Chow Rings of Matroids as Permutation Representations

Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality and the Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.Comment: 21 pages, 3 figure

IBIS soluble linear groups

Let $G$ be a finite permutation group on $\Omega.$ An ordered sequence $(\omega_1,\dots, \omega_t)$ of elements of $\Omega$ is an irredundant base for $G$ if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of $G$ have the same cardinality, $G$ is said to be an IBIS group. In this paper we give a classification of quasi-primitive soluble irreducible IBIS linear groups, and we also describe nilpotent and metacyclic IBIS linear groups and IBIS linear groups of odd order.Comment: arXiv admin note: text overlap with arXiv:2206.01456 by other author

Topological and Geometric Combinatorics

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Hypercellular graphs: partial cubes without Q3−Q_3^- as partial cube minor

We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain finite convex subgraphs contractible to the 3-cube minus one vertex $Q^-_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell -- a property naturally generalizing the notion of median graphs.Comment: 35 pages, 6 figures, added example answering Question 1 from earlier draft (Figure 6.

Commuting quantum circuits: efficient classical simulations versus hardness results

The study of quantum circuits composed of commuting gates is particularly useful to understand the delicate boundary between quantum and classical computation. Indeed, while being a restricted class, commuting circuits exhibit genuine quantum effects such as entanglement. In this paper we show that the computational power of commuting circuits exhibits a surprisingly rich structure. First we show that every 2-local commuting circuit acting on d-level systems and followed by single-qudit measurements can be efficiently simulated classically with high accuracy. In contrast, we prove that such strong simulations are hard for 3-local circuits. Using sampling methods we further show that all commuting circuits composed of exponentiated Pauli operators e^{i\theta P} can be simulated efficiently classically when followed by single-qubit measurements. Finally, we show that commuting circuits can efficiently simulate certain non-commutative processes, related in particular to constant-depth quantum circuits. This gives evidence that the power of commuting circuits goes beyond classical computation.Comment: 19 page

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations

Algorithms for Permutation Groups and Cayley Networks

110 pagesBases, subgroup towers and strong generating sets (SGSs) have played a key role in the development of algorithms for permutation groups. We analyze the computational complexity of several problems involving bases and SGSs, and we use subgroup towers and SGSs to construct dense networks with practical routing schemes. Given generators for G ≤ Sym(n), we prove that the problem of computing a minimum base for G is NP-hard. In fact, the problem is NP-hard for cyclic groups and elementary abelian groups. However for abelian groups with orbits of size less than 8, a polynomial time algorithm is presented for computing minimum bases. For arbitrary permutation groups a greedy algorithm for approximating minimum bases is investigated. We prove that if G ≤ Sym(n) with a minimum base of size k, then the greedy algorithm produces a base of size Ω (k log log n)

Topology and monoid representations

The goal of this paper is to use topological methods to compute $\mathrm{Ext}$ between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the $E_2$-page over fields of good characteristic. For von Neumann regular monoids in which Green's $\mathscr L$- and $\mathscr J$-relations coincide (e.g., left regular bands), the computation of $\mathrm{Ext}$ between arbitrary simple modules reduces to this case, and so our results subsume those of S. Margolis, F. Saliola, and B. Steinberg, Combinatorial topology and the global dimension of algebras arising in combinatorics, J. Eur. Math. Soc. (JEMS), 17, 3037-3080 (2015). Applications include computing $\mathrm{Ext}$ between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered $G$-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product $G\wr S_n$). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. These generalize the results of S. Margolis, F. V. Saliola, and B. Steinberg. Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry, Mem. Amer. Math. Soc., 274, 1-135, (2021). We also determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids

Equivalence of Classical and Quantum Codes

In classical and quantum information theory there are different types of error-correcting codes being used. We study the equivalence of codes via a classification of their isometries. The isometries of various codes over Frobenius alphabets endowed with various weights typically have a rich and predictable structure. On the other hand, when the alphabet is not Frobenius the isometry group behaves unpredictably. We use character theory to develop a duality theory of partitions over Frobenius bimodules, which is then used to study the equivalence of codes. We also consider instances of codes over non-Frobenius alphabets and establish their isometry groups. Secondly, we focus on quantum stabilizer codes over local Frobenius rings. We estimate their minimum distance and conjecture that they do not underperform quantum stabilizer codes over fields. We introduce symplectic isometries. Isometry groups of binary quantum stabilizer codes are established and then applied to the LU-LC conjecture