Breaking down the reduced Kronecker coefficients

We resolve three interrelated problems on reduced Kronecker coefficients g ¯ ( α , β , γ ) . First, we disprove the saturation property which states that g ¯ ( N α , N β , N γ ) > 0 implies g ¯ ( α , β , γ ) > 0 for all N > 1 . Second, we esimate the maximal g ¯ ( α , β , γ ) , over all | α | + | β | + | γ | = n . Finally, we show that computing g ¯ ( λ , μ , ν ) is strongly #P -hard, i.e. #P -hard when the input ( λ , μ , ν ) is in unary.