The HOMFLY Braidor Algebra
Let be the free associative (but non-commutative) algebra generated by the elements of the symmetric group on and by formal variables and , and let be the quotient of by the following "HOMFLY" relations:
- commutes with everything else.
- The product of permutations is as in the symmetric group .
- If is a permutation then .
- , where is the transposition of and .
Finally, declare that while for every and every , and let be the graded completion of .
We say that an element of is "sorted" if it is written in the form where is a permutation and and the 's are all non-negative integer. The HOMFLY relations imply that every element of is a linear combinations of sorted elements. Thus as a vector space, can be identified with the ring of power series in the variables tensored with the group ring of . The product of is of course very different than that of .
- The general element of is where denotes the identity permutation and is a power series in two variables and . is commutative.
- The general element of is where and are power series in three variables and and are the two elements of . is not commutative and its product is non-trivial to describe.
- The general element of is described using power series in 4 variables. The general element of is described using n! power series in variables.
The algebra embeds in in a trivial way by regarding as a subset of in the obvious manner; thus when given an element of we are free to think of it also as an element of . There is also a non-trivial map defined as follows:
- acts on permutations by "shifting them one unit to the right", i.e., by identifying with .
We seek to find a "braidor"; an element of satisfying:
- (higher order terms).
- in .
With the vector space identification of with in mind, we are seeking two power series of three variables each, whose low order behaviour is specified and which are required to satisfy 6 functional equations written in terms of 4 variables.
The Equations in Functional Form
The first few terms of a solution can be computed using a computer, as shown below. But a true solution, written in a functional form, is still missing.
A primitive mathematica program to play with these objects is here.
A Numerology Problem
Question. Can you find nice formulas for the functions and of the variables , and , whose Taylor expansions begin with