ON A PROJECTIVE GENERALIZATION OF ALPERIN’S CONJECTURE Geoffrey R. Robinson Department of Mathematics and Computer Science University of Leicester Leicester LE1 7RH England ( e-mail grr1@mcs.le.ac.uk ).

INTRODUCTION: In this paper, we prove that a “projective” generalization of the Knörr-Robinson formulation ( [3]) of Alperin’s conjecture for a finite group G with a central p-subgroup Z holds for G if the “ordinary” form of the conjecture holds for G/Z. This implies that “on average”, Dade’s projective conjecture [2] , (which may be thought of as a refinement of the formulation in [3] of Alperin’s conjecture which takes account of character defects and linear characters of central p-subgroups) follows if the “ordinary” formulation from [3] of Alperin’s conjecture holds for the appropriate central quotient. More precisely, if G is a finite group with a central p-subgroup Z, and Alperin’s conjecture ( for p) holds in every section of G/Z, then if Dade’s projective conjecture fails for some linear character, λ, of Z, and some defect d, it must also fail at least one other defect ( for the same λ). Even more precisely, the sum ( for a given λ ) over all defects d of the “error terms” is 0. We consider that the results of this paper reinforce the view, already evidenced in [3] and [5], that“module-theoretic” methods are likely to provide insights into Alperin’s conjecture ( and related problems ) which might not be available from more naive counting arguments.

SECTION 1 : NOTATION, DEFINITIONS, GENERAL ARGUMENTS

Let R be a complete discrete valuation ring of characteristic 0, with unique maximal ideal π such that F = R/π is algebraically closed of prime characteristic p, and such that R contains a primitive |X|-th root of unity for each finite group X considered in what ensues. Let 𝕂 be the field of fractions of R. All modules considered here are finitely generated as R-modules unless explicitly stated otherwise.

We recall that a p-permutation module ( over R or F ) for a finite group is one whose restriction to a Sylow p-subgroup of that group is isomorphic to a permutation module. We will have occasion to deal with “linear source” modules ( or p-monomial modules ) over R for a finite group, that is, modules whose restriction to a Sylow p-subgroup of that group is isomorphic to a monomial module.

We recall from [3] that a block B of a finite group H is said to be of Lefschetz type if

∑_{σ∈S_{p}(H)/H}(−1)^{|σ|}k(B_{σ}) = 0,

where S_{p}(H) denotes the p-subgroup complex of H, and B_{σ} is the union of blocks defined as in [3].

Generally speaking, we retain the terminology and notation of [3], [4], and [5].

Let G be a finite group with a central p-subgroup, Z, and let G̃ denote G/Z, and in general we use a tilde to denote images in G̃. We let S_{p}(G, Z) denote the set of chains in S_{p}(G) whose initial subgroup is Z. This may be thought of as the p-subgroup complex of G/Z, with appropriate adjustment of the length of chains. Let B be a block of RG, B̃ denote the corresponding block of RG̃. Then we may view B as an RG-module under conjugation, and clearly Z is in the kernel of the associated action of G. We recall from [3] that

∑_{σ∈S_{p}(G̃)/G̃}(−1)^{|σ|}IndG̃ ; G̃_{σ}(B̃_{σ})

is a virtual projective RG̃-module.

Now let λ be a linear character of Z. Let k(G, λ) denote the number of irreducible characters of 𝕂G which lie over the linear character λ of Z ( and let k(B, λ) have an analogous meaning for characters assigned to B ). Let E_{λ} = ∑_{z∈Z} λ(z^{−1})z. Then E_{λ}RG is also an RG̃-module via conjugation. We first claim that E_{λ}RG is a p-monomial module ( in fact, a monomial module). Let T be a fixed transversal to Z in G. Since E_{λ}zt = λ(z)E_{λ}t for all t ∈ T, z ∈ Z, it follows that each element of E_{λ}RG has a unique expression in the form ∑_{t∈T} α_{t}E_{λ}t, where each α_{t} ∈ R. Furthermore, for any t ∈ T and g ∈ G, there is a unique z ∈ Z and a unique u ∈ T such that t^{g} = uz. Then (E_{λ}t)^{g} = λ(z)E_{λ}u. Hence we have explicitly demonstrated that E_{λ}RG is a monomial module for G̃ under conjugation action. Let Q be a p-subgroup of G which contains Z. We now determine which elements of the given “monomial basis44 are sent to scalar multiples of themselves by every element of Q. If (E_{λ}t)^{x} = αE_{λ}t for some α ∈ R, then we must have t^{x} = tz for some z ∈ Z, while conversely, if t^{x} = tz for some z ∈ Z, then (E_{λ}t)^{x} = λ(z)E_{λ}t. Hence the elements of the monomial basis we are seeking to determine are the elements {E_{λ}t : t̃ ∈ CG̃(Q̃)}. This will enable us to adapt some of the ideas and results of Broué [1] to the current context. If we consider E_{λ}RG as an RN_{G}(Q) module under conjugation, then we see that all the summands with vertex Q must occur as summands of E_{λ}RC_{G}(Q/Z).

We remark that for each λ, E_{λ} ≡ E_{1} ( mod πG), so that E_{λ}RG has reduction (mod π) isomorphic to F G̃ as F G̃-module under conjugation. We will give a detailed proof that

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}Ind_{G_{σ}}^{G}(E_{λ}RG_{σ})

is a virtual projective RG̃-module. By the Burry-Carlson-Puig theorem, we need to prove that whenever Q is a p-subgroup of G which strictly contains Z, then the restriction of the above virtual module to N_{G}(Q) involves no summand with vertex Q ( for the moment, we are considering the above as a virtual RG-module such that Z has trivial action on all summands involved). The restriction in question is

∑_{σ∈S_{p}(G,Z)/N_{G}(Q)}(−1)^{|σ|}Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(RG_{σ}),

and it is clear that only Q-stable chains can give any summands with vertex Q, so we need to determine the summands with vertex Q which occur in

∑_{σ∈S(G,Z)^{Q}/N_{G}(Q)}(−1)^{|σ|}Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(RG_{σ}).

As remarked above, for each σ ∈ S(G, Z)^{Q}, the summands of Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(RG_{σ}) with vertex Q are the summands of Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(RC_{G_{σ}}(Q/Z)) with vertex Q.

Now we choose a chain σ ∈ S(G, Z)^{Q}, say σ = Z < V_{1} < … < V_{n}, and we construct a chain σ′ ≠ σ as follows : choose m maximal such that Q ≰ V_{m} ( regarding Z as V_{0}, there is always such an m ). If V_{m+1} = QV_{m}, let σ′ = Z < … < V_{m} < V_{m+2} < … < V_{n} (if m = n − 1 in this case, just omit V_{n} ). Otherwise, let σ′ = V < … < V_{m} < QV_{m} < V_{m+1} < … < V_{n} ( if m = n in this case, just insert QV_{n} after V_{n}). Then it is easy to see that σ′′ = σ, and that N_{G}(Q)_{σ} = N_{G}(Q)_{σ′}. Thus C_{G_{σ}}(Q/Z) = C_{G_{σ′}} (Q/Z), and the summands with vertex Q from the ( orbits of the ) chains σ and σ′ cancel each other. Thus the virtual module in question involves no indecomposable module with vertex Q, as required to complete the proof that the above virtual module is a virtual projective. Now we may count fixed-points on the virtual module in two ways. On the one hand, by Frobenius reciprocity, the number of fixed points on the above virtual module is

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}rank_{R}(E_{λ}Z(RG_{σ})).

This last number is

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}k(G_{σ}, λ).

On the other hand, since we have a virtual projective module, we may count fixed-points by counting the multiplicity of the projective cover of the trivial RG̃ module. For each σ, the number of occurences of the projective cover of the trivial module as a summand of Ind_{G_{σ}}^{G}(E_{λ}RG_{σ}) is the number of occurences of the projective cover of the trivial module as a summand of E_{λ}RG_{σ} ( as RG̃_{σ}-module ), which is the number of occurences of the projective cover of the trivial F G̃_{σ} module as a summand of the reduction ( mod π) of E_{λ}RG_{σ}. But, as remarked above, the reduction (mod π) of E_{λ}RG_{σ} is isomorphic to F G̃_{σ} as an F G̃_{σ} module ( with conjugation action ). Hence we may conclude that the multiplicity of the projective cover of the trivial module in the virtual module we are considering is

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}dim_{F} (T rG̃_{σ} ; 1(F G̃_{σ})).

This last number is also the number of fixed-points of G̃ on the virtual projective module

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}Ind_{G_{σ}}^{G}(F G̃_{σ}),

which is

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}k(G̃_{σ}).

Hence we have :

THEOREM 1 : ∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}k(G_{σ}, λ) = ∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}k(G̃_{σ}).

SECTION 2 : PROOF OF THE MAIN THEOREM

THEOREM 2 : Let G, Z, B be as in the previous section. Then for each linear character, λ, of Z, we have

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|+1}k(B_{σ}, λ) = ∑_{σ∈S_{p}(G̃)/G̃}(−1)^{|σ|}k(B̃_{σ}).

In particular, if B̃ is of Lefschetz type, then the alternating sum above is zero for each choice of λ.

PROOF : Following the lines of argument of [3], together with those of section 1 above, we need to show that

∑_{σ∈S_{p}(G,Z)/G}(−1)^{|σ|}Ind_{G_{σ}}^{G}(E_{λ}RB_{σ})

is a virtual projective module in the Green ring for RG̃. Once this is done, the proof can be completed along the same lines as the proof of Theorem 1. Since we gave the proof of Theorem 1 in some detail, we will only indicate where the arguments need to be modified to take account of blocks. The first difference is that E_{λ}B is a linear source RG̃-module ( being a direct summand of E_{λ}RG), but does not itself usually have a monomial basis. We let Q be a p-subgroup of G which strictly contains Z. We need to show

∑_{σ∈S(G,Z)^{Q}/N_{G}(Q)}(−1)^{|σ|}Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(B_{σ})

involves no summand with vertex Q. For each chain σ ∈ S_{p}(G, Z)^{Q}, we construct the chain σ′ exactly as we did in theorem 1. We need to note this time that the summands of Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(B_{σ}) which have vertex Q are ( up to isomorphism) the summands of Ind_{N_{G}(Q)_{σ}}^{N_{G}(Q)}(b_{σ}) with vertex Q, where b_{σ} = Br_{Q}(1_{B_{σ}})RC_{G_{σ}}(Q/Z) ( we note that, although we are working over R, Br_{Q}(1_{B_{σ}}) may be meaningfully interpreted as a well-defined sum of block idempotents of RN_{G_{σ}}(Q)). Then we have b_{σ} = b_{σ′} and N_{G_{σ}}(Q) = N_{G_{σ′}} (Q), so the summands with vertex Q from ( the orbits of ) σ and σ′ cancel each other. Thus the virtual module above is a virtual projective for RG̃, as Q is arbitrary.

REMARK : We remark that if B has defect group Z, then B̃ is a block of defect 0 of RG̃, so even this very special case of the above theorem gives an alternative proof of the familiar fact that k(B) = |Z| under these hypotheses. ( For in that case, B has defect group Z, and if σ is a chain in S_{p}(G, Z) other than Z, we have B_{σ} = 0. On the other hand, when σ is the chain Z, we have k(B, 1) = k(B̃) = 1 , so when λ is the trivial character, the alternating sum considered above is 1 ; the theorem then tells us that the alternating sum is 1 for each linear character, λ, of Z ).

ACKNOWLEDGEMENT: The author is indebted to Burkhard Külshammer, who brought an error in an earlier version of this paper to the attention of the author.

References

[1] Broué,M., On Scott modules and p-permutation modules: An approach through the Brauer morphism, Proc AMS,93,(1985),401-408.

[2] Dade,E.C.,Counting Characters in Blocks,II, J. Reine u. Angewandte Mathematik, 448,(1994),97-190.

[3] Knörr,R. and Robinson,G.R., Some remarks on a conjecture of Alperin, J. London Math Soc, (2),39,(1989), 48-60.

[4] Robinson, G.R. , Local Structure, Vertices, and Alperin’s conjecture , Proc. London Math. Soc,(3),72(1996),312-330.

[5] Robinson,G.R., and Staszewski R., More on Alperin’s conjecture , Astérisque,181-82, (1990),237-255.