Matroids and Canonical Forms: Theory and Applications

Motivation: computational homology

• For a typical 250-neuron experimental rig, the resulting chain complex has dimension in the hundreds of billions, while the largest benchmarked complex had n ≈ 3×10^9.These scales motivate an algorithm for computing homology in larger systems, including neuroscience applications.Source: Motivation: computational homology · p. 12

Approach: three ingredients

• The thesis synthesizes matrix factorization, matroid theory, and homological algebra, interpreting Gaussian pivoting through matroid exchange and exact-sequence splitting.It identifies discrete Morse theory as a shared thread and connects it to Schur complements, minimal bases, greedy optimization, and exact sequences.Source: Approach: three ingredients · pp. 12–13

Primal example: Jordan bases

• For a nilpotent operator T, its Jordan bases are exactly the orbits of minimum-weight q-bases, characterized by q(B_m) spanning q(K_m) for every m.Here K_m is the kernel of T^m and B_m is the intersection of the q-basis B with K_m.Source: Primal example: Jordan bases · p. 14
• Jordan bases can be computed by choosing vectors from each K_m that represent a basis in q(K_m)/q(K_{m−1}), then taking their union.The thesis also interprets constructive Jordan-form procedures as matroid greedy optimization or gradient descent.Source: Primal example: Jordan bases · pp. 15–16

Outline and Contributions of the Thesis

• Eirene improves elapsed time and max heap demands by roughly two orders of magnitude against leading competitors on the largest complexes in the literature.The thesis presents this as its principal outcome: a novel algorithm for computing homology and persistent homology.Source: Outline and Contributions of the Thesis · p. 16
• The thesis defines combinatorial homology using matroid bifiltrations, generalizing homology and persistent homology and enabling greedy algorithms to merge with homological algebra.For cellular spaces, it uses a three-way translation among topology, algebra, and combinatorics to develop efficient homology computation.Source: Outline and Contributions of the Thesis · p. 16, canonical forms · pp. 16–17

canonical forms.

• The thesis relates combinatorial exchange to matrix factorization via a homological-algebra lifting of Möbius inversion, and classifies LU and Jordan decompositions through greedy optimization.Its proposed foundation for spectral and algebraic Morse theory lifts the Morse complex to a Jordan complement.Source: canonical forms · p. 17

Notation

• For J ⊆ I, χJ denotes the zero-one characteristic function on I, equal to 1 on J and 0 elsewhere.Source: Notation · p. 19
• The notation specifies R and C, the function space k^I, the null space K(T), Hom(U,V), and the support of a function.Source: Notation · p. 19
• For indexed k-linear spaces, the product contains all indexed families, while the coproduct consists of families with finite support; these constructions usually coincide in cases of interest.Source: Notation · p. 19
• Given families of linear maps, ×f maps into the product and ⊕f maps from the coproduct by summing the component maps.Source: Notation · p. 19

3.1 Independence, rank, and closure

• A matroid is an independence system satisfying the Steinitz Exchange Axiom; its bases have equal cardinality, defining rank, and closure generalizes linear span.Circuits are minimally dependent sets; in graphical matroids they correspond to simple cycles.Source: §3.1 · pp. 21–22, §3.2 · p. 22

3.3 Minors

• Matroid minors arise by deletion and contraction, while dual-matroid independence means that the complement contains a basis.Contraction generalizes quotienting a vector space by a subspace; deletion restricts the matroid to the remaining ground set.Source: §3.3 · p. 24, §3.6 · p. 27

3.4 Modularity

• The rank function is submodular, and equality in its submodular inequality defines a modular pair of sets.For modular pairs, independent sets of S/(S∩T) are also independent in S/T.Source: §3.4 · p. 25

3.5 Basis Exchange

• Basis exchange transforms a based representation through a Schur complement, generalizing singleton clearing to block clearing.For a single-element exchange, the transformation clears the entering column to a unit vector; block clearing does this for multiple columns.Source: §3.5 · p. 26

Base, Rank, and Closure

• After rotating elements of a set S into a basis C, its rank is |S∩C| and its closure is cl(S∩C).The stopping conditions are that C contains a basis for S or that cl(S∩C) contains S.Source: Base, Rank, and Closure · p. 26

Contraction

• To represent M//S, rotate a basis element of S into the standard basis and apply the canonical deletion operator.When S is already in the basis, applying the operator directly gives a representation of M//S.Source: Contraction · p. 27

Deletion

• To delete S while preserving a standard representation, rotate as many elements of S as possible out of the basis, then apply the deletion operator.Restricting the resulting representation to E−S represents M−S with basis C−S.Source: Deletion · p. 27

4.1 Generators

• Freely generated set families are closed under union and finite intersection, and every such family is modular.These results connect free generation by an independent set to modularity.Source: §4.1 · pp. 28–29
• The sets E_T form a pairwise-disjoint partition of the union of S, but modularity alone need not imply free generation.Three Euclidean-plane points, none a scalar multiple of another, give a modular family that cannot be generated by an independent set.Source: §4.1 · p. 29
• A finite family is freely generated exactly when it satisfies (4.1.3) and (4.1.4), and its generating bases are precisely the bases of M/S.Source: §4.1 · pp. 30–31

Filtrations

• Filtrations are modular and freely generated, and the ranks of M and M/S agree for every filtration.The nonempty minors of M/S are Sp/Sp−1, with rank ρ(Sp)−ρ(Sp−1).Source: Filtrations · p. 32

Bifiltrations

• A bifiltration is freely generated if and only if it is modular.Source: Bifiltrations · p. 32

N-filtrations

• A union of two or more filtrations may be modular without being freely generated, including a family of three distinct lines in the Euclidean plane.Source: N-filtrations · p. 32

4.2 Minimal Bases

• F-minimal bases are exactly the bases of M/F, equivalently the bases that generate F.The section assumes weight functions take finitely many values, ensuring minimal bases exist.Source: §4.2 · p. 33
• F-G minimal bases either do not exist or are exactly the bases that generate F∪G; they exist precisely when F∪G is modular.When they exist, they are the bases of M/(F∪G).Source: §4.2 · p. 33

Canonical Forms

• For lattices whose bases satisfy the Exchange Axiom, the main result gives an exact existence condition and a combinatorial characterization of Jordan bases.The discussion notes that bases in lattices lacking the Jordan-Hölder property can have different cardinalities, and that prior work gave only a sufficient existence condition.Source: Canonical Forms · p. 34

5.1 Nilpotent Canonical Forms

• Every complementary strong map factors as a contraction followed by an embedding.In the factorization, the first map is the identity as a set function, and the second preserves rank in the stated minors.Source: §5.1 · p. 35
• Every Jordan basis freely generates the kernel and image filtrations of T.The section also characterizes Jordan bases as orbits of K-minimal bases in M/I(T), under the condition that every power of T is complementary.Source: §5.1 · p. 36
• A nilpotent strong map has a Jordan basis exactly when all its powers are complementary.When this condition holds, its Jordan bases are the orbits of K-minimal bases in M/I(T), and also the preorbits of I-maximal bases in M|K(T).Source: §5.1 · pp. 36–37
• The number of maximal orbits of each length is uniquely determined across Jordan bases, and finding one characterization amounts to matroid greedy optimization.The procedure in §8.2 finds an I-maximal basis in K(T); the text notes that the underlying argument appeared earlier and was revisited.Source: §5.1 · p. 37

5.2 Graded Canonical Forms

• A graded Jordan basis has a barcode uniquely determined by T.The section states that any two such bases determine identical barcodes.Source: §5.2 · p. 38
• T^m is complementary exactly when (Z_p,B_{p+m}) is modular for every p, and T is Jordan exactly when Z∪B is modular.The operator is constructed on the matroid union of graded pieces from closed filtrations Z and B with B_p ⊆ Z_p.Source: §5.2 · pp. 38–39
• The Jordan bases of T are exactly the orbits of the Z-B-minimal bases in M.The construction places copies of each element in graded pieces and sends e(p) to e(p+1).Source: §5.2 · pp. 38–39
• In the linear setting, matroid-theoretic Jordan bases correspond exactly to linear Jordan bases of the induced graded operator.The graded Jordan bases of Q are the orbits of the Z-B-minimal bases in W.Source: §5.2 · p. 40
• The Elder Rule coincides with the matroid greedy algorithm, and returns a Krull-Schmidt decomposition for finitely supported, pointwise finite-dimensional persistence modules.Source: §5.2 · p. 40

5.3 Generalized Canonical Forms

• Every finite-dimensional linear operator over an arbitrary field has a basis in which its matrix is block diagonal with generalized Jordan blocks.The blocks are indexed by irreducible polynomials and positive integer exponents.Source: §5.3 · p. 41
• The generalized Jordan block presentation is unique up to permutation of its blocks.A generalized Jordan basis is the basis corresponding to this block-diagonal form.Source: §5.3 · p. 41
• The construction models multiplication by x on quotient modules and uses a matroid independence system defined by whether projected sets generate submodules of length equal to their size.The resulting generalized Jordan bases of X are built from Jordan bases of the constructed operator P.Source: §5.3 · pp. 42–43
• Generalized Jordan bases of T are the generalized orbits of K-minimal bases of N/PN.Here K is the integer-valued weight function whose sublevel sets are K(P^m).Source: §5.3 · p. 43

Algebraic Foundations

• For categorical terminology, objects and morphisms can be read as vector spaces and linear maps, while monomorphisms and epimorphisms mean injections and surjections.Under these substitutions, references to preadditive and abelian categories may be omitted.Source: Algebraic Foundations · p. 45

6.1 Biproducts

• A product structure on W combines maps out of W into an isomorphism with a product, while a coproduct structure combines maps into W into an isomorphism from a coproduct.The dual structures form a biproduct structure.Source: §6.1 · p. 45

Examples

• For a finite family of vector spaces, the canonical biproduct on their direct sum uses coordinate projections and inclusions.For complementary subspaces in W, replacing the direct sum with W gives the canonical biproduct structure generated by those subspaces.Source: Examples · pp. 45–46
• Families of injective maps into W form a coproduct structure exactly when their images are complementary subspaces.Dually, a family of surjections out of W forms a product structure exactly when its kernels are cocomplementary.Source: Examples · p. 46

6.2 Idempotents

• An idempotent associated with a (co)product component acts as a projection, and idempotents are used to generate new complements from old.Projection along a subspace gives a linear isomorphism between any two of its complements.Source: §6.2 · pp. 46–47

6.3 Arrays

• A map represented relative to (co)product structures has an array representation, and multiplication of these representations agrees with composition.For vector spaces built from one-dimensional components, this is the standard matrix representation in the corresponding bases.Source: §6.3 · p. 48

6.4 Kernels

• In a coproduct structure {f,g}, g is a kernel to f♯, while g♯ is a cokernel to f.The section characterizes kernels by a monomorphism whose image equals the kernel of the map.Source: §6.4 · pp. 50–51

6.4.1 The Splitting Lemma

• The splitting lemma turns the stated kernel or cokernel conditions into coproduct or product structures, respectively.Its formulation is described as a bridge between matroid complementarity and categorical biproducts.Source: §6.4.1 · p. 51

6.4.2 The Exchange Lemma

• The exchange lemma reduces whether a pair forms a product or coproduct structure to invertibility of the corresponding composed map.The text presents this as an exhaustive answer to determining exchange pairs and as mathematically equivalent to the splitting lemma.Source: §6.4.2 · p. 52

Example 2: Preadditive kernels

• When a11 is invertible, the constructed map k is a kernel to λ opT in any preadditive category.The proof checks the kernel definition directly rather than relying on finite-dimensional dimension counting.Source: Example 2: Preadditive kernels · pp. 53–54
• For finite-dimensional k-linear spaces, the kernel property can be confirmed by dimension counting.For arbitrary preadditive categories, the proof instead checks the definition directly.Source: Example 2: Preadditive kernels · p. 54
• The factorization used in the construction is unique because k is a monomorphism.The proof uses λ op 2 k = 1 to establish that k is monic.Source: Example 2: Preadditive kernels · p. 54

Example 3: Cokernels in R2

• When a11 is nonzero, the example constructs kop so that kopTλ1 = 0; the image of Tλ1 has dimension 1.Since k is an epimorphism, kop is a cokernel to Tλ1.Source: Example 2: Preadditive kernels · pp. 54–55

Example 4: Preadditive cokernels

• If a11 is invertible, the constructed map is a cokernel to Tλ1 in any preadditive category.The proof is stated to be dual to the kernel argument.Source: Example 4: Preadditive cokernels · p. 55

6.4.3 Idempotents

• The kernel map k is λ2 composed with a canonical projection onto the kernel of λ op1 T; dually, kop is λ op2 composed with its corresponding projection.The projections act as identity on the respective kernel and vanish on the specified image.Source: §6.4.3 · p. 55

6.4.4 Exchange

• The constructed kernel and cokernel maps are dual morphisms in the biproduct structures obtained through the exchange lemma.The exchange argument uses invertibility of a11.Source: §6.4.4 · pp. 56–57

6.5 The Schur Complement

• Row and column operations produce arrays sharing a block called the Schur complement, and the section gives new characterizations of that object.The submatrix occurs in the forms obtained after row and column operations; entries may be block submatrices or maps in a preadditive category.Source: §6.5 · pp. 57–58

6.5.1 Diagramatic Complements

• In the diagram category [D,C], the Schur complement is a literal complement to α in T, an interpretation the authors say had gone unremarked in published literature.The remark acknowledges at least one earlier category-theoretic treatment.Source: §6.5.1 · p. 60

6.6 Möbius Inversion

• A nonzero path contribution A(p) implies that p increases monotonically with respect to the transitive closure of Supp(A) and the canonical order on {0,…,m}.Source: §6.6 · p. 61
• For A = 1 − t with strictly acyclic support, the inverse is the finite geometric sum because t^N = 0 for an array with N rows and columns.The proof derives the inverse from (1−t)^−1 = ∑ t^k.Source: §6.6 · pp. 61–62
• When A is the incidence array χ⪯ of a partial order, the identity recovers classical Möbius inversion.Source: §6.6 · pp. 61–62
• The theorem is stated to be strictly more general than the preceding lemma.Source: §6.6 · p. 62
• Acyclic support gives the theorem’s generalized Möbius inversion formula for an array arising from an automorphism in a preadditive category.The formula uses the strictly triangular part t of A.Source: §6.6 · pp. 61–62

Exchange Formulae

• The section catalogs algebraic and relational identities involved in structure exchange, with some formulae intended for consultation as needed.The text recommends reading the definitions and results of §7.1 before the later discussion.Source: Exchange Formulae · p. 64, §7.2 · p. 66

7.1 Relations

• A binary relation extends to a partial order exactly when its transitive closure is antisymmetric; for finite chains in a strictly acyclic relation, the set has cardinality greater than the chain length.Source: §7.1 · p. 64
• For a partial matching d, acyclicity is preserved under the relation transformation RdRRd, and a pairing is acyclic on R exactly when it is acyclic on that transformed relation.The induced relations on the two index sets are unchanged by this transformation.Source: §7.1 · p. 65

7.2 Formulae

• A partial matching on an array induces a transformed pairing that is acyclic when the original pairing is cyclic, and transitive relations extending the specified support also extend the transformed support.Source: Relations · pp. 66–67

Relations

• An invertible array with acyclic support has exactly one perfect matching, and its support and inverse support have identical transitive closures.The section also defines a row-first pivot associated with a partial matching.Source: Relations · pp. 66–67

Schur Complements

• The section characterizes Schur-complement blocks after partial matchings, including formulas for dual elements and conditions yielding a Jordan pair.The Jordan-pair result assumes T^2 = 0 and invertibility of g♯T f.Source: Schur Complements · pp. 67–68

Möbius Inversion

• For an acyclic partial matching, the transformed matrix representation is related to the original by the stated Möbius-inversion identity; a corresponding result is given for acyclic Jordan pairings.Source: M¨obius Inversion · pp. 68–69

8.1 LU Decomposition

• An exchange pair is characterized by invertibility of a♯Tb♭, and exchanging it replaces a and b with Tb♭ and a♯T in the coproduct and product structures.The Exchange Lemma gives the corresponding criteria for the updated pair to remain a coproduct and product.Source: §8.1 · pp. 70–71
• Under the rank-one assumption, LU Decomposition repeatedly pivots on an exchange pair and removes that pair from the remaining candidates.The algorithm stops when no exchange pair remains; the text states this is equivalent to (×α♯)T(⊕β♭) = 0.Source: §8.1 · p. 71
• The resulting change-of-structure arrays are lower- and upper-triangular, with nonzero entries appearing on exchange pairs.The pth column and row are drawn from successive representations of T, respectively.Source: §8.1 · p. 72
• For a biproduct of finitely many simple objects in a preadditive category, T(λ,υ) factors into lower- and upper-triangular arrays around a sparse zero-one array.When the morphisms are maps between copies of k, this is an LU decomposition of T(λ,υ).Source: §8.1 · p. 72

8.2 Jordan Decomposition

• If T^(n+1) = 0 and gT^n f = 1, then (k,T^0f,...,T^nf) is a coproduct structure, providing a Jordan complement for f.The section also gives a construction of a Jordan complement as eg, where e projects onto K1 along the earlier iterates of f.Source: §8.2 · pp. 73–75
• The Jordan decomposition algorithm selects a maximal eligible orbit, exchanges it and its necessary duals into the biproduct, and repeats until the complement vanishes.Each iteration increases the number of orbits in the biproduct and decreases the complement.Source: §8.2 · p. 76

8.3 Filtered Exchange

• For Ω = Hom(k,W), closure consists of maps whose images lie in the image of the coproduct, so independence agrees with linear independence of the corresponding vectors.This gives a composition-based characterization of independence in terms of biproducts.Source: §8.3 · p. 77
• A second basis υ is F-minimal exactly when the filtered relation extends a perfect matching on the support of 1(λ,υ).The criterion is equivalent to block-upper-triangular structure with invertible diagonal blocks.Source: §8.3 · p. 78
• Algorithm 3 chooses minimal exchange pairs and returns bases λ∞ and υ∞ that are F-minimal and G-minimal, respectively.The section states that exchange pairs in the preceding algorithms may be chosen minimal because candidates lie in a block of a block-diagonal array.Source: §8.3 · p. 78

8.4 Block exchange

• Set-level exchange yields block lower- and upper-triangular arrays for Algorithm 1 when elements are grouped by iteration.The analysis of earlier sections carries over to set operations with minimal modification.Source: §8.4 · p. 79

Acyclic blocks

• For acyclic pairings, the associated arrays are triangular up to permutation; the same holds for block Jordan elimination using Proposition 7.1.3 and Möbius inversion.Source: Acyclic blocks · p. 79

Minimal blocks

• Filtered versions of Algorithms 1 and 2 replace pair minimality with requiring induced relations to extend to ∼F and ∼G.Their correctness follows as for Algorithms 3 and 4; variants of the Jordan or LU exchange algorithms also return graded bases when initial structures are graded.Source: Minimal blocks · p. 79, Graded blocks · p. 80

Linear blocks

• When ∼F and ∼G are linear orders, minimal pairings coincide with acyclic pairings, and all minimal pairs form a minimal acyclic pairing containing every minimal pairing.For non-linear orders, modified linear orders contained in F and G yield bases minimal with respect to the original functions.Source: Linear blocks · pp. 79–80

9.2 The linear persistence module

• Every finite-dimensional persistence module decomposes into indecomposable submodules, and the multiplicity of each nontrivial interval is independent of the decomposition.Proper decompositions correspond one-to-one with projective classes of graded Jordan bases.Source: §9.2 · pp. 84–85

9.3 Homological Persistence

• Computing homological persistence can use Z-B minimal bases of V rather than the generally much larger direct sum of homology spaces.The graded Jordan bases of the induced map Q are the orbits of those minimal bases; an F-minimal Jordan basis of the differential on C(m) generates the cycles and boundaries.Source: §9.3 · p. 86

9.4.1 Related work

• The standard persistence algorithm has sharp worst-case cubic complexity, while another algorithm computes the same task in O(n^ω) time.Related algorithms with cubic worst-case bounds, including the twist algorithm [18] and the dual algorithm [21, 22], have shown empirical performance improvements.Source: §9.4.1 · p. 86

9.4.2 Basis selection

• Jordan pivots on square-zero graded maps can be represented by row and column clearing, linking their effect on sparsity to Gauss-Jordan elimination.The optimization focus is fill reduction in cellular boundary operators by leveraging topology; because representative benchmarks and broadly effective random models are lacking, the approach relies on simplifying assumptions, including an analogy to spectral graph smoothness.Source: §9.4 · p. 86, §9.4.2 · pp. 87–88
• In the cycle-and-spoke example, fundamental-circuit cardinalities summed to 70 for the relatively normal basis T1, versus 178 for the relatively tangential basis T2.With two exceptions, T1’s circuits had cardinality four; T2’s circuit for the spoke from 1 to 1 had size 24.Source: §9.4.2 · p. 88
• Ordering edges by w and −w favors bases relatively normal and tangential to the circle, respectively, although the resulting trees look qualitatively similar after metric information is discarded.The example graph has two 12-vertex cycles joined by spokes, heuristically treated as normal, while the cycle edges are treated as tangential.Source: §9.4.2 · pp. 88–89

9.4.3 Input reduction

• Because persistence often uses modules of dimension 10 or below, cells above dimension 1+p can be omitted, and acyclic pairing can discard dimension-(p+1) cells before constructing much of the Vietoris–Rips differential matrix.Eirene uses a linear ordering of simplices to determine these pairings before matrix construction begins.Source: §9.4.3 · pp. 89, 92

9.4.2 Basis selection

• Figure 9.2’s upper-basis circuits totaled 7.3 × 10^5 in cardinality (median 11 edges), versus 4.9 × 10^6 (median 52 edges) for the lower basis; the sparsity pattern was robust across examples and metrics.The bases were obtained by ordering edges by w and −w, respectively.Source: §9.4.2 · p. 89, Figure 9.2 · p. 90

9.5 Benchmarks

• Algorithm 4 with the stated optimizations is implemented in Eirene, and benchmark experiments use published sample spaces across cluster, shared-memory, and personal-computer systems.The supplied passages identify Tables 9.1 and 9.2 as reporting time and memory performance, but provide no table values.Source: §9.5 · p. 92, Table 9.2 · p. 93

Morse theory

• The discussion narrows Morse theory to Forman’s discrete Morse theory for algebraic complexes and discrete Morse-Witten theory for cell complexes.It describes the first as fundamental to modern homology computation and the second as modeling the Smale-Witten program.Source: Morse theory · p. 94

10.1 Smooth Morse Theory

• A Morse-Smale function yields a chain complex whose homology is isomorphic to the manifold’s homology, and adjacent-index critical points have finitely many connecting orbits.The Morse complex is formed by counting connecting orbits with signs; the finiteness statement applies when the indices differ by one.Source: §10.1 · pp. 95–96

10.2 Discrete and Algebraic Morse Theory

• Forman’s discrete Morse complex is built from critical cells and gradient paths, and its homology is isomorphic to the cellular complex’s homology.For flat Morse functions, Forman’s Witten-complex construction also relates its differential to the discrete Morse boundary operator.Source: §10.2 · pp. 97–100

• Theorem 10.2.1 extends to arbitrary 2-nilpotent operators on objects in abelian categories, including categories without a biproduct structure.Source: §10.3 · p. 100
• For a chain complex with a graded coproduct and discrete Morse function, the gradient pairing is acyclic and the Morse complex is its Jordan complement in the differential.Source: §10.3 · p. 100
• The next generalization removes both grading and the condition that the operator square to zero, applying to arbitrary operators on finite-dimensional real vector spaces.Source: §10.3 · p. 100
• Acyclic relations are fundamental to both Möbius inversion and classical algebraic Morse theory, and cardinality-one gradient fields coincide with Chen and Kerber’s clearing operations.Source: §10.3 · p. 101
• Matroid duality clarifies how primal and dual exchange operations combine into a Jordan pivot, while the Schur complement is presented as a categorical complement.Source: §10.3 · pp. 101–102
• For sufficiently large t, the Laplacian has a low sector of |ω|−2|df| eigenvalues near zero, separated from a high sector, and critical-cell vectors form a basis for the low eigenspace.Source: §10.3 · p. 103

10.3 Morse-Witten Theory

• For η∈E−d f (♭), ⟨u_τ,T_tk_tη⟩ has leading term e^{τ−η}τT_k₀η, with error o_e(e^{τ−η}).The argument divides the cell contributions into cases and shows that non-leading terms are negligible or vanish.Source: §10.3 · p. 108
• The proof extends the asymptotic to ⟨T_t h_τ,h_σ⟩ by bounding G and using expansions of N_σ₀ and π_σ₁.The expansion separates contributions supported on d f (♭) and d f (♯), after which the desired conclusion follows.Source: §10.3 · pp. 108–109
• For critical σ₀ and σ₁, the pairing is 1+o_e(1) when σ₀=σ₁ and o_e(e^{-|σ₀−σ₁|}) otherwise.The same asymptotic is stated as an immediate consequence.Source: §10.3 · p. 109

Abelian Matroids

• Semisimple abelian representations extend the idea of linear matroid representations.The chapter presents this as a natural extension.Source: Abelian Matroids · p. 110
• Deletion, contraction, and duality have natural analogues for abelian representations.The chapter also characterizes these objects as simpler and more general than their linear counterparts.Source: Abelian Matroids · p. 110
• The relation between primal and dual abelian representations is little more than a restatement of the categorical exchange lemma.The chapter presents this categorical account as evidence of potential interaction between matroid theory and category theory.Source: Abelian Matroids · p. 110

Linear representations

• A linear representation assigns vectors so that a subset is matroid-independent exactly when its indexed vectors are linearly independent.The section distinguishes independence of an indexed family from independence of its image as a set; the converse can fail.Source: Linear representations · pp. 110–111

Matrix representations

• Restriction of a linear representation is canonical, whereas contraction is defined by quotienting after restricting to the complementary set and is non-canonical.The section says there is no defined operation, canonical or otherwise, that produces a matrix representation of the dual from a given matrix.Source: Linear representations · p. 111, Matrix representations · p. 111

Based Representations

• Based representations inherit restriction and contraction operations and also admit a canonical dual operation.The dual is a bona fide representation of the dual matroid, a nontrivial fact of representation theory.Source: Based Representations · p. 112

Standard Representations

• A standard representation records the non-basis columns and uniquely determines a full matrix representation.Standard representations are not themselves matrix representations because their column indices do not range over the entire ground set.Source: Standard Representations · p. 112
• The dual of a standard representation is a standard representation for the dual matroid.The section also relates covariant standard representations to k-linear standard matrix representations by a one-to-one correspondence.Source: Standard Representations · p. 112

11.2 Covariant Matroids

• For finite-length objects, families of simple maps whose coproduct is monic or whose product is epic define independence systems that satisfy the Steinitz Exchange property.The exchange property follows from the Jordan–Hölder theorem and the semisimplicity of subobjects of semisimple objects.Source: §11.2 · p. 113

Abelian representations

• Abelian representations extend linear representation theory, with canonical restriction and, for semisimple representations, non-canonical contractions.The section does not define an operation that produces a matroid-dual representation from a given representation.Source: Abelian representations · p. 114

Morphic Representations

• Covariant morphic representations pair a map with a semisimple coproduct structure, while contravariant ones pair it with a semisimple product structure.Restriction and contraction have analogues for abelian representations; the described contraction uses an invertible block and a projection.Source: Morphic Representations · pp. 114–115

Standard representations

• Covariant standard representations correspond one-to-one with k-linear standard matrix representations, and standard representations transfer to the dual matroid.The section also establishes a dual standard representation for the dual matroid.Source: Standard representations · p. 115