The BMM Symmetrising Trace Conjecture for Families of Complex Reflection Groups of Rank Two

TL;DR

The paper tackles the BMM symmetrising trace conjecture for the exceptional rank-2 complex reflection groups by building explicit matrix models of their generic Hecke algebras and deploying a uniform z-basis framework. By exploiting factorizations W=Z(W)W'X and constructing z_j-bases via coset tables, the authors obtain faithful representations and a computable Gram matrix A_j whose symmetry and determinant ensure a canonical symmetrising trace tau_Bj for each group in the tetrahedral and octahedral families. They carry out detailed computations for j=4..15, establishing that tau_Bj is the desired trace and verifying the conjecture for these families; the approach is designed to extend to the icosahedral family in a uniform manner. The methods rely on automated algebraic computations and show the potential for a systematic treatment of rank-2 exceptional complex reflection groups within the BMM program, with a project-wide resource of computational tools.

Abstract

The exceptional complex reflection groups of rank 2 are partitioned into three families. We construct explicit matrix models for the Hecke algebras associated to the maximal groups in the tetrahedral and octahedral family, and use them to verify the BMM symmetrising trace conjecture for all groups in these two families, providing evidence that a similar strategy might apply for the icosahedral family.

Figures (2)

• Figure 1: Tetrahedral Family.
• Figure 2: Octahedral Family.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 2.1
• Theorem 2.2
• Conjecture 2.3: The BMR freeness conjecture
• Definition 2.4
• Example 2.5
• Conjecture 2.6: The BMM symmetrising trace conjecture
• Conjecture 2.7: The lifting conjecture
• Remark 2.8
• Example 3.1