The Hopf structure of symmetric group characters as symmetric functions

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.Comment: arXiv admin note: text overlap with arXiv:1605.0667

Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract)

We show that the Kronecker coefficients indexed by two two–row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients.Ministerio de Educación y Ciencia MTM2007–64509Junta de Andalucía FQM–33