Kronecker coefficient

In mathematics, Kronecker coefficients \( g_{\mu \nu}^{\lambda} \) 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 \( V_{\lambda} \) for the Specht module associated to λ. Then the Kronecker coefficients \( g_{\mu \nu}^{\lambda} \) 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 GL_n, i.e. if we write W_λ 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.
See also

Littlewood–Richardson coefficient

References

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, MR2721467

Mathematics Encyclopedia