TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The decomposition map is independent of the stable lattice

Statement

If $L$ and $L^{'}$ are two $G$ -stable $O G$ -lattices in the same finite-dimensional $K G$ -module $V$ , then

$[L / m L] = [L^{'} / m L^{'}]$

in the modular Grothendieck group $R_{k} (G)$ .

Facts & Assumptions

Given: A finite-dimensional $K G$ -module $V$ and two $G$ -stable $O G$ -lattices $L, L^{'} \subseteq V$ .

[F1]

The decomposition map is defined by taking the class of the reduction of a stable lattice (Decomposition map from ordinary to modular Grothendieck groups).

[A1]

Because $L$ and $L^{'}$ are full lattices in the same $K$ -space, some integer $r \geq 0$ satisfies $π^{r} L \subseteq L^{'} \subseteq π^{- r} L$ , where $π$ is a uniformizer of $O$ .

Proof

technique · direct

1.1

By [A1], after replacing $(L, L^{'})$ by $(π^{r} L, L^{'})$ if necessary, we may assume $L \subseteq L^{'}$ . Multiplication by the scalar $π^{r}$ does not change the class of the reduction in the Grothendieck group, because $π^{r} L / m π^{r} L ≅ L / m L$ as $k G$ -modules.

A1givenalgebra

2.1

Put $A : = L^{'} / L$ . Reduction of the inclusion $L ↪ L^{'}$ has kernel $(L \cap π L^{'}) / π L$ and cokernel $L^{'} / (L + π L^{'})$ . Multiplication by $π$ identifies the kernel with $A [π] : = {a \in A : π a = 0}$ , while the cokernel is $A / π A$ . Thus $[L / π L] - [L^{'} / π L^{'}] = [A [π]] - [A / π A]$ in $R_{k} (G)$ .

step 1.1algebra

3.1

Multiplication by $π$ on the finite-length $O G$ -module $A$ gives exact sequences $0 \to A [π] \to A \to π A \to 0, 0 \to π A \to A \to A / π A \to 0.$ Their Grothendieck-group identities give $[A [π]] = [A / π A]$ . Both end terms are annihilated by $π$ , so this equality is an equality in $R_{k} (G)$ . Step 2.1 therefore gives $[L / m L] = [L^{'} / m L^{'}]$ .

step 2.1algebra

4.1

Hence the Grothendieck-group class in [F1] is independent of the stable lattice.

F1step 3.1∎

Depends on

Decomposition map from ordinary to modular Grothendieck groups

Used by

Decomposition numbers and the decomposition matrix Definition

Cited to discharge well-definedness by Decomposition map from ordinary to modular Grothendieck groups.

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)