Kronecker coefficient

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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. They play an important role algebraic combinatorics and geometric complexity theory. They were introduced by Murnaghan in 1938.

Definition

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

${\displaystyle V_{\mu }\otimes V_{\nu }=\bigoplus _{\lambda }g_{\mu \nu }^{\lambda }V_{\lambda }.}$

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

${\displaystyle s_{\mu }\star s_{\nu }=\sum _{\lambda }g_{\mu \nu }^{\lambda }s_{\lambda }.}$

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

${\displaystyle \uparrow _{S_{|\mu |}\times S_{|\nu |}}^{S_{|\lambda |}}\left(V_{\mu }\otimes V_{\nu }\right)=\bigoplus _{\lambda }c_{\mu \nu }^{\lambda }V_{\lambda },}$

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

${\displaystyle W_{\mu }\otimes W_{\nu }=\bigoplus _{\lambda }c_{\mu \nu }^{\lambda }W_{\lambda }.}$

Properties

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

A major unsolved problem in representation theory and combinatorics is to give a combinatorial description of the Kronecker coefficients. It has been open since 1938, when Murnaghan asked for such a combinatorial description. [1]

The Kronecker coefficients can be computed as

${\displaystyle g(\lambda ,\mu ,\nu )={\frac {1}{n!}}\sum _{\sigma \in S_{n}}\chi ^{\lambda }(\sigma )\chi ^{\mu }(\sigma )\chi ^{\nu }(\sigma ),}$

where ${\displaystyle \chi ^{\lambda }(\sigma )}$ is the character value of the irreducible representation corresponding to partition ${\displaystyle \lambda }$ on a permutation ${\displaystyle \sigma \in S_{n}}$.

The Kronecker coefficients also appear in the generalized Cauchy identity

${\displaystyle \sum _{\lambda ,\mu ,\nu }g(\lambda ,\mu ,\nu )s_{\lambda }(x)s_{\mu }(y)s_{\nu }(z)=\prod _{i,j,k}{\frac {1}{1-x_{i}y_{j}z_{k}}}.}$

References

1. ^ Murnaghan, D. (1938). "The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups". Amer. J. Math. 60: 44–65.