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Joint binary projections for commuting idempotents

Lax342547.JointProjectors · concepts/Lax342547/JointProjectors.lean · lax-342547

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    Natural Language Statement

    Theorem

    The simultaneous splitting used in Lemma 5.2, with both dimension bounds. The projections are indexed by all binary labels; at most dim(V) are nonzero.

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    1import Lax342547.MomentSpace
    2import Mathlib.LinearAlgebra.Dimension.Finite
    3
    4/-!
    5---
    6title: Joint binary projections for commuting idempotents
    7type: theorem
    8---
    9The simultaneous splitting used in Lemma 5.2, with both dimension bounds.
    10The projections are indexed by all binary labels; at most dim(V) are nonzero.
    11-/
    12
    13namespace Lax342547.JointProjectors
    14
    15open Lax342547.MomentSpace
    16
    17axiom exists_joint_decomposition {ι V : Type} [Fintype ι] [DecidableEq ι]
    18 [AddCommGroup V] [Module Binary V] [FiniteDimensional Binary V]
    19 (M : ι → Module.End Binary V)
    20 (hid : ∀ i, M i * M i = M i) (hcomm : ∀ i j, M i * M j = M j * M i) :
    21 ∃ P : (ι → Binary) → Module.End Binary V,
    22 (∑ s, P s) = 1 ∧ (∀ s, P s * P s = P s) ∧
    23 (∀ s t, s ≠ t → P s * P t = 0) ∧
    24 (∀ i s, M i * P s = s i • P s) ∧
    25 (∑ s, Module.finrank Binary (LinearMap.range (P s))) = Module.finrank Binary V ∧
    26 Nat.card {s // P s ≠ 0} ≤ Module.finrank Binary V
    27
    28end Lax342547.JointProjectors
    29
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