We show that the Grothendieck bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function… (More)
We provide counter–examples to Mulmuley's SH conjecture for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynomial time. We also provide a short proof of the #P–hardness of computing the Kro-necker coefficients. Both… (More)
We provide counter–examples to Mulmuley's strong saturation conjecture (strong SH) for the Kronecker coefficients. This conjecture was proposed in the setting of Geometric Complexity Theory to show that deciding whether or not a Kronecker coefficient is zero can be done in polynomial time. We also provide a short proof of the #P– hardness of computing the… (More)
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… (More)
Quasipolynomial formulas for the Kronecker coefficients indexed by two two–row shapes (extended abstract) Abstract. We show that the Kronecker coefficients indexed by two two–row shapes are given by quadratic quasipoly-nomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipoly-nomial formulas and a… (More)
Using the a noncommutative version of Chevalley's theorem due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius series for their two sets of noncommutative harmonics with respect to the left action of the symmetric group (acting on variables). We use these results to derive the Frobenius series for the enveloping algebra of the… (More)
We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share symmetries related to the operations of taking complements with respect to rectangles and adding rectangles.
I. Matrices totalmente no negativas y funciones simétricas. primera parte presenta una introducci on a las matrices totalmente no negativas, y su relación con las funciones simétricas.
The number of real roots of a system of polynomial equations fitting inside a given box can be counted using a vector symmetric polynomial introduced by P. Milne, the volume function. We provide the expansion of Milne's volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear… (More)
In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For n sufficiently large, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree n do not depend on the first part of the indexing partitions, but only on the values of their remaining… (More)