Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Homomorphism polynomials span matrix-symmetric polynomials

Lax461425.HomomorphismSpanning · concepts/Lax461425/HomomorphismSpanning.lean · lax-461425

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    Over a field of characteristic zero, every matrix-symmetric polynomial in an n×nn × n matrix, for n≥1n ≥ 1, is a finite linear combination of homomorphism polynomials of bipartite multigraph patterns having at most nn vertices on each side.

    Concept map
    5 concepts; 6 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    Each proof establishes this claim relative to its assumptions.

    Lean source view on GitHub

    1import Lax461425.SymmetricFamilies
    2import Lax461425.HomomorphismPolynomials
    3import Mathlib.LinearAlgebra.Span.Defs
    4
    5/-!
    6---
    7title: Homomorphism polynomials span matrix-symmetric polynomials
    8type: theorem
    9---
    10Over a field of characteristic zero, every matrix-symmetric polynomial in an `n × n`
    11matrix, for `n ≥ 1`, is a finite linear combination of homomorphism polynomials of
    12bipartite multigraph patterns having at most `n` vertices on each side.
    13-/
    14
    15namespace Lax461425.HomomorphismSpanning
    16
    17open SymmetricFamilies HomomorphismPolynomials
    18
    19/-- Matrix-invariant polynomials are linear combinations of homomorphism polynomials of
    20patterns with at most `n` vertices on each side. -/
    21axiom invariant_mem_span_hom {K : Type*} [Field K] [CharZero K] {n : ℕ}
    22 (hn : 1 ≤ n) (p : MatrixPolynomial n K) (hp : MatrixInvariant p) :
    23 p ∈ Submodule.span K {q | ∃ F : Pattern,
    24 Fintype.card F.A ≤ n ∧ Fintype.card F.B ≤ n ∧ hom K F n = q}
    25
    26end Lax461425.HomomorphismSpanning
    27
    Show Proof
    Builds on
    Used by
    From Mathlib

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…