18,016 research outputs found
On the complexity of computing Kronecker coefficients
We study the complexity of computing Kronecker coefficients
. We give explicit bounds in terms of the number of parts
in the partitions, their largest part size and the smallest second
part of the three partitions. When , i.e. one of the partitions
is hook-like, the bounds are linear in , but depend exponentially on
. Moreover, similar bounds hold even when . By a separate
argument, we show that the positivity of Kronecker coefficients can be decided
in time for a bounded number of parts and without
restriction on . Related problems of computing Kronecker coefficients when
one partition is a hook, and computing characters of are also considered.Comment: v3: incorporated referee's comments; accepted to Computational
Complexit
Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH
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.Comment: 25 pages. With an appendix by Ketan Mulmuley. To appear in
Computational Complexity. See also
http://emmanuel.jean.briand.free.fr/publications
Representations of the symmetric group are decomposable in polynomial time
We introduce an algorithm to decompose orthogonal matrix representations of
the symmetric group over the reals into irreducible representations, which as a
by-product also computes the multiplicities of the irreducible representations.
The algorithm applied to a -dimensional representation of is shown to
have a complexity of operations for determining multiplicities of
irreducible representations and a further operations to fully
decompose representations with non-trivial multiplicities. These complexity
bounds are pessimistic and in a practical implementation using floating point
arithmetic and exploiting sparsity we observe better complexity. We demonstrate
this algorithm on the problem of computing multiplicities of two tensor
products of irreducible representations (the Kronecker coefficients problem) as
well as higher order tensor products. For hook and hook-like irreducible
representations the algorithm has polynomial complexity as increases
Symbolic-Numeric Tools for Analytic Combinatorics in Several Variables
Analytic combinatorics studies the asymptotic behaviour of sequences through
the analytic properties of their generating functions. This article provides
effective algorithms required for the study of analytic combinatorics in
several variables, together with their complexity analyses. Given a
multivariate rational function we show how to compute its smooth isolated
critical points, with respect to a polynomial map encoding asymptotic
behaviour, in complexity singly exponential in the degree of its denominator.
We introduce a numerical Kronecker representation for solutions of polynomial
systems with rational coefficients and show that it can be used to decide
several properties (0 coordinate, equal coordinates, sign conditions for real
solutions, and vanishing of a polynomial) in good bit complexity. Among the
critical points, those that are minimal---a property governed by inequalities
on the moduli of the coordinates---typically determine the dominant asymptotics
of the diagonal coefficient sequence. When the Taylor expansion at the origin
has all non-negative coefficients (known as the `combinatorial case') and under
regularity conditions, we utilize this Kronecker representation to determine
probabilistically the minimal critical points in complexity singly exponential
in the degree of the denominator, with good control over the exponent in the
bit complexity estimate. Generically in the combinatorial case, this allows one
to automatically and rigorously determine asymptotics for the diagonal
coefficient sequence. Examples obtained with a preliminary implementation show
the wide applicability of this approach.Comment: As accepted to proceedings of ISSAC 201
Kronecker Coefficients For Some Near-Rectangular Partitions
We give formulae for computing Kronecker coefficients occurring in the
expansion of , where both and are nearly
rectangular, and have smallest parts equal to either 1 or 2. In particular, we
study , ,
, and
. Our approach relies on the interplay between
manipulation of symmetric functions and the representation theory of the
symmetric group, mainly employing the Pieri rule and a useful identity of
Littlewood. As a consequence of these formulae, we also derive an expression
enumerating certain standard Young tableaux of bounded height, in terms of the
Motzkin and Catalan numbers
On vanishing of Kronecker coefficients
We show that the problem of deciding positivity of Kronecker coefficients is
NP-hard. Previously, this problem was conjectured to be in P, just as for the
Littlewood-Richardson coefficients. Our result establishes in a formal way that
Kronecker coefficients are more difficult than Littlewood-Richardson
coefficients, unless P=NP.
We also show that there exists a #P-formula for a particular subclass of
Kronecker coefficients whose positivity is NP-hard to decide. This is an
evidence that, despite the hardness of the positivity problem, there may well
exist a positive combinatorial formula for the Kronecker coefficients. Finding
such a formula is a major open problem in representation theory and algebraic
combinatorics.
Finally, we consider the existence of the partition triples such that the Kronecker coefficient but the
Kronecker coefficient for some integer
. Such "holes" are of great interest as they witness the failure of the
saturation property for the Kronecker coefficients, which is still poorly
understood. Using insight from computational complexity theory, we turn our
hardness proof into a positive result: We show that not only do there exist
many such triples, but they can also be found efficiently. Specifically, we
show that, for any , there exists such that, for all
, there exist partition triples in the
Kronecker cone such that: (a) the Kronecker coefficient
is zero, (b) the height of is , (c) the height of is , and (d) . The proof of the last result
illustrates the effectiveness of the explicit proof strategy of GCT.Comment: 43 pages, 1 figur
The Saxl Conjecture and the Dominance Order
In 2012 Jan Saxl conjectured that all irreducible representations of the
symmetric group occur in the decomposition of the tensor square of the
irreducible representation corresponding to the staircase partition. We make
progress on this conjecture by proving the occurrence of all those irreducibles
which correspond to partitions that are comparable to the staircase partition
in the dominance order. Moreover, we use our result to show the occurrence of
all irreducibles corresponding to hook partitions. This generalizes results by
Pak, Panova, and Vallejo from 2013.Comment: 11 page
Membership in moment polytopes is in NP and coNP
We show that the problem of deciding membership in the moment polytope
associated with a finite-dimensional unitary representation of a compact,
connected Lie group is in NP and coNP. This is the first non-trivial result on
the computational complexity of this problem, which naively amounts to a
quadratically-constrained program. Our result applies in particular to the
Kronecker polytopes, and therefore to the problem of deciding positivity of the
stretched Kronecker coefficients. In contrast, it has recently been shown that
deciding positivity of a single Kronecker coefficient is NP-hard in general
[Ikenmeyer, Mulmuley and Walter, arXiv:1507.02955]. We discuss the consequences
of our work in the context of complexity theory and the quantum marginal
problem.Comment: 20 page
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
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