(back) TO DO Combinatorics section

Geometry/set partition Notes

There is no low-dimensional geometry in the partition algebra. But there are substructures and enhancements with geometrical features . This leads to various interesting bits of maths - in particular some combinatorics.

The line and the plane and non-crossing partitions

As usual we represent set partitions by graphs.
We then specify a graph (and hence a set partition) by a picture drawn in the plane.
Vertices are represented by points of the plane, and edges are lines between points.

We say that a graph picture is non-crossing if no two lines meet except possibly at their endpoints.

In particular suppose we consider partitions of {1,2,...,n} (with {1,2,...,n} considered as an ordered set in the natural way). Then we may draw our graph with lines restricted to the upper half-plane, and represent the elements as points on the boundary line (the x-axis) in the natural way. (There are several examples in the main figure here .)
In this case we have the notion of a non-crossing partition - a partition having a non-crossing representing graph picture.
We write NCP_n for the set of non-crossing partitions of {1,2,...,n}.

The following figure shows how to map a non-crossing partition on {1,2,...,n} to a non-crossing pair partition on {1,2,...,2n}. The figure first shows a neighbourhood of a single vertex in a picture of a non-crossing partition as above, and shows edges arriving at that single vertex. The figure then illustrates how to convert this to edges arriving at a pair of vertices.

 

 



The case of a vertex with no incoming edges is the natural degeneration of this. Thus altogether we have something like the following image of the partition {{1,2,4},{3}}:

 

It is straightforward to see that this defines a bijection
beta: NCP_n --> NCPP_2n.



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To do

This is a work in progress. Here we note some jobs that need doing (entirely for my own organisational benefit).

  • Suggested further reading

    On representation theory of the symmetric group
    Algebras and Representations Leeds undergraduate module
    References on representation theory of the symmetric group
    Khovanov's rep theory resources
    Benson, Representations and Cohomology: Basic representation theory of finite groups and associative algebras
    J A Green, Polynomial representations of GLn (Springer) (version with K Erdmann and M Schocker)
    Assem, Simson and Skowronski, Vol.1 (LMS ST65)
    Curtis and Reiner, Representation theory of finite groups and associative algebras, Wiley Interscience
    Curtis and Reiner, Methods of representation theory I, Wiley Interscience
    Hamermesh, Group Theory
    Gabriel and Roiter, Representations of finite-dimensional algebras
    Jain and Parvathi, Noncommutative rings, group rings, diagram algebras, and their applications
    ...

    Organiser: Paul