A remark on the Dade group and the Burnside group

Serge Bouc

Abstract: The object of this note is to show that the formula for tensor induction of relative syzygies in the Dade group, stated in [3], can be viewed as a special case of a functorial homomorphism from the dual B^{∗} of the Burnside group to the subgroup D^{Ω} of the Dade group generated by relative syzygies. It follows that there exists a short exact sequence of functors

0 → R_{ℚ}^{∗} → B^{∗} → D^{Ω}/D_{tors}^{Ω} → 0 ,

where R_{ℚ} is the functor of rational representations. This may be viewed as an improvement (from ℚ to ℤ) of Theorem D of [5].

1. Introduction

1.1. Let p be a prime number, and k be a field of characteristic p. In [6],[7], E.C. Dade defined a group structure on the set D(P) = D_{k}(P) of equivalence classes of endo-permutation kP -modules, which is now called the Dade group of P. This group can also be viewed as the set of equivalence classes of permutation P -algebras over k, under a suitable relation.

Most examples of endo-permutation modules are provided by relative syzygies, defined as follows : let X be a finite P -set, then the relative syzygy Ω_{X} of the trivial module with respect to X is defined as the kernel of the augmentation map kX → k sending each element of the set X to 1.

In [3], it was shown that the subgroup D^{Ω}(P) generated by these relative syzygies is invariant under the natural functorial operations of restriction, inflation, deflation, and tensor induction on the Dade group. These operations can be defined using the corresponding operations of restriction, inflation, Brauer quotient, and tensor induction of permutation algebras. In particular, a rather complicated formula for tensor induction of relative syzygies was stated :

1.2. Theorem : [[3] Theorem 5.1.2] Let Q ⊆ P be p-groups. Let X be a non-empty finite Q-set. Then in the Dade group D(P)

Ten_{Q}^{P}Ω_{X} = ∑_{{U,V ∈[s_{P}] \atop U≤_{P} V}} μ_{P} (U, V) |\{a ∈ V \P/Q | X^{V^{a}∩Q} ≠ ∅\}| Ω_{P/U} ,

where [s_{P}] is a set of representatives of the poset s_{P} of conjugacy classes of subgroups of P, where μ_{P} is the Möbius function of the poset s_{P}, and X^{V^{a}∩Q} is the set of fixed points of X under V^{a} ∩ Q.

1.3. The object of this note is to give an interpretation of the previous formula in perhaps more conceptual terms, using the following definition :

Date : April 17, 2003

1.4. Definition : Let B(P) denote the Burnside group of P, and let

B^{∗}(P) = Hom_{ℤ}(B(P), ℤ)

denote the dual group.

If X is a finite P -set, let ω_{X} be the element of B^{∗}(P) defined on the canonical basis of B(P) by

ω_{X}(P/Q) = \{ \begin{matrix}1 & if X^{Q} ≠ ∅ \\ 0 & otherwise \end{matrix} ,

where Q is a subgroup of P.

1.5. Now let C_{p} denote the following category :

• The objects of C_{p} are the finite p-groups.

• If P and Q are finite p-groups, then Hom_{C_{p}}(P, Q) is the Grothendieck group of finite (Q, P)-bisets, or equivalently the Burnside group B(Q × P^{op}).

• The composition in C_{p} is bilinear, and if U : P → Q is a finite (Q, P)-biset and V : Q → R is a finite (R, Q)-biset, then the composition V ∘U is equal to V ×_{Q} U.

The category C_{p} is preadditive, in the sense of Mac Lane ([8]), and the correspondence sending a p-group P to its Burnside group B(P) is a functor from C_{p} to the category Ab of abelian groups : if Q is another finite p-group and U is a finite (Q, P)-biset, then the map B(U) : B(P) → B(Q) is the linear map sending the class of the P -set X to the class of the Q-set U ×_{P} X.

Similarly, the correspondence P ↦ B^{∗}(P) is an additive functor from C_{p} to Ab : with the same notation, the map B^{∗}(U) : B^{∗}(P) → B^{∗}(Q) is defined by

B^{∗}(U) = ^{t}B(U^{op}) ,

where ^{t}B(U^{op}) denote the transposed map of B(U^{op}), and U^{op} is the (P, Q)-biset with underlying set equal to U, and (P, Q)-action defined by

∀(g, h) ∈ P × Q, ∀u ∈ U, g.u.h (in U^{op}) = h^{−1}ug^{−1} (in U) .

1.6. In [5] Section 3, we attached to any endomorphism a of the field k, and to any finite p-group P, a map γ_{a} : D(P) → D(P), and we observed in Example (3.3) that the element Ω_{P/1} of D(P) is invariant under γ_{a}. The same argument shows more generally that if X is a finite P -set, then γ_{a}(Ω_{X}) = Ω_{X}. It now follows from Proposition (3.10) of [5], and from Sections 4 and 5 of [3], that the correspondence sending P to D^{Ω}(P) is also an additive functor from C_{p} to Ab.

The main results of this note can now be stated : 1.7. Theorem : There is a unique natural transformation Θ : B^{∗} → D^{Ω} of additive functors from C_{p} to Ab, with the property that

Θ_{P} (ω_{X}) = Ω_{X}

for any finite p-group P and any finite P -set X.

Let R_{ℚ} denote the functor from C_{p} to Ab sending a p-group P to the group R_{ℚ}(P) of its rational representions, and let R_{ℚ}^{∗} denote the dual functor. It follows from a theorem of Ritter and Segal ([4]) that the natural transformation B → R_{ℚ} sending the finite P -set X to the permutation module ℚX is surjective. This gives by duality an injective natural transformation i : R_{ℚ}^{∗} → B^{∗}.

1.8. Theorem : The image of the natural transformation Θ ∘ i is equal to the torsion part D_{tors}^{Ω} of D^{Ω}. In other words, there is an exact sequence of functors from C_{p} to Ab

0 → R_{ℚ}^{∗} → B^{∗} → D^{Ω}/D_{tors}^{Ω} → 0 .

1.9. Remark: Theorem 1.8 gives in some sense an explanation for the exact sequence of functors of Theorem D of [5] (see also Proposition 7.6.2 of [3]) : this sequence can be obtained by applying the functor Hom_{Ab}(−, ℚ) to the previous one, giving finally

0 → ℚD → ℚB → ℚR_{ℚ} → 0 ,

since moreover ℚD ≅ ℚD^{∗} (see Remark 10.3 of [5]), and since ℚD ≅ ℚD^{Ω} (by Proposition 7.4.9 of [3]).

2. Proof of Theorem 1.7

2.1. Theorem 1.2 follows easily from Theorem 1.7, by expressing that the map Θ commutes with (tensor) induction. Unfortunately, I couldn’t find any direct proof of Theorem 1.7, and the only proof I know uses Theorem 1.2.

2.2. Lemma : The set of elements ω_{P/Q}, for Q ∈ [s_{P}], is a ℤ-basis of B^{∗}(P). Proof: Let δ_{P/Q} be the element of B^{∗}(P) defined by

δ_{P/Q}(P/R) = \{ \begin{matrix}1 & if Q and R are conjugate in P \\ 0 & otherwise \end{matrix} .

Then the set of elements δ_{P/Q}, for Q ∈ [s_{P}], is the dual basis of the canonical basis of B(P). Now if R is a subgroup of P, one has that

ω_{P/R} = ∑_{{Q∈[s_{P}] \atop Q≤_{P} R}} δ_{P/Q} ,

since for any subgroup S of P, the set (P/R)^{S} is non empty if and only if S ≤_{P} R. Hence the set of elements ω_{P/R}, for R ∈ [s_{P}], is obtained from the basis (δ_{P/R})_{R∈[s_{P}]} of B^{∗}(P) by a matrix which is triangular with 1 on the diagonal, for a suitable ordering of the set [s_{P}]. The lemma follows.

2.3. Remark: By definition of the Möbius function of s_{P}, it follows that

δ_{P/R} = ∑_{{Q∈[s_{P}] \atop Q≤_{P} R}} μ_{P} (Q, R)ω_{P/Q} ,

thus for any φ ∈ B^{∗}(P)

φ = ∑_{{Q,R∈[s_{P}] \atop Q≤_{P} R}} φ(P/R)μ_{P} (Q, R)ω_{P/Q} .

2.4. The uniqueness assumption in Theorem 1.7 is now obvious : indeed by Lemma 2.2, the map Θ_{P} : B^{∗}(P) → D^{Ω}(P) is uniquely defined by

Θ_{P} (ω_{P/Q}) = Ω_{P/Q} .

It remains to check that this definition implies

(2.5)  Θ_{P} (ω_{X}) = Ω_{X}

for any finite P -set X, and that the maps Θ_{P} define a natural transformation of functors from B^{∗} to D^{Ω}.

2.6. In order to check relation 2.5, first note that by Remark 2.3

ω_{X} = ∑_{U,V ∈[s_{P}] ; {U≤_{P} V \atop X^{V} ≠∅}} μ_{P} (U, V)ω_{P/U}

in B^{∗}(P). Hence, it remains to check that a similar relation holds in D^{Ω}(P), namely

Ω_{X} = ∑_{U,V ∈[s_{P}] ; {U≤_{P} V \atop X^{V} ≠∅}} μ_{P} (U, V)Ω_{P/U} .

But this is precisely the content of Lemma 5.2.3 of [3], i.e. the case Q = P of Theorem 1.2.

2.7. Since every morphism in C_{p} is a linear combination of transitive bisets, and since any transitive biset is the composition of a restriction, followed by a deflation, followed by an isomorphism, followed by an inflation, and followed by an induction (see Lemme 3 of [1] or Lemma 7.4 of [5]), it is enough to check that

F (Θ_{P} (ω_{X})) = Θ_{Q}(F (ω_{X})) ,

whenever P and Q are finite p-groups, when X is any finite P -set, and F : P → Q is one of restriction, deflation, isomorphism, inflation, or induction. Hence there are essentially three cases :

• There is a group homomorphism f : Q → P, and F is restriction along f. This case involves restriction, inflation and isomorphism. It corresponds to the morphism from P to Q in C_{p} defined by the (Q, P)-biset P, acted on the right by multiplication by P, and on the left by first taking image by f and multiplying on the left in P. If R is any subgroup of Q, one has that

(Res_{f}ω_{X})(Q/R) = ω_{X}(P^{op} ×_{Q} Q/R) .

Now the map from P^{op} ×_{Q} Q/R to P/f(R) sending (g, qR) to gf(q)f(R), for g ∈ P and q ∈ Q, is an isomorphism of P -sets. Thus

(Res_{f}ω_{X})(Q/R) = ω_{X}(P/f(R)) = \{ \begin{matrix}1 & if X^{f(R)} ≠ ∅ \\ 0 & otherwise \end{matrix} .

Thus Res_{f}ω_{X} = ω_{Res_{f} X} in this case, as was to be shown, since Res_{f}Ω_{X} = Ω_{Res_{f} X} by Lemma 4.1.1 of [3].

• The group Q is equal to P/N, for some normal subgroup N of P, and F is deflation from P to Q. This case corresponds to the morphism from P to Q in C_{p} defined by the (Q, P)-biset Q, acted on the left by multiplication by Q, and on the right by first taking image in Q and multiplying on the right in Q. If R is any subgroup of P containing N, and if \overline{R} = R/N, one has that

(Def_{Q}^{P}ω_{X})(Q/\overline{R}) = ω_{X}(Q ×_{Q} Q/\overline{R}) = ω_{X}(P/R) = \{ \begin{matrix}1 & if X^{R} ≠ ∅ \\ 0 & otherwise \end{matrix} .

Thus (Def_{Q}^{P}ω_{X}) = ω_{X^{N}} in this case, as was to be shown, since Def_{Q}^{P}Ω_{X} = Ω_{X^{N}} by Lemma 4.2.1 of [3].

• The group P is a subgroup of Q, and F is induction from P to Q. This case corresponds to the morphism from P to Q in C_{p} defined by the (Q, P)-biset Q, acted on the left by multiplication by Q, and on the right by multiplication by P. If R is any subgroup of Q, one has that

(Ind_{P}^{Q}ω_{X})(Q/R) = ω_{X}(Q ×_{Q} Q/R) = ω_{X}(Res_{P}^{Q}Q/R)
= |{x ∈ P \Q/R | X^{P ∩^{x}R} ≠ ∅}| .

By Theorem 1.2, the equality to check in this case is

Ind_{P}^{Q}ω_{X} = ∑_{{U,V ∈[s_{Q}] \atop U≤_{Q}V}} μ_{Q}(U, V) |\{a ∈ V \Q/P | X^{V^{a}∩P} ≠ ∅\}| ω_{Q/U} ,

Let ω denote the right hand side of this relation. One has that

ω(Q/R) = ∑_{{U,V ∈[s_{Q}] \atop R≤_{Q}U≤_{Q}V}} μ_{Q}(U, V) |\{a ∈ V \Q/P | X^{V^{a}∩P} ≠ ∅\}|

= ∑_{V ∈[s_{Q}]} δ_{R,V}^{Q} |{a ∈ V \Q/P | X^{V^{a}∩P} ≠ ∅}| ,

where δ_{R,V}^{Q} = 1 if R and V are conjugate in Q, and δ_{R,V}^{Q} = 0 otherwise. Thus

ω(Q/R) = |{a ∈ R\Q/P | X^{R^{a}∩P} ≠ ∅}| ,

and ω = Ind_{P}^{Q}ω_{X}, as was to be shown.

This completes the proof of Theorem 1.7

2.8. Remark: The last part of this proof shows that conversely, Theorem 1.7 implies Theorem 1.2.

3. Proof of Theorem 1.8

3.1. If P is a finite p-group, and let φ ∈ B^{∗}(P). I must show that Θ_{P} (φ) is a torsion element of D(P) if and only if φ belongs to i_{P} (R_{ℚ}^{∗}(P)).

By Remark 2.3

φ = ∑_{{Q,R∈[s_{P}] \atop Q≤_{P} R}} φ(P/R)μ_{P} (Q, R)ω_{P/Q} ,

thus

Θ_{P} (φ) = ∑_{{Q,R∈[s_{P}] \atop Q≤_{P} R}} φ(P/R)μ_{P} (Q, R)Ω_{P/Q}

in D(P). The element Θ_{P} (φ) is a torsion element of D(P) if and only if the element 2|P |Θ_{P} (φ) is, i.e. if there exists an integer n > 0 such that

2n ∑_{{Q,R∈[s_{P}] \atop Q≤_{P} R}} φ(P/R)μ_{P} (Q, R)|P |Ω_{P/Q} = 0 .

By Proposition 6.5.1 of [3], for any finite P -set X, one has that

|P |Ω_{X} = − ∑_{{U⊆V ⊆P \atop X^{U} ≠∅}} |U|μ(U, V)Ten_{V}^{P} Δ(V) ,

where μ is the Möbius function of the poset of all subgroups of P, ordered by inclusion, and Δ(P) is defined in Notation 6.2.1 of [3] by

Δ(P) = Ω_{M(P)} ,

where M(P) is the disjoint union of sets P/Q, for maximal proper subgroups Q of P. It follows that

2n ∑_{{Q,R∈[s_{P}] \atop Q≤_{P} R}} φ(P/R)μ_{P} (Q, R) ∑_{{U⊆V ⊆P \atop U≤_{P} Q}} |U|μ(U, V)Ten_{V}^{P} Δ(V) = 0 .

Summing first on Q gives, by the defining property of the Möbius function μ_{P}

2n ∑_{U⊆V ⊆P} φ(P/U)|U|μ(U, V)Ten_{V}^{P} Δ(V) = 0 .

In this expression, for V ⊆ P, the coefficient of Ten_{V}^{P} Δ(V) is equal to

2n ∑_{U⊆V} φ(P/U)|U|μ(U, V) ,

hence it is constant on the conjugacy class of V in P. It follows that the whole previous summation can be written as

∑_{V ∈[s_{P}]} 2n|P : N_{P} (V)|( ∑_{U⊆V} φ(P/U)|U|μ(U, V))Ten_{V}^{P} Δ(V) = 0 .

By Corollary 6.5.2 and Proposition 6.4.1 of [3], this is equivalent to requiring that for any non-cyclic subgroup V of P

2n|P : N_{P} (V)| ∑_{U⊆V} φ(P/U)|U|μ(U, V) = 0 ,

or equivalently, since n > 0

∑_{U⊆V} φ(P/U)|U|μ(U, V) = 0 .

Now Theorem 1.8 follows from the following :

3.2. Lemma : Let P be a finite p-group, and let φ ∈ B^{∗}(P). Then φ ∈ i_{P} (R_{ℚ}^{∗}(P)) if and only if, for any non-cyclic subgroup Q of P

∑_{U⊆Q} φ(P/U)|U|μ(U, Q) = 0 .

Proof: The group R_{ℚ}^{∗}(P) is a free abelian group, with canonical basis (V^{∗})_{V ∈Irr_{ℚ}(P)} indexed by the set Irr_{ℚ}(P) of rational irreducible representations of P. The value of the linear form V^{∗} on a (finitely generated) ℚP -module W is equal to the multiplicity m(V, W) of V as a summand of W. Since B^{∗}(P) is also a free abelian group, there is a commutative diagram

(3.3)  \begin{matrix}R_{ℚ}^{∗}(P) & \underrightarrow{i_{P}} & B^{∗}(P) \\ r_{P} ↓ & & ↓ b_{P} \\ Hom_{ℤ}(R_{ℚ}(P), ℚ) & \overrightarrow{I_{P}} & Hom_{ℤ}(B(P), ℚ) \end{matrix} ,

where r_{P} and b_{P} are the canonical maps, which may be viewed as inclusions, and I_{P} = Hom_{ℤ}(i_{P} , ℚ). This square is obviously cartesian : if s : B(P) → R_{ℚ}(P) is the canonical map sending a P -set to the corresponding permutation ℚP -module, and if i : ℤ → ℚ is the canonical injection, then for α ∈ Hom_{ℤ}(R_{ℚ}(P), ℚ) and β ∈ B^{∗}(P), the equality α ∘ s = i ∘ β implies α(Im(s)) ⊆ Im(i) = ℤ, hence Im(α) ⊆ ℤ since s is surjective.

Hence to complete the proof of Lemma 3.2, it suffices to prove the following claim :

3.4. Claim : An element φ ∈ Hom_{ℤ}(B(P), ℚ), i.e. a linear form on B(P) with values in ℚ, lies in the image of the map I_{P} if and only if for any non-cyclic subgroup Q of P

(3.5)  ∑_{U⊆Q} φ(P/U)|U|μ(U, Q) = 0 .

Suppose first that φ ∈ Im(I_{P}). Then φ is a linear combination with rational coefficients of the elements I_{P} (V^{∗}), for V ∈ Irr_{ℚ}(P). It suffices to prove that Equation 3.5 holds for these elements.

If E and F are ℚP -modules, define

⟨E, F ⟩_{P} = dim_{ℚ} Hom_{ℚP} (E, F) ,

and extend this scalar product to a bilinear form on R_{ℚ}(P), with values in ℤ. With this notation, one has that

⟨V, V ⟩_{P} V^{∗}(W) = ⟨V, W ⟩_{P} ,

for any finite dimensional ℚP -module W. Thus if φ = I_{P} (V^{∗})

⟨V, V ⟩_{P} ∑_{U⊆Q} φ(P/U)|U|μ(U, Q) = ∑_{U⊆Q} |U|μ(U, Q)⟨V, ℚP/U⟩_{P}
= ∑_{U⊆Q} |U|μ(U, Q)⟨Res_{U}^{P}V, ℚ⟩_{U}
= ∑_{U⊆Q} μ(U, Q) ∑_{x∈U} χ_{V} (x) ,

where χ_{V} is the character of V. This can also be written as

⟨V, V ⟩_{P} ∑_{U⊆Q} φ(P/U)|U|μ(U, Q) = ∑_{x∈Q} χ_{V} (x)( ∑_{<x>⊆U⊆Q} μ(U, Q)) ,

and this is zero if Q is not cyclic. Hence Equation 3.5 holds.

Now Hom_{ℤ}(R_{ℚ}, ℚ) and Hom_{ℤ}(B, ℚ) both are Mackey functors for P over ℚ, and the maps I_{Q}, for Q ⊆ P, form a morphism of Mackey functors I : Hom_{ℤ}(R_{ℚ}, ℚ) → Hom_{ℤ}(B, ℚ). In particular, there is a natural action of the Burnside algebra ℚ⊗_{ℤ} B(P) on Hom_{ℤ}(R_{ℚ}(P), ℚ) and Hom_{ℤ}(B(P), ℚ). For example, the action of ℚ ⊗_{ℤ} B(P) on Hom_{ℤ}(B(P), ℚ) is given by

(Y φ)(X) = φ(X × Y) ,

for X and Y in B(P), and φ ∈ Hom_{ℤ}(B(P), ℚ).

This algebra ℚ ⊗_{ℤ} B(P) is a split semi-simple commutative algebra, with primitive idempotents

e_{Q}^{P} = \frac{1}{|N_{P} (Q)|} ∑_{R⊆Q} |R|μ(R, Q)P/R ,

for Q ∈ [s_{P}] (see for example [2] Theorem 3.3.2).

The proof of Claim 3.4 can be completed by induction on the order of P : first observe that there is nothing to prove if P is cyclic, since in that case i_{P} and I_{P} are isomorphisms, and since the condition of the claim is void if P is cyclic. This starts induction.

Now suppose that P is non-cyclic, and that Claim 3.4 holds for any proper subgroup of P. Let φ ∈ Hom_{ℤ}(B(P), ℚ), and suppose that relation 3.5 holds for any non-cyclic subgroup Q of P. If R ⊆ Q are subgroups of P, then

Res_{Q}^{P}φ(Q/R) = φ(P/R) .

Hence the induction hypothesis implies that Res_{Q}^{P}φ is in the image of the map I_{Q}, for any proper subgroup Q of P. Moreover

(e_{P}^{P} φ)(X) = φ(e_{P}^{P} X) = |X^{P} |φ(e_{P}^{P}) ,

by the defining property of the idempotent e_{P}^{P}. This shows that

e_{P}^{P} φ = φ(e_{P}^{P})δ_{P/P} .

Moreover

φ(e_{P}^{P}) = \frac{1}{|P |} ∑_{R⊆P} |R|μ(R, Q)φ(P/R) = 0

by assumption, since P is not cyclic. Hence e_{P}^{P} φ = 0, and φ = ∑_{Q∈[s_{P}]−{P }} e_{Q}^{P}φ. But

e_{Q}^{P}φ is a linear combination with rational coefficients of elements of Hom_{ℤ}(B(P), ℚ) of the form Ind_{R}^{P}Res_{R}^{P}φ, for R ⊆ Q. Hence e_{Q}^{P}φ ∈ Im(I_{P}), for any proper subgroup Q of P. Hence φ ∈ Im(I_{P}), as was to be shown.

References

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[4] S. Bouc. A remark on a theorem of Ritter and Segal. J. of Group Theory, 4:11–18, 2001.

[5] S. Bouc and J. Thévenaz. The group of endo-permutation modules. Invent. Math., 139:275–349, 2000.

[6] E. Dade. Une extension de la théorie de Hall et Higman,. J. Algebra, 20:570–609, 1972.

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[8] S. Mac Lane. Categories for the working mathematician, volume 5 of Graduate texts in Mathematics. Springer, 1971.

Serge Bouc UFR de Mathématiques Université Paris 7-Denis Diderot 75251. Paris Cedex 05 , France email: bouc@math.jussieu.fr