Mercedes Rosas

The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes

The Kronecker product of two Schur functions sμ and sν, denoted by sμ * sν, is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions μ and ν. The coefficient of sλ in this product is denoted by γλμν, and corresponds to the multiplicity of the irreducible character χλ in χμχν.We use Sergeevs Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for sλ[XY] to find closed formulas for the Kronecker coefficients γλμν when λ is an arbitrary shape and μ and ν are hook shapes or two-row shapes.Remmel (J.B. Remmel, J. Algebra120 (1989), 100–118; Discrete Math.99 (1992), 265–287) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, Bull. Belg. Math. Soc. Simon Stiven1 (1994), 649–683) derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach. We believe that the approach of this paper is more natural. The formulas obtained are simpler and reflect the symmetry of the Kronecker product.

Symmetric functions in noncommuting variables

Consider the algebra Q >of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x 1 , x 2 ,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

We introduce a natural Hopf algebra structure on the space of noncommutative symmetric functions. The bases for this algebra are indexed by set partitions. We show that there exists a nat- ural inclusion of the Hopf algebra of noncommutative symmetric functions in this larger space. We also consider this algebra as a subspace of noncommutative polynomials and use it to understand the structure of the spaces of harmonics and coinvariants with respect to this collection of noncommuta- tive polynomials and conclude two analogues of Chevalleys theorem in the noncommutative setting.

Reduced Kronecker Coefficients and Counter–Examples to Mulmuley’s Strong Saturation Conjecture SH: With an Appendix by Ketan Mulmuley

Abstract.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 Kronecker coefficients. Both results rely on the connections between the Kronecker coefficients and another family of structural constants in the representation theory of the symmetric groups, Murnaghan’s reduced Kronecker coefficients.An appendix by Mulmuley introduces a relaxed form of the saturation hypothesis SH, still strong enough for the aims of Geometric Complexity Theory.

MacMahon Symmetric Functions, the Partition Lattice, and Young Subgroups

A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we show that the MacMahon symmetric functions are the generating functions for the orbits of sets of functions indexed by partitions under the diagonal action of a Young subgroup of a symmetric group. We define a MacMahon chromatic symmetric function that generalizes Stanleys chromatic symmetric function. Then, we study some of the properties of this new function through its connection with the noncommutative chromatic symmetric function of Gebhard and Sagan.

Inequalities between Littlewood-Richardson coefficients

We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture holds for a finite number of pairs, for any given height. Moreover, we propose a natural generalization of the conjecture to the case of skew shapes.

Milne's volume function and vector symmetric polynomials

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 Milnes volume function in the basis of monomial vector symmetric functions, and observe that only monomial functions of a particular kind appear in the expansion, the squarefree monomial functions. By means of an appropriate specialization of the vector symmetric Newton identities, we derive an inductive formula that expresses the squarefree monomial functions in the power sums basis. As a corollary, we obtain an inductive formula that writes Milnes volume function in the power sums basis. The lattice of the sub-hypergraphs of a hypergraph appears in a natural way in this setting.

Resolutions of De Concini-Procesi ideals of hooks

We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study its Hilbert series and regularity.

A Combinatorial Overview of the Hopf Algebra of MacMahon Symmetric Functions

Abstract. A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we give a combinatorial overview of the Hopf algebra structure of the MacMahon symmetric functions relying on the construction of a Hopf algebra from any alphabet of neutral letters obtained in [18, 19].

On the Sn-module structure of the noncommutative harmonics

Using a noncommutative analog of Chevalleys decomposition of polynomials into symmetric polynomials times coinvariants due to Bergeron, Reutenauer, Rosas, and Zabrocki we compute the graded Frobenius characteristic 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 derived free Lie algebra in n variables.