Character Table Spectracones & the $n=5$ SNIEP: From $(1,a,a,b,b)$ to the 3D $(1,a,a,b,c)$ Frontier
Recent work by Johnson & Paparella (2025) and Gershnik, Lewis, & Paparella (2026) revealed that the character tables of finite groups act as ideal Perron similarities, transforming the nonlinear eigenvalue realization problem into explicit systems of linear halfspaces governed by group character values and conjugacy class sizes. In this interactive exploration, we test how much of the $n=5$ Symmetric NIEP can be carved out by finite group character tables alone.
Starting with the 2D benchmark $\sigma = (1, a, a, b, b)$ conjectured by Loewy & McDonald (2004), we plot the polyhedral cones generated by $D_8, Q_8, S_4, A_5$, and the cyclic group $\mathbb{Z}_5$, diagnose why a large wedge near the McDonald–Newman boundary $1 + a + 2b = 0$ is missing from all irreducible $5 \times 5$ character tables, and resolve it by adjoining a $1 \times 1$ eigenvalue block to the $4 \times 4$ Klein four-group table ($V_4 \oplus [1]$)—achieving 99.97% coverage of the Loewy–McDonald benchmark region. We then lift the entire machinery to the unsolved 3D frontier $\sigma = (1, a, a, b, c)$ inside the trace polytope $\Delta_3$.