Kronecker coefficient

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

$V_\mu \otimes V_\nu = \bigoplus_\lambda g_{\mu \nu}^{\lambda} V_\lambda.$

One can interpret this on the level of symmetric functions, giving a formula for the Kronecker product of two Schur polynomials:

$s_\mu \star s_\nu = \sum_{\lambda} g_{\mu \nu}^{\lambda} s_\lambda.$

This is to be compared with Littlewood–Richardson coefficients, where one instead considers the induced representation

$\uparrow_{S_{|\mu|} \times S_{|\nu|}}^{S_{|\lambda|}} \left ( V_\mu \otimes V_\nu \right ) = \bigoplus_\lambda c_{\mu \nu}^{\lambda} V_\lambda,$

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

$W_\mu \otimes W_\nu = \bigoplus_\lambda c_{\mu \nu}^{\lambda} W_\lambda.$

Bürgisser & Ikenmeyer (2008) showed that Kronecker coefficients are hard to compute.