q-Pascal Matrices and Irreducible Representations of the Braid Group B₃

An explicit computational framework for constructing finite-dimensional irreducible representations of the 3-strand braid group B₃ using q-deformed Pascal matrices. Following the algebraic approach of Albeverio–Kosyak, we define: • σ₁(q,n) via Gaussian binomial coefficients in upper-triangular form • σ₂(q,n) through anti-diagonal reflection of σ₁(q⁻¹,n)⁻¹ • A diagonal twist Λ that ensures the braid relation ρ(σ₁)ρ(σ₂)ρ(σ₁) = ρ(σ₂)ρ(σ₁)ρ(σ₂) The construction is verified numerically to machine precision for q=2 across dimensions 3–6, with complete matrices provided. Included is an interactive React implementation that allows real-time exploration of the representation matrices and braid relation verification. Key features: - Explicit matrix families for dimensions 3–6 at q=2 - Machine-precision verification of braid relations - Self-contained implementation requiring only basic arithmetic - Reproducible construction algorithm suitable for computer algebra systems This work provides concrete computational access to the representation theory of B₃, connecting q-combinatorics with braid group structure through explicitly checkable matrix realizations. Author Brian Simpson licensed under the octomorphic open license version 1.1. braid group, q-Pascal matrices, representation theory, Gaussian binomial coefficients, B₃, quantum groups, computational algebra