Theorem 1.

Let G be a finite group, and let U,V be indecomposable O-free O⁢G-modules having sources of O-ranks prime to p. Suppose that W=U⊗OV is indecomposable and that W has a simple multiplicity module. Then the sources of W have O-rank prime to p, and there exist vertices Q,R of U,V, respectively, such that Q∩R is a vertex of W and such that QR is a Sylow p-subgroup of G.

Corollary 2.

Let G be a finite group, and let U,V be simple kG-modules with sources of dimensions prime to p such that W=U⊗kV is simple. Then the sources of W have dimension prime to p, and there exist vertices Q,R of U,V, respectively, such that Q∩R is a vertex of W and such that QR is a Sylow p-subgroup of G.

Corollary 3.

Suppose that the field of fractions K of O has characteristic zero. Let U,V be O-free O⁢G-modules with irreducible characters and sources of O-ranks prime to p. Suppose that the character of W=U⊗OV is irreducible. Then the sources of W have O-rank prime to p, and there exist vertices Q,R of U,V, respectively, such that Q∩R is a vertex of W and such that QR is a Sylow p-subgroup of G.

Corollary 4 (Navarro [13, Theorem A]).

Let G be a finite p-solvable group, and let U,V be simple kG-modules such that W=U⊗kV is simple. Then there exist vertices Q,R of U,V, respectively, such that Q∩R is a vertex of W and such that QR is a Sylow p-subgroup of G.

Lemma 5 (see e.g. [9, Theorem 5.1.11] or [12, Chapter 3, Theorem 1.17]).

Let G be a finite group, and let U,V be indecomposable O-free O⁢G-modules, and let Q,R be vertices of U,V, respectively. Then, for every indecomposable direct summand W of U⊗OV, there is x∈G such that Q∩Rx contains a vertex of W.

Lemma 6.

Let G be a finite group, and let U,V be indecomposable O-free O⁢G-modules having sources of O-ranks prime to p.

1. For any x∈G, there exists an indecomposable direct summand

   W⁢𝑜𝑓⁢U⊗𝒪V

   such that Q∩Rx is contained in a vertex of W.

2. If x∈G is chosen such that Q∩Rx has maximal order amongst the subgroups of the form Q∩Ry, where y∈G, then U⊗𝒪V has an indecomposable direct summand W with vertex Q∩Rx and with sources of 𝒪-rank prime to p.

3. Suppose that U⊗𝒪V is indecomposable. Then the sources of U⊗𝒪V have 𝒪-rank prime to p, and for any x∈G, the subgroup Q∩Rx is contained in a vertex of U⊗𝒪V. If Q∩Rx has maximal order amongst all subgroups of this form, then Q∩Rx is a vertex of U⊗𝒪V.

Proof.

Let Q,R be vertices of U,V, respectively. Let x∈G. Then Q∩Rx is contained in a vertex of U and in a vertex of V. The assumptions on U and V imply that ResQ∩RxG⁢(U) and ResQ∩RxG⁢(V) have indecomposable direct summands of 𝒪-ranks prime to p. Thus their tensor product ResQ∩RxG⁢(U⊗𝒪V) has an indecomposable direct summand Y of 𝒪-rank prime to p. In particular, Q∩Rx is the vertex of Y. Moreover, since Y is indecomposable, it follows that Y is isomorphic to a direct summand of ResQ∩RxG⁢(W) for some indecomposable direct summand W of U⊗𝒪V, and hence Q∩Rx is contained in a vertex of W. This shows (i). The same argument, assuming in addition that Q∩Rx has maximal order amongst the subgroups of this form, in conjunction with Lemma 5 shows (ii). Statement (iii) follows from (i) and (ii). ∎

Lemma 7 (Green [6, Theorem 9]).

Let G be a finite group, U an O-free indecomposable O⁢G-module, and Q a vertex of U. Then rkO(U)p≥|G:Q|p.

Theorem 8 ([7, Theorem 4.5]).

Let G be a finite group, U an indecomposable O-free O⁢G-module, and Q a vertex of U. Suppose that the sources of U have O-rank prime to p and that U has a simple multiplicity module. Then

Proof of Theorem 1.

Let Q,R be vertices of U,V, respectively, chosen such that Q∩R has maximal order amongst all intersections of a vertex of U and a vertex of V. It follows from Lemma 6 (iii) that Q∩R is a vertex of W=U⊗𝒪V and that the sources of W have 𝒪-rank prime to p.

Since W has a simple multiplicity module, it follows from Theorem 8 and Lemma 7 that we have

The left side is also equal to |G:Q|p⋅|Q:Q∩R|; hence cancelling |G:Q|p yields

Now |Q:Q∩R|=|QR|/|R|, so together this yields |Q⁢R|≥|G|p.

Let P be a Sylow p-subgroup of G containing Q. Let x∈G such that Rx⊆P. It follows from Lemma 6 (ii) that Q∩Rx is contained in a vertex of W. In particular, we have |Q∩Rx|≤|Q∩R|. Thus |Q⁢(Rx)|≥|Q⁢R|≥|P|. But since both Q,Rx are contained in P, we also have |Q⁢(Rx)|≤|P|. Thus all inequalities in this proof are equalities. This forces Q⁢(Rx)=P and |Q∩Rx|=|Q∩R|, so Q and Rx (instead of R) satisfy all conclusions. ∎

Remark 9.

The proof of Theorem 1 shows a bit more: with the notation and hypotheses of Theorem 1, any intersection of a vertex of U and a vertex of V is contained in a vertex of W, and any intersection of maximal order of a vertex of U and a vertex of V is a vertex of W. Moreover, for any choice of vertices Q,R of U,V, respectively, such that both Q,R are contained in a Sylow p-subgroup P of G, we have P=Q⁢R, and Q∩R is a vertex of W. This points to some information about fusion in G: if x∈G is chosen such that Q∩Rx has maximal order amongst all subgroups of this form, then Q∩Rx is a vertex of W, and by Lemma 6 (iii), for any y∈G, the group Q∩Ry is G-conjugate to a subgroup of Q∩Rx. In particular, if x, y are two elements in G such that Q∩Rx and Q∩Ry both have the same maximal order amongst all groups of this form, then Q∩Rx and Q∩Ry are G-conjugate since they both are vertices of W.

Remark 10.

With the notation and hypotheses of Theorem 1, denote by (Q,X), (R,Y), (S,Z) vertex-source pairs of U,V,W, respectively. The fact that the displayed inequalities in the proof of Theorem 1 are all equalities implies that we have

Moreover, by [9, Theorem 5.12.15], the associated multiplicity modules MU, MV, MW of U,V,W, respectively, satisfy

Remark 11.

Theorem 1 holds with slightly weaker hypotheses on the multiplicity modules of W. Instead of requiring the simplicity of a multiplicity module of W, it suffices to require the equality

for the multiplicity module MW of W associated with a vertex-source pair (S,Z) of W (this is the last equality in Remark 10). By a result of Knörr [7, Proposition 4.2], this equality holds if MW is simple (essentially because MW is then a projective simple module of a finite central p′-extension of NG⁢(S,Z)/S), but the simplicity of MW is not necessary in general for this equality. In particular, even if MW is simple, we do not know whether this implies that MU,MV are necessarily simple as well.

Remark 12.

With the notation and hypotheses of Theorem 1, if D,E are defect groups of the blocks to which U,V belong, respectively, such that D,E are both contained in a fixed Sylow p-subgroup P of G, then D and E contain vertices of U and V, respectively, and hence P=D⁢E by Remark 9. It would be interesting to investigate whether there are more precise relationships between the defect groups of the three blocks to which U,V,W belong. For the remainder of this Remark, suppose that all three modules U,V,W have simple multiplicity modules (this is for instance the case in any of the Corollaries 2, 3, 4). Then, as a consequence of the results of Knörr in [7], the vertices Q,R, Q∩R of U,V,W in Theorem 1 are centric in the fusion systems of the blocks to which these modules belong (see [10, Theorem 10.3.1] for a formulation of [7, Theorem 3.3] in terms of fusion systems of blocks). Thus, for instance, if the block of 𝒪⁢G to which U (resp. V,W) belongs has abelian defect groups, then Q (resp. R, Q∩R) is a defect groups of this block. Furthermore, by a slight generalisation [10, Theorem 10.3.6] of a result of Erdmann [5], if Q (resp. R, Q∩R) in Theorem 1 is cyclic, then it is a defect group of the block to which U (resp. V,W) belongs.

Remark 13.

E. Giannelli pointed out that the situation is well understood for simple tensor products of simple modules over symmetric groups. Let n be a positive integer. Let U,V be simple k⁢Sn-modules of dimensions greater than 1 such that W=U⊗kV is simple. Then, by [2, Main Theorem], we have p=2 and n is even. The exact tensor products which can arise in this way are described in [11], proving a conjecture of Gow and Kleshchev. It follows from that description that then n=2⁢m for some odd integer m and that U can be chosen to be the basic spin module (labelled by the partition (m+1,m-1)). By [4, Theorem 1.1], the vertices of U are the Sylow 2-subgroups of Sn. It is, however, not always possible to choose vertices Q,R of U,V, respectively, such that QR is a Sylow 2-subgroup and such that Q∩R is a vertex of W. Note that U has sources of even dimension (cf. [1, Lemma 5.3] and [4, Theorem 6.1]), so the hypotheses of Theorem 1 are not satisfied. The following example is due to E. Giannelli. Consider the case n=6, with simple k⁢S6-modules U,V,W labelled by the partitions (4,2), (5,1), (3,2,1), respectively. It follows from [11, Theorem 1.1] that we have W≅U⊗kV. As mentioned above, [4, Theorem 1.1] implies that U has a Sylow 2-subgroup Q of S6 as a vertex. Moreover, [3, Theorem 1.2] implies that also V has a Sylow 2-subgroups R of S6 as a vertex. The product QR is therefore a Sylow 2-subgroup of S6 if and only if Q=R. In that case, we have Q∩R=Q=R, but W has a trivial vertex. In general, for G an arbitrary finite group and U,V,W simple kG-modules such that W≅U⊗kV, it seems to be unknown at present whether there is always a choice of vertices Q,R of U,V, respectively, such that Q and R generate a Sylow p-subgroup of G.