Kronecker coefficient

Template:Short description In mathematics, Kronecker coefficients g^λ_μν 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 in algebraic combinatorics and geometric complexity theory. They were introduced by Murnaghan in 1938.

Definition

Given a partition λ of n, write V_λ for the Specht module associated to λ. Then the Kronecker coefficients g^λ_μν are given by the rule

$V_{\mu }\otimes V_{\nu }=\underset{\lambda }{⨁}{g}_{\mu \nu }^{\lambda }{V}_{\lambda }.$

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

${\uparrow }_{S_{|\mu |}\times S_{|\nu |}}^{S_{|\lambda |}}(V_{\mu }\otimes V_{\nu })=\underset{\lambda }{⨁}{c}_{\mu \nu }^{\lambda }{V}_{\lambda },$

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

$W_{\mu }\otimes W_{\nu }=\underset{\lambda }{⨁}{c}_{\mu \nu }^{\lambda }{W}_{\lambda }.$

Properties

Template:Harvtxt showed that computing Kronecker coefficients is #P-hard and contained in GapP. A recent work by Template:Harvtxt shows that deciding whether a given Kronecker coefficient is non-zero is NP-hard. This recent interest in computational complexity of these coefficients arises from its relevance in the Geometric Complexity Theory program.

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. A combinatorial description would also imply that the problem is # P-complete in light of the above result.

The Kronecker coefficients can be computed as $$g(\lambda ,\mu ,\nu )=\frac{1}{n!}\sum _{\sigma \in S_n}{\chi }^{\lambda }(\sigma ){\chi }^{\mu }(\sigma ){\chi }^{\nu }(\sigma ),$$ where ${\chi }^{\lambda }(\sigma )$ is the character value of the irreducible representation corresponding to integer partition $\lambda$ on a permutation $\sigma \in S_n$.

The Kronecker coefficients also appear in the generalized Cauchy identity $$\sum _{\lambda ,\mu ,\nu }g(\lambda ,\mu ,\nu )s_{\lambda }(x)s_{\mu }(y)s_{\nu }(z)=\prod _{i,j,k}\frac{1}{1-x_i{y}_j{z}_k}.$$

See also

• Littlewood–Richardson coefficient

References

• Template:Citation