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 γ λ µν , and corresponds to the multiplicity of the irreducible character χ λ in χ µχ ν We use Sergeev’s 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 γ λ µν when λ is an arbitrary shape and µ and ν are hook shapes or two-row shapes. Remmel [9 J.B. Remmel, “A formula for the Kronecker product of Schur functions of hook shapes,” J. Algebra 120, 1989, pp. 100–118, 10 J.B. Remmel, “Formulas for the expansion of the Kronecker products S(m,n) ⊗ S(1p−r,r) and S(1k2 l) ⊗ S(1p−r,r) ,” Discrete Math. 99, 1992, pp. 265–287] and Remmel and Whitehead [11] J.B. Remmel and T. Whitehead, “On the Kronecker product of Schur functions of two row shapes,” Bull. Belg. Math. Soc. Simon Stevin 1, 1994, pp. 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

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

On the growth of the Kronecker coefficients: accompanying appendices

This text is an appendix to our work ”On the growth of Kronecker coefficients” [1]. Here, we provide some complementary theorems, re- marks, and calculations that for the sake of space are not going to appear into the final version of our paper. We follow the same terminology and notation. External references to numbered equations, theorems, etc. are pointers to [1]. This file is not meant to be read independently of the main text.Ministerio de Economía y Competitividad MTM2013–40455–PJunta de Andalucía FQM–333Junta de Andalucía P12-FQM-269

Normally ordered forms of powers of differential operators and their combinatorics

We investigate the combinatorics of the general formulas for the powers of the operator h∂k, where h is a central element of a ring and ∂ is a differential operator. This generalizes previous work on the powers of operators h∂. New formulas for the generalized Stirling numbers are obtained.Ministerio de Economía y competitividad MTM2016-75024-PJunta de Andalucía P12-FQM-2696Junta de Andalucía FQM–33

Symmetric functions in noncommuting variables

Consider the algebra Qhhx1, x2, . . .ii of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x1, x2, . . .) 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 will as investigating their properties

Combinatorics on a family of reduced Kronecker coefficients

The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also gives its connection to the plane partitions, which allows us to check that this family satisfies the saturation conjecture for reduced Kronecker coefficients, and that they are weakly increasing. Thanks to its generating function we can describe our family by a quasipolynomial, specifying its degree and period.Ministerio de Economía y CompetitividadJunta de AndalucíaFondo Europeo de Desarrollo Regiona

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

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.Natural Sciences and Engineering Research Council of CanadaFonds Québécois de la Recherche sur la Nature et les Technologie

The defining ideals of conjugacy classes of nilpotent matrices and a conjecture of Weyman

Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition λ, we define several methods to produce a reduced generating set for the associated ideal Iλ. For particular shapes we find nice generating sets. By comparing our sets with some generating sets of Iλ arising from a work of Weyman, we find a counterexample to a related conjecture of Weyman.Natural Sciences and Engineering Research Council of CanadaMinisterio de Educación y Cienci