88 research outputs found
Specht Polytopes and Specht Matroids
The generators of the classical Specht module satisfy intricate relations. We
introduce the Specht matroid, which keeps track of these relations, and the
Specht polytope, which also keeps track of convexity relations. We establish
basic facts about the Specht polytope, for example, that the symmetric group
acts transitively on its vertices and irreducibly on its ambient real vector
space. A similar construction builds a matroid and polytope for a tensor
product of Specht modules, giving "Kronecker matroids" and "Kronecker
polytopes" instead of the usual Kronecker coefficients. We dub this process of
upgrading numbers to matroids and polytopes "matroidification," giving two more
examples. In the course of describing these objects, we also give an elementary
account of the construction of Specht modules different from the standard one.
Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure
Representing matroids via pasture morphisms
Using the framework of pastures and foundations of matroids developed by
Baker-Lorscheid, we give algorithms to: (i) compute the foundation of a
matroid, and (ii) compute all morphisms between two pastures. Together, these
provide an efficient method of solving many questions of interest in matroid
representations, including orientability, non-representability, and computing
all representations of a matroid over a finite field.Comment: 22 pages, 2 figures, 1 table. Comments welcom
Binary Matroids and Quantum Probability Distributions
We characterise the probability distributions that arise from quantum
circuits all of whose gates commute, and show when these distributions can be
classically simulated efficiently. We consider also marginal distributions and
the computation of correlation coefficients, and draw connections between the
simulation of stabiliser circuits and the combinatorics of representable
matroids, as developed in the 1990s.Comment: 24 pages (inc appendix & refs
Foundations of matroids -- Part 2: Further theory, examples, and computational methods
In this sequel to "Foundations of matroids - Part 1", we establish several
presentations of the foundation of a matroid in terms of small building blocks.
For example, we show that the foundation of a matroid M is the colimit of the
foundations of all embedded minors of M isomorphic to one of the matroids
, , , , , , ,
, and we show that this list is minimal. We establish similar minimal
lists of building blocks for the classes of 2-connected and 3-connected
matroids. We also establish a presentation for the foundation of a matroid in
terms of its lattice of flats. Each of these presentations provides a useful
method to compute the foundation of certain matroids, as we illustrate with a
number of concrete examples. Combining these techniques with other results in
the literature, we are able to compute the foundations of several interesting
classes of matroids, including whirls, rank-2 uniform matroids, and projective
geometries. In an appendix, we catalogue various 'small' pastures which occur
as foundations of matroids, most of which were found with the assistance of a
computer, and we discuss some of their interesting properties.Comment: 69 page
K-classes of matroids and equivariant localization
To every matroid, we associate a class in the K-theory of the Grassmannian.
We study this class using the method of equivariant localization. In
particular, we provide a geometric interpretation of the Tutte polynomial. We
also extend results of the second author concerning the behavior of such
classes under direct sum, series and parallel connection and two-sum; these
results were previously only established for realizable matroids, and their
earlier proofs were more difficult.Comment: v2: added a starting point for combinatorialists in Section 2.4, +
minor change
Ordinary and Generalized Circulation Algebras for Regular Matroids
Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (x_e)_{e in E}, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of degree 1 have been studied by many authors and can be understood combinatorially in terms of the matroid represented by the linear equations satisfied by these generators. Such an algebra is related to algebras associated to deletions and contractions of the matroid by a short exact sequence, and can also be written as the quotient of a polynomial algebra by certain powers of linear forms.
We study such algebras in the case that the matroid is regular, which we term circulation algebras following Wagner. In addition to surveying the existing results on these algebras, we give a new proof of Wagner's result that the structure of the algebra determines the matroid, and construct an explicit basis in terms of basis activities in the matroid. We then consider generalized circulation algebras in which we mod out by a fixed power of each variable, not necessarily equal to 2. We show that such an algebra is isomorphic to the circulation algebra of a "subdivided" matroid, a variation on a result of Nenashev, and derive from this generalized versions of many of the results on ordinary circulation algebras, including our basis result. We also construct a family of short exact sequences generalizing the deletion-contraction decomposition
Of matroid polytopes, chow rings and character polynomials
Matroids are combinatorial structures that capture various notions of independence. Recently there has been great interest in studying various matroid invariants. In this thesis, we study two such invariants: Volume of matroid base polytopes and the Tutte polynomial. We gave an approach to computing volume of matroid base polytopes using cyclic flats and apply it to the case of sparse paving matroids. For the Tutte polynomial, we recover (some of) its coefficients as degrees of certain forms in the Chow ring of underlying matroid. Lastly, we study the stability of characters of the symmetric group via character polynomials. We show a combinatorial identity in the ring of class functions that implies stability results for certain class of Kronecker coefficients
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