Small integral relation algebras generated by a partial order

1991 ◽  
Vol 23 (2) ◽  
pp. 129-138 ◽  
Author(s):  
I. Düntsch
Keyword(s):  
Partial Order ◽  
Integral Relation ◽  
Relation Algebras

scholarly journals Representations of integral relation algebras.

1970 ◽  
Vol 17 (3) ◽  
pp. 279-287 ◽  
Author(s):  
Ralph McKenzie
Keyword(s):  
Integral Relation ◽  
Relation Algebras

Deriving the Partial Order of Documents to Extend Clustering Applications

2019 ◽  
Vol 7 (1) ◽  
pp. 424-430
Author(s):  
A. George Louis Raja ◽  
F. Sagayaraj Francis ◽  
P. Sugumar
Keyword(s):  
Partial Order

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 $\mathfrak{sl}_3$.

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