Decomposition of large combinatorial structures

1989 ◽  
Vol 52 (3) ◽  
pp. 289-297 ◽  
Author(s):  
Yair Caro
Keyword(s):  
Combinatorial Structures

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

2020 ◽  
Author(s):  
Heather M Russell ◽  
Julianna Tymoczko
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Transition Matrix ◽  
Group Representation ◽  
Jones Polynomial ◽  
Basis Matrix ◽  
Combinatorial Structures ◽  
Quantum Invariants ◽  
One Dimensional ◽  
Link Diagrams

Abstract Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

scholarly journals Teaching Combinatorial Principles Using Relations through the Placemat Method

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1825
Author(s):  
Viliam Ďuriš ◽  
Gabriela Pavlovičová ◽  
Dalibor Gonda ◽  
Anna Tirpáková
Keyword(s):  
Graph Theory ◽  
Complexity Theory ◽  
Slovak Republic ◽  
Teaching Mathematics ◽  
Combinatorial Structures ◽  
Algorithmic Approach ◽  
Scientific Disciplines ◽  
Combinatorial Principles ◽  
The Subject ◽  
As Graph

The presented paper is devoted to an innovative way of teaching mathematics, specifically the subject combinatorics in high schools. This is because combinatorics is closely connected with the beginnings of informatics and several other scientific disciplines such as graph theory and complexity theory. It is important in solving many practical tasks that require the compilation of an object with certain properties, proves the existence or non-existence of some properties, or specifies the number of objects of certain properties. This paper examines the basic combinatorial structures and presents their use and learning using relations through the Placemat method in teaching process. The effectiveness of the presented innovative way of teaching combinatorics was also verified experimentally at a selected high school in the Slovak Republic. Our experiment has confirmed that teaching combinatorics through relationships among talented children in mathematics is more effective than teaching by a standard algorithmic approach.

scholarly journals Study of Lie algebras by using combinatorial structures

2012 ◽  
Vol 436 (2) ◽  
pp. 349-363 ◽  
Author(s):  
Manuel Ceballos ◽  
Juan Núñez ◽  
Ángel F. Tenorio
Keyword(s):  
Lie Algebras ◽  
Combinatorial Structures

scholarly journals Algorithms for combinatorial structures: Well-founded systems and Newton iterations

2012 ◽  
Vol 119 (8) ◽  
pp. 1711-1773 ◽  
Author(s):  
Carine Pivoteau ◽  
Bruno Salvy ◽  
Michèle Soria
Keyword(s):  
Combinatorial Structures ◽  
Newton Iterations

scholarly journals The ‘Burnside Process’ Converges Slowly

2002 ◽  
Vol 11 (1) ◽  
pp. 21-34 ◽  
Author(s):  
LESLIE ANN GOLDBERG ◽  
MARK JERRUM
Keyword(s):  
Markov Chain ◽  
Mixing Time ◽  
Significant Proportion ◽  
Main Tool ◽  
Infinite Family ◽  
Combinatorial Structures ◽  
Special Groups ◽  
Sampling Problem ◽  
Burnside's Lemma ◽  
Burnside’S Lemma

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

scholarly journals Biquasiles and dual graph diagrams

2017 ◽  
Vol 26 (08) ◽  
pp. 1750048 ◽  
Author(s):  
Deanna Needell ◽  
Sam Nelson
Keyword(s):  
Dual Graph ◽  
Algebraic Structures ◽  
Combinatorial Structures ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
The Way

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

scholarly journals Small covers, infra-solvmanifolds and curvature

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Shintarô Kuroki ◽  
Mikiya Masuda ◽  
Li Yu
Keyword(s):  
Sectional Curvature ◽  
Riemannian Metrics ◽  
Flat Torus ◽  
Combinatorial Structures ◽  
Simple Polytope ◽  
Simple Polytopes ◽  
Small Cover ◽  
Equivalent Conditions

AbstractIt is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

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