Wp by the “dot” action w ·v = w(v +ρ)−ρ (w ∈ Wp) where 2ρ is the sum of the positive roots.
< v +ρ,α∨ < p, all positive roots α}; this is called the fundamental alcove. Then its closure A 1 is a fundamental domain for the action of Wp. The space X(T)⊗ Z R is “tiled” by (closures of) alcoves, and Wp acts simply transitively on them. If π is a subset of X(T) we say that a module M belongs to π is all of its composition factors have highest weight in π. Let pr λ V = sum of all submodules of V which belong to Wp ·λ. Jantzen’s translation principle: There is an equivalence of categories between pr λ and pr µ , for any λ,µ in the same facet.
Z and X(T)+ = Z≥0. So X(T)⊗ Z R = R. The fundamental alcove A1 is the open interval (−1,p −1) and its closure A 1 is [−1,p −1]. In this case the restricted region X1(T) = X(T)+ ∩A 1. (This does not generalize.) It follows from the strong linkage principle that ∇(λ) = L(λ) is simple (and hence that T(λ) = ∇(λ) = L(λ)) for any λ ∈ X1(T). So the tensor product L(λ)⊗L(µ) of restricted simples is tilting. Easy to see that the family F(SL2) consists precisely of the tilting modules T(λ) for 0 ≤ λ ≤ 2p −2. These modules have structure: T(λ) = L(λ) for 0 ≤ λ ≤ p −1; [L(2p −2−λ),L(λ),L(2p −2−λ)] for p ≤ λ ≤ 2p −2.
denote by i|j the point on the wall A i ∩A j between consecutive alcoves i,j. −1 p −1 2p −1 3p −1 A1 A2 A3 Instead of labeling modules by their highest weight λ we sometimes use the alcove number i such that λ ∈ Ai or the facet label i|j such that λ ∈ A i ∩A j when λ is on a wall. In this notation, we have that F(SL2) = {T(1) = L(1),T(1|2) = L(1|2),T(2) = [L(1),L(2),L(1)]}.
summands of any L⊗L (not necessarily restricted) are twisted tensor products of members of F(SL2). Multiplicities can be easily worked out. If p = 2 it turns out that any L⊗L is indecomposable. Furthermore, in this case every indecomposable tilting module can be factored as a tensor product of two simples (usually in many ways). For any p, we classify the cases where L⊗L is indecomposable. We also classify the indecomposable tilting direct summands. Thus for any p, we classify the indecomposable tilting modules that can be factored as a tensor product of two simples.
number the alcoves as in the following picture. 1 2 3 4 3 4 6 8 6 8 9 9 5 7 The dashed lines across the top indicate the upper bound on highest weights that can appear in a restricted tensor product L⊗L . Note that this is the set of dominant weights λ ≤ 2(p −1)ρ.
= 2,3. Alcoves 6,7,8,9,6 ,8 ,9 don’t appear when p = 2 and alcoves 8,9,8 ,9 don’t appear when p = 3. (For p = 2 all weights are on walls.) Integral weights are located at intersection points of the light lines, and the heavy lines deﬁne the walls of the alcoves. The center of the black circle is the origin in the Euclidean plane, while the southernmost vertex in each ﬁgure is at the point −ρ.
For any λ ∈ A2 there is a unique indecomposable module (not a highest weight module) with module structure diagram M(2) = 2 3 1 3 2 in terms of the alcove (not highest weight) notation. The module M(λ) and the simple module L(λ) with λ ∈ A2 are the only indecomposable non-tilting direct summands of a restricted L⊗L for G = SL3 in characteristic p ≥ 3.
tensor product L⊗L consists of the following indecomposable modules: A T(λ) for any λ ∈ X(T) such that 0 ≤ λ,α∨ ≤ 2p −2, for all simple roots α. B L(λ) and M(λ) for any λ ∈ A2. C A ﬁnite list, depending on p, of ‘exceptional’ tilting modules of the form T(λ), for various λ not already listed. For p = 2 there are no exceptional tilting modules; for p = 3 there are precisely four of highest weight lying on the wall 4|6 or 4 |6 ; for p > 3 the number of exceptional modules grows with p with the highest weight of such modules lying in the closure of the union of alcoves 6,8,9,6 ,8 ,9 . In case p = 2 the alcove A2 is empty, so (B) is vacuous. So for p = 2 the members of F(SL3) are just the tilting modules listed in (A).
restricted L⊗L are rigid (socle and radical series coincide) but this fails to hold for certain tilting modules for p = 3. The ﬁrst such example is T(4,3). The structure of T(4,3) is analyzed in detail by Ringel in his appendix to [BDM1], by determining the quiver and relations for the relevant block of the smallest generalized Schur algebra containing the weight. The recent paper [AK] by Andersen and Kaneda ﬁnds many more examples of non-rigid tilting modules.
modules in the Weyl ﬁltration of T(4,3). Look at the ﬁve copies of L(1,0). and the fact that two of them appear in the third radical layer. (1,0) (0,5) (5,1) (1,0) (1,0) (4,3) (1,0) (0,5) (5,1) (1,0) Since T(4,3) is contravariantly self-dual, we can see from this data that it must be a non-rigid module.
have restricted simple G1T-socle. Thus equation (∗) gives the complete decomposition of a tensor product of unrestricted simples. Thus the case p = 2 for SL3 behaves similarly to the SL2 case. If p = 3 the four exceptional tilting modules in F(SL3) are T(5,0), T(5,2), T(0,5), and T(2,5). The two modules T(5,2) and T(2,5) are simple tilting modules, and do not have restricted simple G1T-socles. All other members of F(SL3) have restricted simple G1T-socle. Thus, even though the individual summands in the decomposition (∗) are not all indecomposable, many of them are, and the ones that are not are still manageable through ’secondary’ decompositions. For p ≥ 5 the number and complexity of such secondary decompositions increases with p. Thus, we did not succeed in understanding unrestricted tensor product decompositions for large primes. But we do obtain in (∗) a ﬁrst approximation which gives some information.
∼ = L(1,0)⊗L(1,1)(1) ⊗L(1,0)(2) ⊗ L(0,1)⊗L(1,1)(1) ⊗L(1,0)(2) ∼ = L(1,0)⊗L(0,1) ⊗ L(1,1)⊗L(1,1) (1) ⊗ L(1,0)⊗L(1,0) (2) ∼ = T(1,1)⊕T(0,0) ⊗ T(2,2)⊕2T(1,1) (1) ⊗T(2,0)(2) ∼ = T(13,5)⊕2T(6,2)(1) ⊕2T(11,3)⊕2T(5,1)(1). In the calculation, the ﬁrst line follows from Steinberg’s tensor product theorem. To get the last line we applied Donkin’s tensor product theorem, after interchanging the order of sums and products.
tilting modules, Moscow Math. J. 11 (2011). [DH] S.R.D. and A.E. Henke, Quart. J. Math. 56 (2005). [BDM1] C. Bowman, S.R.D., and S. Martin, Int. Elec. J. of Algebra 9 (2011). [Ringel] C.M. Ringel, Appendix to [BDM1]. [BDM2] C. Bowman, S.R.D., and S. Martin, preprint, arXiv:1111.5811.