The Homogeneous Weight Partition and its Character-Theoretic Dual

The values of the normalized homogeneous weight are determined for arbitrary finite Frobenius rings and expressed in a form that is independent from a generating character and the M\"obius function on the ring. The weight naturally induces a partition of the ring, which is invariant under left or right multiplication by units. It is shown that the character-theoretic left-sided dual of this partition coincides with the right-sided dual, and even more, the left- and right-sided Krawtchouk coefficients coincide. An example is provided showing that this is not the case for general invariant partitions if the ring is not semisimple.