Kronecker coefficient

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 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

[math]\displaystyle{ V_\mu \otimes V_\nu = \bigoplus_\lambda g_{\mu \nu}^{\lambda} V_\lambda. }[/math]

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

[math]\displaystyle{ s_\mu \star s_\nu = \sum_{\lambda} g_{\mu \nu}^{\lambda} s_\lambda. }[/math]

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

[math]\displaystyle{ \uparrow_{S_{|\mu|} \times S_{|\nu|}}^{S_{|\lambda|}} \left ( V_\mu \otimes V_\nu \right ) = \bigoplus_\lambda c_{\mu \nu}^{\lambda} V_\lambda, }[/math]

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

[math]\displaystyle{ W_\mu \otimes W_\nu = \bigoplus_\lambda c_{\mu \nu}^{\lambda} W_\lambda. }[/math]

Properties

(Bürgisser Ikenmeyer) showed that computing Kronecker coefficients is #P-hard and contained in GapP. A recent work by (Ikenmeyer Mulmuley) 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

[math]\displaystyle{ g(\lambda, \mu, \nu) = \frac{1}{n!} \sum_{\sigma \in S_n} \chi^{\lambda}(\sigma)\chi^{\mu}(\sigma)\chi^{\nu}(\sigma), }[/math]

where [math]\displaystyle{ \chi^{\lambda}(\sigma) }[/math] is the character value of the irreducible representation corresponding to partition [math]\displaystyle{ \lambda }[/math] on a permutation [math]\displaystyle{ \sigma \in S_n }[/math].

The Kronecker coefficients also appear in the generalized Cauchy identity

[math]\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_iy_jz_k}. }[/math]

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, http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/viewFile/dmAJ0131/2487