The Hopf monoid of orbit polytopes

Supina, Mariel

doi:10.4310/joc.2020.v11.n4.a1

Cited by 6 publications

(7 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. Proposition 4.19 of [Sup20] implies that for an orbit polytope with composition α, any face can be described as a product of orbit polytopes with compositions β (1) , . .…”

Section: Orbit Polytopesmentioning

confidence: 99%

Techniques in equivariant Ehrhart theory

Sophia¹,

Kim²,

Supina³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the h * -polynomial, and certificates for the (non)existence of invariant non-degenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their equivariant Ehrhart theory is known.

show abstract

“…Proof. Proposition 4.19 of [Sup20] implies that for an orbit polytope with composition α, any face can be described as a product of orbit polytopes with compositions β (1) , . .…”

Section: Orbit Polytopesmentioning

confidence: 99%

Techniques in equivariant Ehrhart theory

Sophia¹,

Kim²,

Supina³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The EGF of (f n ) coincides with the one in [31, A307389], proving that both sequences are indeed the same. This latter sequence counts the number of elements in the so-called species of orbit polytopes in dimension n, see [33].…”

Section: 2mentioning

confidence: 99%

Invariant Synchrony and Anti-Synchrony Subspaces of Weighted Networks

Nijholt¹,

Sieben²,

Swift³

2022

Preprint

View full text Add to dashboard Cite

The internal state of a cell in a coupled cell network is often described by an element of a vector space. Synchrony or anti-synchrony occurs when some of the cells are in the same or the opposite state. Subspaces of the state space containing cells in synchrony or anti-synchrony are called polydiagonal subspaces. We study the properties of several types of polydiagonal subspaces of weighted coupled cell networks.In particular, we count the number of such subspaces and study when they are dynamically invariant. Of special interest are the evenly tagged anti-synchrony subspaces in which the number of cells in a certain state is equal to the number of cells in the opposite state. Our main theorem shows that the dynamically invariant polydiagonal subspaces determined by certain types of couplings are either synchrony subspaces or evenly tagged anti-synchrony subspaces. A special case of this result confirms a conjecture about difference-coupled graph network systems.

show abstract

“…The Hopf algebra of matroids was introduced by Crapo and Schmitt [CS05a,CS05b,CS05c]. The corresponding Hopf monoid was described by Aguiar and Mahajan [AM10, §13.8.2] and has attracted recent interest; see, e.g., [San20,Sup20,AS20,Bas20]. There are many definitions of a matroid (see, e.g., [Oxl11, Section 1]), but for our purposes the most convenient definition is that a matroid is a simplicial complex Γ on vertex set I such that the induced subcomplex Γ|S " tσ P Γ : σ Ď Su is pure for every S Ď I (i.e., every facet of Γ|S has the same size).…”

Section: Introductionmentioning

confidence: 99%

Hopf monoids of ordered simplicial complexes

Castillo,

Martin,

Samper

2020

Preprint

View full text Add to dashboard Cite

We study pure ordered simplicial complexes (i.e., simplicial complexes with a linear order on their ground sets) from the Hopf-theoretic point of view. We define a Hopf class to be a family of pure ordered simplicial complexes that give rise to a Hopf monoid under join and deletion/contraction. The prototypical Hopf class is the family of ordered matroids. The idea of a Hopf class allows us to give a systematic study of simplicial complexes related to matroids, including shifted complexes, broken-circuit complexes, and unbounded matroids (which arise from unbounded generalized permutohedra with 0/1 coordinates).We compute the antipodes in two cases: facet-initial complexes (a much larger class than shifted complexes) and unbounded ordered matroids. In the latter case, we embed the Hopf monoid of ordered matroids into the Hopf monoid of ordered generalized permutohedra, enabling us to compute the antipode using the topological method of Aguiar and Ardila. The calculation is complicated by the appearance of certain auxiliary simplicial complexes that we call Scrope complexes, whose Euler characteristics control certain coefficients of the antipode. The resulting antipode formula is multiplicity-free and cancellationfree.

show abstract

The Hopf monoid of orbit polytopes

Cited by 6 publications

References 6 publications

Techniques in equivariant Ehrhart theory

Techniques in equivariant Ehrhart theory

Invariant Synchrony and Anti-Synchrony Subspaces of Weighted Networks

Hopf monoids of ordered simplicial complexes

Contact Info

Product

Resources

About