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Integral Mackey decomposition and Higman's criterion for group lattices

Statement

Let $(K, O, k)$ be a splitting $p$ -modular system and let $H$ be finite.

For subgroups $A, B \leq H$ and an $O B$ -lattice $V$ , with $L_{x} = A \cap x B x^{- 1}$ , there is a natural Mackey decomposition

${Res}_{A}^{H} {Ind}_{B}^{H} V ≅ ⨁_{x \in A \ H / B} {Ind}_{L_{x}}^{A} (^{x} {Res}_{B \cap x^{- 1} A x}^{B} V) .$

Induction is transitive and preserves finite-free lattices and direct summands. For an $O H$ -lattice $M$ and $Q \leq H$ , define

${Tr}_{Q}^{H} (α) = \sum_{t \in H / Q} t α t^{- 1} (α \in {End}_{O Q} (M)) .$

Then $M$ is relatively $Q$ -projective if and only if ${id}_{M} = {Tr}_{Q}^{H} (α)$ for some $α$ . If $M$ is nonzero indecomposable, is relatively $R$ -projective, and $P$ is a vertex of $M$ , then $P$ is contained in an $H$ -conjugate of $R$ .

Facts & Assumptions

Given: The modular system, finite groups, subgroups, and finite-free lattices in the Statement.

[F1]

Relative projectivity and vertices for these lattices are defined by the induction-summand condition (Relative projectivity and vertices for integral group lattices).

[F2]

A nonzero indecomposable $O H$ -lattice has a local endomorphism ring (Krull-Schmidt holds for finite-rank OH-lattices).

Proof

1.1

Decompose $H$ into its finite $A$ - $B$ double cosets. The summand of $O H \otimes_{O B} V$ supported on $A x B$ is identified with $O A \otimes_{O L_{x}}^{x} V, a \otimes v ⟼ a x \otimes v,$ where $q \in L_{x}$ acts on $^{x} V$ as $x^{- 1} q x$ acts on $V$ . These maps and their inverses are well-defined on the tensor relations, and their finite direct sum is the displayed Mackey isomorphism. Tensor associativity gives ${Ind}_{A}^{H} {Ind}_{B}^{A} ≅ {Ind}_{B}^{H}$ for $B \leq A \leq H$ . Since $O H$ is finite free as a right subgroup algebra, these operations preserve finite-free lattices; functoriality preserves split inclusions and retractions.

F1algebra

1.2

First connect F1's summand definition to a split induction counit. Put $Y = {Ind}_{Q}^{H} V = O H \otimes_{O Q} V$ . The counit $ε_{Y} : {Ind}_{Q}^{H} {Res}_{Q}^{H} Y \to Y$ has the explicit $H$ -linear section $s_{Y} (h \otimes v) = h \otimes (1 \otimes v) .$ It respects the relation $h q \otimes v = h \otimes q v$ because $q \otimes v = 1 \otimes q v$ in $Y$ , and $ε_{Y} s_{Y} = 1_{Y}$ . If $M$ is a summand of $Y$ with inclusion $i : M \to Y$ and retraction $r : Y \to M$ , then $s_{M} = ({Ind}_{Q}^{H} {Res}_{Q}^{H} r) s_{Y} i$ splits the counit for $M$ , by naturality of the counit and $r i = 1_{M}$ . Conversely, a split counit displays $M$ as a summand of its induced module.

Now let $T$ be left-coset representatives for $H / Q$ . The induction counit $ε : O H \otimes_{O Q} M \to M$ is $ε (h \otimes m) = h m$ . Given an $H$ -linear section $s$ , write $s (m)$ in the direct sum indexed by $T$ and let $α (m)$ be its coefficient in the identity-coset component. Equivariance makes $α$ $Q$ -linear, and $ε s = 1$ says $1_{M} = \sum_{t \in T} t α t^{- 1} = {Tr}_{Q}^{H} (α) .$ Conversely this trace identity makes $s (m) = \sum_{t \in T} t \otimes α (t^{- 1} m)$ an $H$ -linear section of $ε$ . This proves the integral Higman criterion. [F1, algebra]

2.1

The image of every relative trace is a two-sided ideal of $E = {End}_{O H} (M)$ : an $H$ -endomorphism can be moved inside either side of the finite trace sum. Suppose now that $M$ is indecomposable, relatively $P$ -projective and relatively $R$ -projective. By step 1.2 choose trace expressions for $1_{M}$ from $P$ and from $R$ and multiply them. The diagonal $H$ -orbits on the finite set $H / P \times H / R$ regroup the product as a finite sum of relative traces from their stabilizers $s P s^{- 1} \cap t R t^{- 1} .$ The element inside each orbit trace is fixed by that stabilizer, so every summand belongs to the corresponding trace ideal.

step 1.2algebra

3.1

By F2 the ring $E$ is local. If every summand from step 2.1 were a nonunit, their sum could not be $1_{M}$ ; hence one is a unit. Its two-sided trace ideal then contains $1_{M}$ , and step 1.2 makes $M$ relatively projective for the associated intersection. Conjugating that subgroup by $s^{- 1}$ shows that $M$ is relatively $P \cap s^{- 1} t R t^{- 1} s$ , a subgroup of $P$ . If $P$ is a vertex, its minimality forces this intersection to be $P$ . Thus $P \leq s^{- 1} t R t^{- 1} s$ , as required. Empty double-coset sets cannot occur because $H$ is nonempty; trivial subgroups and $P = 1$ are included. All coset sets and sums are finite, so no choice principle is used.

F1F2step 1.2step 2.1∎

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Sources

Craven, The Brauer Correspondence, Proposition 2.3 and sections 2.1–2.2, pp. 19–22; Proposition 3.11, p. 36 (standard reference, not scraped)