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 G_{12}, G_{13} and G_{22}.
The BMM symmetrising trace conjecture has been proved for G_{12} and G_{22} by
Malle and Michel here.
In our article, we focus on the group G_{13}. We know that the generic Hecke algebra H(G_{13})
associated with this group is generated by 3 elements, s, t and u, and that
it admits a basis B_{13} = {b_{1},...,b_{96}}, with b_{1}=1.

Briefly put, we prove the BMM symmetrising trace conjecture for G_{13} as follows:

**Step 1:** We define the linear map τ on H(G_{13}),
which takes the value 1 on 1 and 0 on all other elements of the basis.

**Step 2:** We expand the basis B_{13} to a spanning set C_{13} of H(G_{13}), which
includes four particular elements, denoted by b_{97}, b_{98}, b_{99}, and b_{100}.

**Step 3:**
We express sb, tb and ub as a linear combination of elements of C_{13} for all b ϵ B_{13}.

**Step 4:** We express sb, tb and ub as a linear combination of elements of B_{13} for all b ϵ B_{13},
by expressing b_{97}, b_{98}, b_{99}, and b_{100} as a linear

combination of elements of B_{13}.

**Step 5:** The element z := b_{25} is a central element of H(G_{13}). 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 z^{4}
as a linear combination of the elements of B_{13}.

**Step 6:** Using Steps 4 and 5, we
calculate the matrix A := (τ (bb')) for all b, b' ϵ B_{13}.

**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(G_{13}).

**Step 8:** We prove that τ (b^{−1}z^{4} ) = 0 for all b ϵ B_{13} \ {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:

**I _{1}.** The set C

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 G_{13}.