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This page shows the Cayley graph of the symmetric group on elements, and makes explicit some of its properties.
In all of the following we abbreviate permutations of 3 elements as
The following shows the Cayley graph of the symmetric groups on 3 elements, , with edges corresponding to any transposition (not necessarily adjacent), hence whose graph distance is the Cayley distance:
If we order the 6 elements of as
then the Cayley distance, regarded as a matrix
is
For , write
for the Cayley distance kernel on , hence the matrix which is the exponential of the Cayley distance-matrix (1) multiplied by .
The eigenvalues of this matrix are (from here)
where the first one appears with multiplicity 4 and is positive since is positive. The only eigenvalue that can become non-positive for is:
In conclusion:
The Cayley distance kernel (2), regarded as a bilinear form on the linear span is:
(relation to weight systems on horizontal chord diagrams)
Incidentally, the critical value in (3) is that corresponding to the fundamental sl(2,C)-Lie algebra weight systems on horizontal chord diagrams (under the canonical identification of the latter with permutations and using Cayley’s observation to express Cayley distance in terms of numbers of permutation cycles), see CSS21.
See the references at:
Last revised on April 21, 2021 at 08:57:13. See the history of this page for a list of all contributions to it.