Maths problem – which \beta satisfy \beta(i+j, k)\beta(j+k, i)\beta(k+i, j) = 1?

A problem has recently arisen when attempting to define colour traces on square traces using minimal assumptions. The problem is as follows.

Let i, j, k = 0, 1, 2, …, m-1, and let + be additive addition modulo m, i.e. we identify m and 0.

Let \beta(i, k) -> C be a complex-valued function of i and k, where \beta satisfies:

  • \beta(i, k) \beta(k, i) = 1, for all i, k, and
  • \beta(i+j, k) \beta(j+k, i) \beta(k+i, j) = 1.

The problem is: what are the allowed values of \beta(i, k) ?

Note we have: \beta(n, 0) = \beta(0, n) = 1 for all n which can be seen from fixing i=n-1, j=1, k=0, which gives \beta(n, 0) \beta(1, n-1) \beta(n-1, 1) = 1. We also have \beta(1, 1)^{2}=1.

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