In mathematics, Kronecker coefficients gλμν describe the decomposition of the tensor product (= Kronecker product) of two irreducible representations of a symmetric group into irreducible representations. More explicitly, given a partition λ of n, write Vλ for the Specht module associated to λ. Then the Kronecker coefficients gλμν are given by the rule
This is to be compared with Littlewood–Richardson coefficients, where one instead considers the induced representation
and the corresponding operation of symmetric functions is the usual product. Also note that the Littlewood–Richardson coefficients are the analogue of the Kronecker coefficients for representations of GLn, i.e. if we write Wλ for the irreducible representation corresponding to λ (where λ has at most n parts), one gets that
Bürgisser & Ikenmeyer (2008) showed that Kronecker coefficients are hard to compute.
- Bürgisser, Peter; Ikenmeyer, Christian (2008), "The complexity of computing Kronecker coefficients", 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), Discrete Math. Theor. Comput. Sci. Proc., AJ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, pp. 357–368, MR 2721467