scholarly journals The simultaneous conjugacy problem in the symmetric group

2021 ◽  
pp. 1
Author(s):  
Andrej Brodnik ◽  
Aleksander Malnič ◽  
Rok Požar
Keyword(s):  
Symmetric Group ◽  
Conjugacy Problem

ON THE CONJUGACY PROBLEM OF POSITIVE BRAIDS

2004 ◽  
Vol 13 (03) ◽  
pp. 311-324 ◽  
Author(s):  
E. A. ELRIFAI ◽  
M. BENKHALIFA
Keyword(s):  
Symmetric Group ◽  
Conjugacy Problem ◽  
Crossing Number ◽  
Unique Factorization ◽  
Link Type ◽  
Linking Numbers ◽  
Complete Calculation

In the symmetric group Sn, we introduced the notion of crossing and linking numbers to each permutation. Then a unique factorization of a permutation is given due to its crossing number of its factors and how the factors are linked. Consequently we introduced a matrix associated to each permutation, which we used it as a tool to prove that positive braids with different matrices are not conjugate braids. Up to n≤5 it is proved that two positive permutation braids are conjugate if and only if they have the same matrix, and a complete calculation of these matrices with the associated link type is given.

Representations of the Infinite Symmetric Group

2016 ◽  
Author(s):  
Alexei Borodin ◽  
Grigori Olshanski
Keyword(s):  
Symmetric Group ◽  
Infinite Symmetric Group

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 $\mathfrak{sl}_3$.

SYMMETRIC GROUP BLOCKS OF DEFECT TWO

1995 ◽  
Vol 46 (2) ◽  
pp. 201-234 ◽  
Author(s):  
JOANNA SCOPES
Keyword(s):  
Symmetric Group

Electron localisation, Lyapunov exponents and the symmetric group

1989 ◽  
Vol 1 (19) ◽  
pp. 3073-3082 ◽  
Author(s):  
K Slevin ◽  
E Castano ◽  
J B Pendry
Keyword(s):  
Symmetric Group ◽  
Lyapunov Exponents

scholarly journals Relations between the Clausen and Kazhdan–Lusztig representations of the symmetric group

2010 ◽  
Vol 214 (5) ◽  
pp. 689-700
Author(s):  
Charles Buehrle ◽  
Mark Skandera
Keyword(s):  
Symmetric Group

scholarly journals The Round Functions of Cryptosystem PGM Generate the Symmetric Group

2006 ◽  
Vol 38 (1) ◽  
pp. 147-155 ◽  
Author(s):  
A. Caranti ◽  
F. Dalla. Volta
Keyword(s):  
Symmetric Group

Objects for the symmetric group

1981 ◽  
Vol 22 (6) ◽  
pp. 1144-1148 ◽  
Author(s):  
M. F. Soto ◽  
R. Mirman
Keyword(s):  
Symmetric Group

scholarly journals Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1330
Author(s):  
Raeyong Kim
Keyword(s):  
Group Structure ◽  
Conjugacy Problem ◽  
High Dimensional ◽  
Fundamental Groups ◽  
Graph Manifold ◽  
Solvable Word Problem ◽  
Algorithmic Problems ◽  
Graph Manifolds ◽  
Dimensional Graph ◽  
Solvable Word

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

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