of the -dimensional representation is reducible – the permutation representation coincides with the regular representation, and thus breaks up into the three one-dimensional representations, as is abelian; see the discrete Fourier transform for representation theory of cyclic groups. For , there is just one irreducible representation, but there are the exceptional irreducible representations of dimension 1. For , there are two dual irreducible representations of dimension 3, corresponding to its action as icosahedral symmetry. For , there is an extra irreducible representation of dimension 5 corresponding to the exceptional transitive embedding of A5 in A6. Tensor products of representations Kronecker coefficients The tensor product of two representations of corresponding to the Young diagrams is a combination of irreducible representations of , The coefficients are called the Kronecker coefficients of the symmetric group. They can be computed from the characters of the representations : The sum is over partitions of , with the corresponding conjugacy classes. The values of the characters can be computed using the Frobenius formula. The coefficients are where is the number of times appears in , so that . A few examples, written in terms of Young diagrams : There is a simple rule for computing for any Young diagram : the result is the sum of all Young diagrams that are obtained from by removing one box and then adding one box, where the coefficients are one except for itself, whose coefficient is , i.e., the number of different row lengths minus one. A constraint on the irreducible constituents of is where the depth of a Young diagram is the number of boxes that do not belong to the first row. Reduced Kronecker coefficients For a Young diagram and , is a Young diagram of size . Then is a bounded, non-decreasing function of , and is called a reduced Kronecker coefficient or stable Kronecker coefficient. There are known bounds on the value of where reaches its limit. The reduced Kronecker coefficients are structure constants of Deligne categories of representations of with . In contrast to Kronecker coefficients, reduced Kronecker coefficients are defined for any triple of Young diagrams, not necessarily of the same size. If , then coincides with the Littlewood-Richardson coefficient . Reduced Kronecker coefficients can be written as linear combinations of Littlewood-Richardson coefficients via a change of bases in the space of symmetric functions, giving rise to expressions that are manifestly integral although