CATALOG ...

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$M_{13B}^{(1)}$

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A set of three Butson type matrices found by R. Nicoară, A. LaClair, N. Geist and A. Wintenberg in May 2016 (UT, Knoxville).

• $M_{13A}^{(0)}$
• $M_{13B}^{(1)}$
• $M_{13C}^{(2)}$

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$${\rm LOG}\left(M_{13B}\right)=\frac{1}{3}\pi \left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & 5 & 5 & 5 & 3 & 3 & 3 & 3 & 2 & 2 & 1 & \bullet & \bullet\\ \bullet & 1 & 1 & 3 & 1 & 3 & 4 & 5 & 1 & 3 & 5 & 3 & 5\\ \bullet & 3 & \bullet & 1 & 3 & 1 & 5 & 3 & \bullet & 4 & 1 & 4 & 3\\ \bullet & 1 & 3 & 4 & 5 & 1 & 1 & 3 & 4 & 4 & 1 & 2 & 5\\ \bullet & 3 & 5 & 3 & 5 & 5 & 3 & \bullet & 2 & \bullet & 1 & 2 & 3\\ \bullet & 5 & 3 & 1 & 1 & 4 & 1 & 5 & 4 & 2 & 1 & 4 & 3\\ \bullet & 5 & 3 & 1 & 5 & 1 & 3 & 1 & 2 & 4 & 4 & 4 & 1\\ \bullet & 2 & \bullet & 4 & 4 & 2 & 1 & \bullet & 4 & 2 & 4 & 5 & 2\\ \bullet & 4 & \bullet & 2 & 2 & 4 & 1 & 2 & 4 & 5 & 4 & 2 & \bullet\\ \bullet & \bullet & 3 & 5 & 3 & 3 & 5 & 1 & \bullet & \bullet & 3 & 2 & 3\\ \bullet & 2 & 2 & \bullet & 2 & \bullet & 3 & 4 & 3 & \bullet & 4 & \bullet & 4\\ \bullet & 3 & 3 & 3 & \bullet & 5 & 5 & 3 & \bullet & 2 & 3 & \bullet & 1\\ \end{array}\right].$$

Defect is $d(M_{13B})=1$.

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Extension of $M_{13B}$ to a $1$-parametric (maximal) affine family:

$$R_{M_{13B}^{(1)}}(a)=\left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & a & \bullet & a & \bullet & \bullet & \bullet & a & \bullet & a & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & a & \bullet & a & \bullet & \bullet & \bullet & a & \bullet & a & \bullet\\ \bullet & -a & -a & \bullet & -a & \bullet & \bullet & -a & \bullet & \bullet & -a & \bullet & -a \\ \bullet & \bullet & \bullet & a & \bullet & a & \bullet & \bullet & \bullet & a & \bullet & a & \bullet\\ \bullet & -a & -a & \bullet & -a & \bullet & \bullet & -a & \bullet & \bullet & -a & \bullet & -a \\ \bullet & \bullet & \bullet & a & \bullet & a & \bullet & \bullet & \bullet & a & \bullet & a & \bullet\\ \bullet & -a & -a & \bullet & -a & \bullet & \bullet & -a & \bullet & \bullet & -a & \bullet & -a \\ \bullet & -a & -a & \bullet & -a & \bullet & \bullet & -a & \bullet & \bullet & -a & \bullet & -a \\ \bullet & \bullet & \bullet & a & \bullet & a & \bullet & \bullet & \bullet & a & \bullet & a & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & a & \bullet & a & \bullet & \bullet & \bullet & a & \bullet & a & \bullet\\ \end{array}\right]$$

so that, for $a\in[0,1)$ one has

$$M_{13B}\mapsto M_{13B}^{(1)}(a) = M_{13B} \circ {\rm EXP}\left(2 \pi i R_{M_{13B}^{(1)}}(a)\right).$$