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Semidirect sums of matroids

Joseph E. Bonin, Joseph P. S. Kung

Abstract

For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is a matroid K on the union of S and T that, like the direct sum and the free product, has the restriction of K to S equal to M and the contraction of K to T equal to N. We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension of M on the union of S and T with the direct sum of N and the rank-0 matroid on S is a semidirect sum of M and N. We study principal sums in depth; these are such matroid unions where the extension of M has each element of T added either as a loop or freely on a fixed flat of M. A second construction of semidirect sums, defined by a Higgs lift, also specializes to principal sums. We also explore what can be deduced if M and N, or certain of their semidirect sums, are transversal or fundamental transversal matroids.

Semidirect sums of matroids

Abstract

For matroids M and N on disjoint sets S and T, a semidirect sum of M and N is a matroid K on the union of S and T that, like the direct sum and the free product, has the restriction of K to S equal to M and the contraction of K to T equal to N. We abstract a matrix construction to get a general matroid construction: the matroid union of any rank-preserving extension of M on the union of S and T with the direct sum of N and the rank-0 matroid on S is a semidirect sum of M and N. We study principal sums in depth; these are such matroid unions where the extension of M has each element of T added either as a loop or freely on a fixed flat of M. A second construction of semidirect sums, defined by a Higgs lift, also specializes to principal sums. We also explore what can be deduced if M and N, or certain of their semidirect sums, are transversal or fundamental transversal matroids.

Paper Structure

This paper contains 5 sections, 35 theorems, 60 equations, 2 figures.

Table of Contents

  1. Block upper-triangular matrices and semidirect sums
  2. Semidirect sums from matroid unions
  3. Principal sums
  4. Semidirect sums from Higgs lifts
  5. Principal sums and transversal matroids

Key Result

Proposition 1.2

Let $A$ and $B$ be matrices over a field $\mathbb{F}$, with disjoint sets $S$ and $T$ indexing the columns, with column matroids $M$ and $N$, and where $A$ has $r(M)$ rows. The column matroid of any matrix $(A,B;U)$ over $\mathbb{F}$ is a semidirect sum of $M$ and $N$. Conversely, if a semidirect su

Figures (2)

  • Figure 1: The $3$-whirl as the matroid union of extensions of $U_{2,3}$ and $U_{1,3}$.
  • Figure 2: A principal sum.

Theorems & Definitions (53)

  • Definition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 43 more