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 b_24k+m = z^k b_m, 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.