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diamond principle
Contents
Context
Homological algebra
Representation theory
representation theory

geometric representation theory

Ingredients
representation , 2-representation , ∞-representation

group , ∞-group

group algebra , algebraic group , Lie algebra

vector space , n-vector space

affine space , symplectic vector space

action , ∞-action

module , equivariant object

bimodule , Morita equivalence

induced representation , Frobenius reciprocity

Hilbert space , Banach space , Fourier transform , functional analysis

orbit , coadjoint orbit , Killing form

unitary representation

geometric quantization , coherent state

socle , quiver

module algebra , comodule algebra , Hopf action , measuring

Geometric representation theory
D-module , perverse sheaf ,

Grothendieck group , lambda-ring , symmetric function , formal group

principal bundle , torsor , vector bundle , Atiyah Lie algebroid

geometric function theory , groupoidification

Eilenberg-Moore category , algebra over an operad , actegory , crossed module

reconstruction theorems

Contents
Idea
In zig-zag persistent homology -theory, the diamond principle (Carlsson & de Silva 2010, Sec. 5.2 ) states that persistence diagrams of pairs of zigzag persistence modules are bijective if the modules coincide away from two consecutive edges where their difference forms an “exact diamond”.

Applied specifically to persistence modules given by the homology groups of zig-zags of topological spaces , the diamond principle is an incarnation of the Mayer-Vietoris sequence in algebraic topology .

In this case it says that the two canonical zigzag persistence modules associated with a filtered topological space (one built from union -zigzags , the other from intersection -zigzags ) in fact yield the same persistence diagrams .

Notice how the diamond principle is a result genuinely about persistent invariants, in that the two persistence modules forming an exact diamond do not have the same shape as diagrams and hence are non-comparable when regarded, say, as quiver representations (since they are objects of distinct categories ).

References
The diamond principle originates in

Review:

Last revised on May 23, 2022 at 07:09:48.
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