not manifestly positive. Reduced Kronecker coefficients can also be written in terms of Kronecker and Littlewood-Richardson coefficients via Littlewood's formula Conversely, it is possible to recover the Kronecker coefficients as linear combinations of reduced Kronecker coefficients. Reduced Kronecker coefficients are implemented in the computer algebra system SageMath. Eigenvalues of complex representations Given an element of cycle-type and order , the eigenvalues of in a complex representation of are of the type with , where the integers are called the cyclic exponents of with respect to the representation. There is a combinatorial description of the cyclic exponents of the symmetric group (and wreath products thereof). Defining , let the -index of a standard Young tableau be the sum of the values of over the tableau's descents, . Then the cyclic exponents of the representation of described by the Young diagram are the -indices of the corresponding Young tableaux. In particular, if is of order , then , and coincides with the major index of (the sum of the descents). The cyclic exponents of an irreducible representation of then describe how it decomposes into representations of the cyclic group , with being interpreted as the image of in the (one-dimensional) representation characterized by . See also Alternating polynomials Symmetric polynomials Schur functor Robinson–Schensted correspondence Schur–Weyl duality Jucys–Murphy element Garnir relations References Cited Publications Representation theory of finite groups Permutations Integer partitions