230 research outputs found
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.Comment: 8 page
On the growth of the Kronecker coefficients
We study the rate of growth experienced by the Kronecker coefficients as we
add cells to the rows and columns indexing partitions. We do this by moving to
the setting of the reduced Kronecker coefficients.Comment: Extended version, Containing 4 appendice
Symmetric Functions in Noncommuting Variables
Consider the algebra Q> of formal power series in countably
many noncommuting variables over the rationals. The subalgebra Pi(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 will as
investigating their properties.Comment: 16 pages, Latex, see related papers at
http://www.math.msu.edu/~sagan, to appear in Transactions of the American
Mathematical Societ
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
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
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