Generating a Transversal for Young Double Cosets in a Symmetric Group using
Maple

   Bill Pletsch (bpletsch@cnm.edu) CNM, United States

For years I have been working on Young double cosets embedded in an overall
symmetric group.  Two years ago at ACA I presented a recursion formula for
a Young group and some resulting conjectures this recursion revealed.  The
conjectures were about counting the number of Young Double Cosets embedded
in an overall symmetric group.  The CAS used there was MACSYMA.  This
work was subsequently published by ACA.

The talk I propose for ACA-2011 will extend the research presented two
years ago.  This will include a new canonical form to be introduced for
Young double cosets; a Maple program using the canonical form to generate
a transversal of the Young double cosets in the symmetric group; and the
sketch of a proof of one of the conjectures from two years ago.

The new canonical form in itself represents a kind of break through.  In the
past researchers in this field such as Kramer and Seligman used what they
called DC-symbols which are symbols for a given class of double cosets. 
These DC-symbols are a particular kind of matrix used to represent a given
class of double coset.  My canonical forms are n-tuples and can be easily
ordered.  They too represent a class of double cosets.  The ability to
order my double coset symbols confers many hitherto unavailable advantages. 
One of which is it is easy to see if a class of double cosets has been missed.

A transversal is a set containing a single representative member of each
double coset generated by two subgroups of an overarching group.  In this
talk the subgroups will be Young Groups in a symmetric group.  With the
canonical form a transversal of the double cosets embedded in the symmetric
group can be generated using Maple. 

The transversal so generated points the way to a proof of a conjecture
made at ACA-2009.  Using Maple and some subroutines I have written, some
transversals will be generated and compared.  In the process it will be
made clear how a few simple inductions will prove the conjecture.  Also
it will be easy to see how the technique itself can be readily adapted to
prove the other conjectures presented in 2009.

It is expected these results will pave the way to further results down the
road.  In the process a greater understanding of double cosets is likely. 
Since double cosets are singularly poorly behaved they are for the most part
poorly understood. It is hoped this research will contribute to a greater
understanding of this fascinating if elusive mathematical construct: the
double coset.