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Symmetric polynomial families

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

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    Natural Language Statement

    Definition

    Polynomials and families of polynomials in square matrix variables, and their invariance under independent row and column permutations.

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    DefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.

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    1import Lax461425.MatrixSymmetry
    2import Mathlib.Algebra.MvPolynomial.Rename
    3
    4/-!
    5---
    6title: Symmetric polynomial families
    7type: definition
    8---
    9Polynomials and families of polynomials in square matrix variables, and their invariance
    10under independent row and column permutations.
    11-/
    12
    13namespace Lax461425.SymmetricFamilies
    14
    15open MatrixSymmetry
    16
    17/-- Polynomials in the variables of an `n × n` matrix. -/
    18abbrev MatrixPolynomial (n : ℕ) (K : Type*) [CommSemiring K] :=
    19 MvPolynomial (Fin n × Fin n) K
    20
    21/-- A one-parameter family of polynomials in square matrix variables. -/
    22abbrev MatrixFamily (K : Type*) [CommSemiring K] :=
    23 ∀ n : ℕ, MatrixPolynomial n K
    24
    25/-- Invariance under independent row and column permutations. -/
    26def MatrixInvariant {K : Type*} [CommSemiring K] {n : ℕ} (p : MatrixPolynomial n K) : Prop :=
    27 ∀ γ : Γ n, MvPolynomial.rename (actVar γ) p = p
    28
    29end Lax461425.SymmetricFamilies
    30
    Builds on
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    From Mathlib

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