Folded Skew-Representable Matroids

• Folded skew-representable matroids are defined as matroids whose rank functions, when scaled by a positive integer, become skew‐representable, preserving underlying combinatorial structure.
• This topic involves two complementary approaches: rank-theoretic folding via tensor products and coefficient-theoretic folding using push-forward along skew tract morphisms.
• Recent advances introduce tensor product techniques and new linear rank inequalities that offer certificates for skew-representability and insights into modular extension properties.

Searching arXiv for the cited foundational and papers on skew-representability, skew tracts, multilinear representations, and folded skew-representable matroids. arXiv search query: all:"folded skew-representable matroids" OR all:"skew-representable matroids tensor products extension properties rank inequalities" OR id:(Bérczi et al., 14 Jul 2025) OR id:(Pendavingh et al., 2011) OR id:(Kühne et al., 2020) OR id:(Su, 2020) OR id:(Pendavingh, 2018) Folded skew-representable matroids are studied in two closely related senses. In the tensor-product framework, a matroid $M$ with rank function $r$ is folded skew-representable if there exists $k\in \mathbb{Z}_+$ such that $k\cdot r$ is skew-representable; equivalently, $M$ is $k$-folded skew-representable (Bérczi et al., 14 Jul 2025). Earlier literature does not usually define “folding” as a standalone primitive, but it develops the mechanisms that realize it: chain-group representations over skew partial fields, wrapping and unwrapping between multilinear representations and matrix-valued coefficients, push-forward along morphisms of skew tracts, and Frobenius-twisted coefficient systems over skew hyperfields (Pendavingh et al., 2011, Su, 2020, Pendavingh, 2018). Across these settings, the common theme is preservation of the underlying matroidal combinatorics while coefficient structure is compressed, transported, or rescaled.

1. Basic notions and the two main uses of folding

A skew-linear representation of a rank-$r$ matroid $M=(E,r)$ over a division ring $D$ is a function $E\to A$, where $r$0 is a right module over $r$1, such that $r$2 is independent in $r$3 if and only if its image is skew-linearly independent in $r$4. In coordinates one may take $r$5 and a matrix $r$6, with independence detected by column rank over $r$7 (Kühne et al., 2020). A multilinear representation over a field $r$8 is given by an order $r$9, a vector space $k\in \mathbb{Z}_+$0, and a map $k\in \mathbb{Z}_+$1 such that independent sets correspond to direct sums of $k\in \mathbb{Z}_+$2-dimensional subspaces and every finite sum of image subspaces has dimension a multiple of $k\in \mathbb{Z}_+$3 (Kühne et al., 2020).

One formal use of folding is therefore rank-theoretic: replace $k\in \mathbb{Z}_+$4 by $k\in \mathbb{Z}_+$5 and ask whether the scaled polymatroid is skew-representable. This is the explicit definition used in the tensor-product treatment of folded skew-representability (Bérczi et al., 14 Jul 2025). A second use is coefficient-theoretic: one starts with a representation over a noncommutative or enriched coefficient system and pushes it into a different algebraic environment while preserving the underlying matroid. The tract-theoretic push-forward, the skew-partial-field homomorphism theorem, and the multilinear wrapping/unwrapping correspondence are all examples of this second use (Pendavingh et al., 2011, Su, 2020).

These two uses are compatible rather than competing. The first is phrased at the level of rank functions and tensor products; the second is phrased at the level of coordinatizations, circuits, cocircuits, and coefficient transport. The later tensor-product results can therefore be read as an abstract rank-theoretic completion of operations that were already implicit in the representation-theoretic literature.

2. Chain groups and skew partial fields

The foundational determinant-free framework is the theory of skew partial fields. A skew partial field is a pair $k\in \mathbb{Z}_+$6, where $k\in \mathbb{Z}_+$7 is a ring, $k\in \mathbb{Z}_+$8, and $k\in \mathbb{Z}_+$9. If $k\cdot r$0 is finite, an $k\cdot r$1-chain group on $k\cdot r$2 is a submodule $k\cdot r$3; its elementary chains are the nonzero chains with inclusion-minimal support, and a chain is $k\cdot r$4-primitive if each coordinate lies in $k\cdot r$5. An $k\cdot r$6-chain group is one in which every elementary chain is a scalar multiple of a $k\cdot r$7-primitive chain. The supports of elementary chains are then the cocircuits of a matroid $k\cdot r$8, and a matroid is $k\cdot r$9-representable when it arises in this way (Pendavingh et al., 2011).

This construction extends Tutte’s chain-group approach to noncommutative rings. Duality is handled by the opposite ring $M$0: if $M$1, then $M$2 is a chain group over $M$3, $M$4, and $M$5. Deletion and contraction are defined directly on chain groups and satisfy

$M$6

As a result, skew-partial-field representability is closed under duals and minors (Pendavingh et al., 2011).

Tutte’s representability criterion also survives in skew form. If one chooses a $M$7-primitive chain $M$8 for each cocircuit $M$9, then the span of these chains represents the matroid exactly when every modular triple $k$0 admits $k$1 with

$k$2

Generator matrices, invertible row operations, pivoting, and column scaling by elements of $k$3 preserve representability and the represented matroid. In particular, if $k$4 is a ring homomorphism with $k$5, then $k$6 is an $k$7-chain group and $k$8. This is one of the basic coefficient-folding mechanisms in the theory (Pendavingh et al., 2011).

A central consequence is that skew partial fields properly enlarge skew fields as representation domains. The non-Pappus matroid is representable over a skew partial field, and the direct sum $k$9 is representable over a skew partial field $r$0 with $r$1, but not over any skew field because $r$2 has zero divisors (Pendavingh et al., 2011).

3. Multilinear conversion and the limits of matrix folding

One canonical folding operation is the passage between multilinear representations over a field and matrix-valued representations over a skew partial field. If $r$3 is a skew field and $r$4, the unwrapping operator $r$5 sends a matrix over $r$6 to an $r$7-matrix obtained by expanding each entry into an $r$8 block, while $r$9 wraps block matrices back into $M=(E,r)$0. These operations preserve addition, multiplication, and invertibility. The fundamental equivalence states that a matroid $M=(E,r)$1 has an $M=(E,r)$2-multilinear representation over $M=(E,r)$3 if and only if $M=(E,r)$4 is representable over the skew partial field $M=(E,r)$5 (Pendavingh et al., 2011).

In this sense, folding compresses an $M=(E,r)$6 block matrix over $M=(E,r)$7 into an $M=(E,r)$8 matrix with entries in $M=(E,r)$9, without changing the represented matroid. The same paper treats this correspondence as a representation-theoretic equivalence rather than a new object class, but it is one of the clearest formal realizations of a folded representation (Pendavingh et al., 2011).

A different use of folding appears in the comparison between skew-linear and multilinear matroids. There, folding means attempting to convert a skew-linear representation over a division ring $D$0 into a multilinear representation over a field by embedding $D$1 into a matrix algebra $D$2 and transporting independence via block columns. This can succeed. For the Weyl algebra in characteristic $D$3, the division ring of fractions is finite-dimensional over its center and hence embeds into a matrix algebra over a field extension; concretely, the matrices $D$4 over $D$5 satisfy $D$6, and the resulting block matrix gives a multilinear representation of order $D$7 (Kühne et al., 2020).

The conversion can also fail. The same Weyl matroid is not multilinear over any field of characteristic $D$8, since a multilinear realization would yield matrices $D$9 with $E\to A$0, forcing $E\to A$1, impossible in characteristic $E\to A$2. More strongly, there exists a skew-linear matroid arising from a Baumslag–Solitar construction that is representable over a division ring $E\to A$3 but has no multilinear representation over any field. The obstruction is group-theoretic: for invertible matrices $E\to A$4 over a field, the relation $E\to A$5 forces

$E\to A$6

whereas the division-ring realization can make the corresponding commutator nontrivial (Kühne et al., 2020).

These examples rule out the common expectation that every skew-linear representation can be folded into a multilinear one. They also show that the multilinear/skew-partial-field bridge of $E\to A$7 is exact for multilinear data but not universal for arbitrary skew-field representations (Pendavingh et al., 2011, Kühne et al., 2020).

4. Push-forward over skew tracts and Frobenius-twisted folding

Baker–Bowler’s tract formalism was extended to the noncommutative setting by the theory of skew tracts. A skew tract $E\to A$8 consists of a multiplicative group $E\to A$9 together with a null set $r$00 encoding additive relations. Matroids over skew tracts admit cryptomorphic descriptions by circuits, quasi-Plücker coordinates, and dual pairs, in weak and strong variants. Duality and minors are built into the formalism, and for perfect skew tracts, including skew fields, weak and strong notions coincide (Su, 2020).

For folding, the key construction is push-forward along a tract morphism $r$01. If $r$02 is a left $r$03-matroid with circuit set $r$04, then

$r$05

is the circuit set of a left $r$06-matroid $r$07, and the underlying ordinary matroid is preserved: $r$08 Quasi-Plücker coordinates fold entrywise, $r$09. The special morphism to the Krasner hyperfield $r$10 sends every nonzero coefficient to $r$11, and the folded image is exactly the underlying ordinary matroid (Su, 2020).

This push-forward framework isolates an important subtlety: dualization need not commute with folding unless the morphism intertwines involutions. By contrast, deletion and contraction behave well under folding because the quasi-Plücker formulas for minors are themselves functorial (Su, 2020).

A second coefficient-folding construction appears in positive-characteristic algebraic matroid theory. Given $r$12 of characteristic $r$13 and elements $r$14, one forms a skew hyperfield $r$15 from $r$16 and the Frobenius automorphism $r$17. The resulting left $r$18-matroid $r$19 has underlying matroid equal to the algebraic matroid $r$20, but its coefficients encode $r$21-derivatives and the full Frobenius flock of derivation spaces. Rescaling by $r$22 corresponds exactly to replacing $r$23 by $r$24, and push-forward along the degree homomorphism $r$25 yields Lindström’s valuated matroid (Pendavingh, 2018).

The $r$26 construction shows that folding need not merely forget structure. It can also package commutative algebraic dependence together with Frobenius and derivation data into a genuinely skew coefficient system, from which ordinary and valuated matroids are later recovered by push-forward or by taking boundary matroids (Pendavingh, 2018).

5. Examples, closure phenomena, and persistent obstructions

The available examples show both the reach and the limitations of skew and folded representability. The non-Pappus matroid is representable over skew fields and also over the skew partial field $r$27. By contrast, the Vámos matroid $r$28 is not representable over any skew partial field. Between these extremes lies the direct sum $r$29, which is representable over a skew partial field with zero divisors but over no skew field. These examples establish that skew-partial-field representability strictly extends skew-field representability, while still leaving classical nonrepresentability phenomena intact (Pendavingh et al., 2011).

Quaternionic unimodular matroids provide a distinguished subclass. With

$r$30

a matroid is QU if it is represented by a $r$31-QU chain group. This class contains the regular and sixth-roots-of-unity classes and strictly extends them; for example, $r$32 is QU but not SRU. The quaternionic theory admits a determinant-like map

$r$33

where $r$34 is the standard complex embedding. For a strong matrix $r$35, $r$36 equals the number of bases of $r$37, giving a generalized Matrix-Tree theorem in the QU setting (Pendavingh et al., 2011).

At the tract level, perfect skew tracts—including skew fields, $r$38, $r$39, and $r$40—satisfy weak $r$41 strong, whereas nonperfect settings can separate the two notions. Folding to $r$42 erases this distinction, since only the underlying ordinary matroid survives (Su, 2020). This is another sense in which folding is structurally lossy: it preserves the matroid, but not necessarily the distinction between weak and strong coefficient data.

Obstructions also appear in algorithmic form. It is undecidable to determine, from a matroid, whether it is skew-linear, and for fixed prime $r$43 or $r$44, it is undecidable whether the matroid is representable over some division ring of characteristic $r$45. There also exists a division ring $r$46 for which representability over $r$47 is undecidable (Kühne et al., 2020). Thus even when folding suggests a route from one coefficient system to another, there is no general decision procedure for detecting when a skew representation exists in the first place.

6. Tensor products, folded rank functions, and new rank inequalities

The 2025 tensor-product framework turns folded skew-representability into a property of rank functions. If $r$48 is a polymatroid function on $r$49 and $r$50 on $r$51, then $r$52 consists of polymatroids on $r$53 satisfying

$r$54

For matroids, this specializes to a tensor product $r$55 on $r$56 with

$r$57

A matroid is $r$58-tensor-compatible with $r$59 when the $r$60-fold iterated tensor product $r$61 is nonempty (Bérczi et al., 14 Jul 2025).

This viewpoint yields a characterization of skew-representability. Let $r$62 be a connected skew-representable matroid of rank at least $r$63, and let $r$64 be its skew characteristic set such that $r$65 is representable over all infinite fields of characteristic $r$66 for each $r$67. Then, for any matroid $r$68, the following are equivalent: $r$69 is $r$70-tensor-compatible with $r$71 for every $r$72; and $r$73 is a direct sum of matroids each representable over a skew field whose characteristic lies in $r$74. In particular, for connected $r$75, skew-representability is equivalent to $r$76-tensor-compatibility with $r$77 for all $r$78, and representability over some skew field of characteristic $r$79 is equivalent to $r$80-tensor-compatibility with $r$81 for all $r$82 (Bérczi et al., 14 Jul 2025).

These tensor products control extension properties. If $r$83 exists, then for any $r$84, some minor of the product is a one-step modular extension of $r$85 with respect to $r$86. If $r$87 is $r$88-tensor-compatible with $r$89, then $r$90 is $r$91-modular extendable, and if this holds for all $r$92, then $r$93 is fully modular extendable. Conversely, a connected matroid of rank at least $r$94 is skew-representable if and only if it is fully modular extendable (Bérczi et al., 14 Jul 2025).

The same framework gives negative certificates. For connected $r$95, non-skew-representability is co-recursively enumerable: a certificate is a $r$96 with $r$97, and for fixed prime $r$98, nonrepresentability over skew fields of characteristic $r$99 is certified by a $k\in \mathbb{Z}_+$00 with $k\in \mathbb{Z}_+$01. This complements, rather than contradicts, the earlier undecidability results: the general problems are co-RE but not decidable in general (Bérczi et al., 14 Jul 2025, Kühne et al., 2020).

Most significantly for folded skew-representable matroids, the tensor-product method yields the first known linear rank inequality for folded skew-representable polymatroids that does not follow from the common information property. If $k\in \mathbb{Z}_+$02 is the rank function of $k\in \mathbb{Z}_+$03 and $k\in \mathbb{Z}_+$04 admits a tensor product with $k\in \mathbb{Z}_+$05, then for cyclic indices,

$k\in \mathbb{Z}_+$06

$k\in \mathbb{Z}_+$07

This inequality holds for folded skew-representable polymatroids because if $k\in \mathbb{Z}_+$08 is skew-representable, then $k\in \mathbb{Z}_+$09 exists and the inequality scales back to $k\in \mathbb{Z}_+$10. It is independent of the common information property: all rank-3 matroids satisfy the known CI-type extension properties, but the non-Desargues matroid violates this inequality under the singleton assignment $k\in \mathbb{Z}_+$11, $k\in \mathbb{Z}_+$12, $k\in \mathbb{Z}_+$13, $k\in \mathbb{Z}_+$14 (Bérczi et al., 14 Jul 2025).

Several open problems remain. Among them are whether every skew partial field admits a homomorphism to $k\in \mathbb{Z}_+$15 for some $k\in \mathbb{Z}_+$16 and field $k\in \mathbb{Z}_+$17; whether a connected matroid whose rank function is $k\in \mathbb{Z}_+$18-tensor-compatible with $k\in \mathbb{Z}_+$19 for all $k\in \mathbb{Z}_+$20 must be folded skew-representable; how to characterize the groups $k\in \mathbb{Z}_+$21 for which $k\in \mathbb{Z}_+$22 is representable over some skew partial field; and which left $k\in \mathbb{Z}_+$23-matroids arise from algebraic data $k\in \mathbb{Z}_+$24 in positive characteristic (Pendavingh et al., 2011, Bérczi et al., 14 Jul 2025, Pendavingh, 2018). These questions indicate that folded skew-representability is not a single isolated notion, but a nexus joining noncommutative coordinatization, tensorial rank geometry, and functorial passage between coefficient systems.