This is the webpage of the project The BMM symmetrising trace conjecture for the exceptional 2-reflection groups of rank 2. The theory behind the conjecture and the algorithms behind the proof are better explained in the article of the same name, which can be found here.

The 2-reflection groups of rank 2 are G12, G13 and G22. The BMM symmetrising trace conjecture has been proved for G12 and G22 by Malle and Michel here. In our article, we focus on the group G13. We know that the generic Hecke algebra H(G13) associated with this group is generated by 3 elements, s, t and u, and that it admits a basis B13 = {b1,...,b96}, with b1=1.

Briefly put, we prove the BMM symmetrising trace conjecture for G13 as follows:

Step 1: We define the linear map τ on H(G13), which takes the value 1 on 1 and 0 on all other elements of the basis.
Step 2: We expand the basis B13 to a spanning set C13 of H(G13), which includes four particular elements, denoted by b97, b98, b99, and b100.
Step 3: We express sb, tb and ub as a linear combination of elements of C13 for all b ϵ B13.
Step 4: We express sb, tb and ub as a linear combination of elements of B13 for all b ϵ B13, by expressing b97, b98, b99, and b100 as a linear
combination of elements of B13.
Step 5: The element z := b25 is a central element of H(G13). We have b24k+m = zk bm, for all k=0,1,2,3 and m=1,2,...,24. We use Step 4 to express the
element z4 as a linear combination of the elements of B13.
Step 6: Using Steps 4 and 5, we calculate the matrix A := (τ (bb')) for all b, b' ϵ B13.
Step 7: We check that A is symmetric and that the determinant of A is a unit, thus proving that τ is a symmetrising trace on H(G13).
Step 8: We prove that τ (b−1z4 ) = 0 for all b ϵ B13 \ {1}. This extra condition ensures that τ is indeed the trace of the BMM conjecture.

In order to deal with Step 3, we created a computer program in the language C++. This program contains many different classes, each dedicated to the representation of different algebraic objects: braid words, linear combinations of braid words, etc. Various functions have been developed to work on these objects, such as adding braid words or multiplying their coefficients. For more details on the implementation or to see the source code please contact Christina Boura.

The inputs of the C++ program are:
I1. The set C13.
I2. The positive and inverse Hecke relations.
I3. The 25 special cases.
One can find a list of all these inputs here.

In order to deal with Steps 4-7, we created a computer program in the language SAGE (we switched programming language, because otherwise we would have needed to create programs in C++ to do what SAGE can already do). The inputs are the outputs of the C++ program. Here is the SAGE program for G13.