Skew-Representable Matroids Theory

• Skew-representable matroids are defined by coordinatization over noncommutative systems, using division rings and generalized coefficient frameworks to capture dependencies beyond ordinary fields.
• They employ noncommutative linear algebra, projective geometry, and tensor product methods to achieve new structural results and rank inequalities.
• Decidability and separation results highlight that skew-representability differs sharply from multilinear representability, presenting unique algorithmic and extension challenges.

Skew-representable matroids are matroids whose dependence structure is coordinatized over a noncommutative coefficient system. In the narrowest and most standard usage, this means skew-linear representability over a division ring: if $D$ is a division ring and $M$ is a matroid of rank $r$ on $E$, a representation is a map $E\to A$, with $A$ a right $D$-module, such that a subset of $E$ is independent in $M$ if and only if its image is linearly independent over $D$ (Kühne et al., 2020). In broader frameworks, skew partial fields, skew hyperfields, and skew tracts extend the same idea to chain groups, circuit signatures, and quasi-Plücker coordinates, thereby encompassing ordinary matroids, oriented matroids, phased matroids, valuated matroids, and genuinely noncommutative phenomena (Pendavingh et al., 2011, Su, 2020, Pendavingh, 2018). The subject is characterized by the interaction of noncommutative linear algebra, projective and incidence geometry, algebraic dependence, and increasingly sharp structural and algorithmic results.

1. Definitions and scope of the term

In the division-ring setting, the basic convention is that all $M$0-modules are right vector spaces. Thus $M$1 denotes the right $M$2-vector space of dimension $M$3, and a finite-dimensional representation may be coordinatized by a matrix $M$4, with columns indexed by the ground set $M$5. A subset is independent precisely when the corresponding columns are right-linearly independent. Because $M$6 may be noncommutative, determinant criteria are not used; the foundational notions are column rank, linear independence, and invertibility (Kühne et al., 2020).

The literature surveyed here also uses broader coefficient systems. A skew partial field is a pair $M$7, where $M$8 is a ring, possibly noncommutative, and $M$9 contains $r$0; a matroid is represented by a $r$1-chain group whose elementary supports are the cocircuits of the underlying matroid (Pendavingh et al., 2011). A skew tract is a group $r$2 together with a nullset $r$3, and matroids over skew tracts admit circuit, dual-pair, and quasi-Plücker axiom systems (Su, 2020). A skew hyperfield is a skew hyperring in which every nonzero element is multiplicatively invertible, and matroids over skew hyperfields are expressed through left or right $r$4-valued circuit signatures together with skew modular elimination (Pendavingh, 2018).

This broader terminology is not completely uniform. In particular, the skew-polynomial papers construct specific matroids from noncommutative polynomial evaluation, but explicitly do not present a general abstract theory of “skew-representable matroids” parallel to division-ring representability. In one case the resulting $r$5-matroids are shown to be ordinarily $r$6-representable, even though their origin is genuinely skew (Liu et al., 2016).

2. Division rings, multilinear representations, and projective geometry

Skew-linear representation generalizes ordinary field representability by allowing the coefficient field to be replaced by a division ring. Every field is a division ring, so every linearly representable matroid is skew-linear. A second generalization, multilinear representability, is defined over a field $r$7 by a positive integer $r$8, a vector space $r$9, and a map from the ground set into $E$0 such that independence corresponds to direct-sum behavior of the chosen $E$1-dimensional subspaces, together with the requirement that all ranks scale by $E$2. In coordinates, a multilinear representation of a rank-$E$3 matroid is encoded by a block matrix $E$4 (Kühne et al., 2020).

A central result is that skew-linear and multilinear representability are incomparable. Earlier work had shown that not every multilinear matroid is skew-linear, and a construction based on a Baumslag–Solitar relation shows the converse fails as well: there exists a skew-linear matroid that is not multilinear (Kühne et al., 2020). This establishes that noncommutative division-ring behavior is not exhausted by block-matrix models over fields.

The main geometric tool for this comparison is a variant of the von Staudt construction. In rank $E$5, a skew-linear representation over a division ring $E$6 can be viewed as a point configuration in the projective plane $E$7, with points written $E$8. The construction reduces arbitrary noncommutative polynomial systems to atomic equations

$E$9

together with $E\to A$0 and $E\to A$1, and then realizes these equations by collinearity gadgets. The resulting finite family $E\to A$2 of von Staudt matroids satisfies a bidirectional correspondence: solvability over a division ring yields representability of some $E\to A$3, and representability of such an $E\to A$4 forces solvability of the original system; multilinear representations of order $E\to A$5 similarly force solutions in $E\to A$6 (Kühne et al., 2020). This provides a noncommutative universality mechanism for matroid representation problems.

3. Skew partial fields, skew tracts, and skew hyperfields

Representation over skew partial fields replaces determinant-based matrix theory by chain groups and supports. If $E\to A$7 is a $E\to A$8-chain group, then the supports of its elementary chains form the cocircuit set of a matroid $E\to A$9. This theory is closed under duality and minors, and it extends Tutte’s representability criterion: primitive cocircuit chains represent a matroid exactly when every modular triple satisfies a $A$0-unit linear relation $A$1 (Pendavingh et al., 2011). Within this framework, multilinear representations are not merely analogous to skew-partial-field representations: a matroid has an $A$2-multilinear representation over a skew field $A$3 if and only if it is representable over the skew partial field $A$4 (Pendavingh et al., 2011).

Skew partial fields are strictly more general than skew fields. The matroid $A$5, where $A$6 is the ternary Reid geometry and $A$7 is the rank-3 Dowling geometry of the quaternion group $A$8, is representable over a skew partial field but not over any skew field (Pendavingh et al., 2011). The same paper also records the non-Pappus matroid as representable over a skew field but not over any commutative field, situating skew representability strictly between ordinary field representability and the broader skew-partial-field setting.

Matroids over skew tracts generalize both Baker–Bowler tracts and Pendavingh’s weak matroids over skew hyperfields. Weak and strong versions are defined by circuit elimination axioms, and the theory is cryptomorphic in terms of circuits, quasi-Plücker coordinates, and dual pairs. Duality interchanges left and right, minors are well behaved, and skew fields appear as special cases because every skew field determines a skew hyperfield and hence a skew tract (Su, 2020). For perfect skew tracts, weak and strong notions coincide; the paper lists skew fields among the perfect examples (Su, 2020).

Single-element extension theory becomes subtler in the skew setting. For weak matroids over skew tracts, the exact obstruction to extending the classical rank-2 local characterization of Crapo and Las Vergnas is the Pathetic Cancellation Property. The resulting theorem states that a right- or left-equivariant cocircuit map defines a weak single-element extension if and only if it does so on every rank-2 contraction, precisely when the skew tract satisfies Pathetic Cancellation (Su, 2020). For stringent skew hyperfields, a stronger theorem holds for strong matroids, and stringent skew hyperfields include the coefficient systems closest to skew fields (Su, 2020).

A further extension appears in positive characteristic. If $A$9 is a field extension of characteristic $D$0, with Frobenius $D$1, then the skew hyperfield $D$2 supports a left $D$3-matroid $D$4 whose underlying ordinary matroid is the algebraic matroid of a family $D$5. Its circuits are built from $D$6-derivatives of circuit polynomials, its cocircuits from derivations, and it refines both the algebraic matroid and the Lindström valuation/Frobenius-flock data (Pendavingh, 2018). In this sense, algebraic matroids in positive characteristic become skew-hyperfield representable.

4. Matroids induced by skew polynomial rings

A distinct but closely related line of work studies matroids induced by skew polynomial evaluation over finite fields. Let $D$7 and $D$8 with $D$9. In the skew polynomial ring $E$0, multiplication is twisted by $E$1, right division defines evaluation, and minimal skew polynomials define a notion of $E$2-independence. The resulting matroid has ground set $E$3, independent sets

$E$4

rank $E$5, and closure $E$6 (Liu et al., 2016).

The essential structure is classwise projective geometry. Conjugacy classes under $E$7-conjugation are the components on which skew independence becomes ordinary $E$8-linear independence. If $E$9 with $M$0, then $M$1 is $M$2-independent if and only if $M$3 are linearly independent over $M$4, and the $M$5-closure is the image of the $M$6-span of the $M$7 under the warping map $M$8 (Liu et al., 2016). The paper proves that the resulting $M$9-matroid is $D$0-representable, and that a single conjugacy-class submatroid is bijectively isometric to projective geometry over $D$1 equipped with the subspace metric (Liu et al., 2016).

A companion construction starts from left and right roots of skew polynomials in $D$2. Right and left minimal polynomials give rise to two matroids $D$3 and $D$4 on the ground set $D$5, defined by right- and left-independence. Again, a skew polynomial may have more roots than its degree, so dependence is controlled by closure under minimal polynomials rather than by naive root counting. Inside each conjugacy class, right and left independence are equivalent to $D$6-linear independence after parametrization by $D$7 or $D$8, and the two global matroids are isomorphic via

$D$9

on the basic conjugacy class (Baumbaugh et al., 2017). The same paper shows that, after adjoining the splitting field of the right evaluation polynomial, all roots of a skew polynomial lie in a single conjugacy class (Baumbaugh et al., 2017).

These constructions are often discussed alongside skew-representability because they arise from genuinely noncommutative algebra. However, the strongest structural statements in this direction reduce the induced dependence to ordinary $M$00-linear algebra on conjugacy classes, and one paper explicitly emphasizes that its $M$01-matroids are classically $M$02-representable rather than instances of a new general abstract representability notion (Liu et al., 2016).

5. Separations, characteristic behavior, and decision problems

The strongest negative results concern decidability. Using the von Staudt construction together with Macintyre’s transfer of undecidable Horn theories from groups to division rings, one obtains that the following problems are undecidable: deciding whether a matroid is skew-linear; deciding, for fixed $M$03, whether a matroid is representable over some division ring of characteristic $M$04; and deciding representability over a certain fixed division ring $M$05 (Kühne et al., 2020). These results show that division-ring representability is not merely a mild extension of field representability.

The skew-linear versus multilinear separation is made explicit by a group-theoretic example based on the Baumslag–Solitar group

$M$06

A noncommutative polynomial system encoding the relation $M$07 together with the nontriviality of a specific commutator has a solution in a division ring but no solution in any matrix ring over any field. Via the von Staudt theorem, this yields a skew-linear matroid that is not multilinear (Kühne et al., 2020). The same paper also contrasts characteristic sets: skew-linear characteristic sets can be highly irregular, and earlier work had already produced a matroid with skew-linear characteristic set $M$08, whereas multilinear characteristic sets satisfy a finiteness phenomenon when $M$09 occurs (Kühne et al., 2020).

More recent work gives a complementary structural viewpoint. For connected matroids, skew-representability can be characterized by iterated tensor products: a connected matroid $M$10 is skew-representable if and only if it is $M$11-tensor-compatible with $M$12 for each $M$13, and representability over some skew field of characteristic $M$14 is characterized similarly by $M$15-tensor-compatibility with $M$16 for every $M$17 (Bérczi et al., 14 Jul 2025). This does not contradict undecidability: it implies instead that non-skew-representability is certifiable. The same paper proves that deciding skew-representability, or representability over a skew field of fixed prime characteristic, is co-recursively enumerable, because certificates of non-skew-representability can be verified (Bérczi et al., 14 Jul 2025).

6. Tensor products, rank inequalities, and current directions

The tensor-product framework links skew-representability to extension properties and rank inequalities. A polymatroid function $M$18 is a tensor product of $M$19 and $M$20 if

$M$21

for all subsets $M$22. Iterating this operation with $M$23 forces modular extension properties, and for connected matroids of rank at least $M$24 full modular extendability is equivalent to skew-representability (Bérczi et al., 14 Jul 2025). One consequence is that tensor compatibility provides a bridge between noncommutative linear algebra and the extension-theoretic language of modular pairs and common-information-type constructions.

Rank $M$25 remains exceptional. Every rank-3 matroid admits a freest tensor product with every uniform matroid, and this freest tensor product is maximal in the weak order among all such tensor products (Bérczi et al., 14 Jul 2025). This explains why rank-3 obstructions to skew-representability are subtle: simple one-step tensor tests with uniform matroids do not detect them. The non-Desargues matroid, which is rank $M$26 and not skew-representable, is therefore a central test case in the theory (Bérczi et al., 14 Jul 2025).

Tensor methods also produce linear rank inequalities. A tensor-product argument yields a new proof of Ingleton’s inequality, and further characteristic-dependent inequalities arise from tensoring with the Fano and non-Fano matroids (Bérczi et al., 14 Jul 2025). More significantly, the same paper derives the first known linear rank inequality for folded skew-representable matroids that does not follow from the common information property, and shows that the non-Desargues matroid violates it (Bérczi et al., 14 Jul 2025). Here “folded skew-representable” means that $M$27 is skew-representable for some positive integer $M$28.

Several open directions remain explicit in the literature. One paper asks whether there exists a matroid whose multilinear characteristic set is infinite with infinite complement (Kühne et al., 2020). The tensor-product paper asks, among other problems, whether $M$29-modular extendability implies $M$30-tensor-compatibility with $M$31, whether polymatroid tensor compatibility with $M$32 for all $M$33 forces folded skew-representability, and whether different tensor-generation orders yield genuinely new inequalities (Bérczi et al., 14 Jul 2025). These questions indicate that the subject is still balancing three partly competing perspectives: direct coordinatization over division rings, generalized coefficient theories over skew tracts and hyperfields, and indirect structural tests based on extensions, tensor products, and inequalities.