scholarly journals THE COLUMN GROUP AND ITS LINK INVARIANTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250063 ◽  
Author(s):  
JOHANNA HENNIG ◽  
SAM NELSON
Keyword(s):  
Symmetric Group ◽  
Link Invariants ◽  
Column Group

The column group is a subgroup of the symmetric group on the elements of a finite birack generated by the column permutations in the birack matrix. We use subgroups of the column group associated to birack homomorphisms to define an enhancement of the integral birack counting invariant and give examples which show that the enhanced invariant is stronger than the unenhanced invariant.

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 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

scholarly journals FROM RIBBON CATEGORIES TO GENERALIZED YANG–BAXTER OPERATORS AND LINK INVARIANTS (AFTER KITAEV AND WANG)

2013 ◽  
Vol 24 (01) ◽  
pp. 1250126 ◽  
Author(s):  
SEUNG-MOON HONG
Keyword(s):  
The Other ◽  
Polynomial Invariant ◽  
Link Invariants ◽  
Fusion Categories ◽  
New Family ◽  
Kauffman Polynomial ◽  
Twist Structure

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO (N)2, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

Objects for the symmetric group

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