15 research outputs found
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 be a finite permutation group on An ordered sequence
of elements of is an irredundant base for
if the pointwise stabilizer is trivial and no point is fixed by the
stabilizer of its predecessors. If all irredundant bases of have the same
cardinality, 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
Hypercellular graphs: partial cubes without 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 (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
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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
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 -page over fields of good characteristic. For von Neumann
regular monoids in which Green's - and -relations
coincide (e.g., left regular bands), the computation of 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 between arbitrary simple
modules and computing a quiver presentation for the algebra of Hsiao's monoid
of ordered -partitions (connected to the Mantaci-Reutenauer descent algebra
for the wreath product ). 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