4,694 research outputs found
No occurrence obstructions in geometric complexity theory
The permanent versus determinant conjecture is a major problem in complexity
theory that is equivalent to the separation of the complexity classes VP_{ws}
and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a
strengthened version of this conjecture over the complex numbers that amounts
to separating the orbit closures of the determinant and padded permanent
polynomials. In that paper it was also proposed to separate these orbit
closures by exhibiting occurrence obstructions, which are irreducible
representations of GL_{n^2}(C), which occur in one coordinate ring of the orbit
closure, but not in the other. We prove that this approach is impossible.
However, we do not rule out the general approach to the permanent versus
determinant problem via multiplicity obstructions as proposed by Mulmuley and
Sohoni.Comment: Substantial revision. This version contains an overview of the proof
of the main result. Added material on the model of power sums. Theorem 4.14
in the old version, which had a complicated proof, became the easy Theorem
5.4. To appear in the Journal of the AM
Geometric complexity theory and matrix powering
Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016). Following a classical homogenization result of Nisan (STOC 1991) we replace the determinant in geometric complexity theory with the trace of a variable matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent. This is the first no-go result in geometric complexity theory that rules out superlinear lower bounds in some model. Interestingly---in contrast to the determinant---the trace of a variable matrix power is not uniquely determined by its stabilizer
On Geometric Complexity Theory: Multiplicity Obstructions Are Stronger Than Occurrence Obstructions
Geometric Complexity Theory as initiated by Mulmuley and Sohoni in two papers
(SIAM J Comput 2001, 2008) aims to separate algebraic complexity classes via
representation theoretic multiplicities in coordinate rings of specific group
varieties. The papers also conjecture that the vanishing behavior of these
multiplicities would be sufficient to separate complexity classes (so-called
occurrence obstructions). The existence of such strong occurrence obstructions
has been recently disproven in 2016 in two successive papers, Ikenmeyer-Panova
(Adv. Math.) and B\"urgisser-Ikenmeyer-Panova (J. AMS). This raises the
question whether separating group varieties via representation theoretic
multiplicities is stronger than separating them via occurrences. This paper
provides for the first time a setting where separating with multiplicities can
be achieved, while the separation with occurrences is provably impossible. Our
setting is surprisingly simple and natural: We study the variety of products of
homogeneous linear forms (the so-called Chow variety) and the variety of
polynomials of bounded border Waring rank (i.e. a higher secant variety of the
Veronese variety). As a side result we prove a slight generalization of
Hermite's reciprocity theorem, which proves Foulkes' conjecture for a new
infinite family of cases.Comment: 24 page
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
Even Partitions in Plethysms
We prove that for all natural numbers k,n,d with k <= d and every partition
lambda of size kn with at most k parts there exists an irreducible GL(d,
C)-representation of highest weight 2*lambda in the plethysm Sym^k(Sym^(2n)
(C^d)). This gives an affirmative answer to a conjecture by Weintraub (J.
Algebra, 129 (1):103-114, 1990). Our investigation is motivated by questions of
geometric complexity theory and uses ideas from quantum information theory.Comment: 9 page
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
Implementing Geometric Complexity Theory: On the Separation of Orbit Closures via Symmetries
Understanding the difference between group orbits and their closures is a key
difficulty in geometric complexity theory (GCT): While the GCT program is set
up to separate certain orbit closures, many beautiful mathematical properties
are only known for the group orbits, in particular close relations with
symmetry groups and invariant spaces, while the orbit closures seem much more
difficult to understand. However, in order to prove lower bounds in algebraic
complexity theory, considering group orbits is not enough.
In this paper we tighten the relationship between the orbit of the power sum
polynomial and its closure, so that we can separate this orbit closure from the
orbit closure of the product of variables by just considering the symmetry
groups of both polynomials and their representation theoretic decomposition
coefficients. In a natural way our construction yields a multiplicity
obstruction that is neither an occurrence obstruction, nor a so-called
vanishing ideal occurrence obstruction. All multiplicity obstructions so far
have been of one of these two types.
Our paper is the first implementation of the ambitious approach that was
originally suggested in the first papers on geometric complexity theory by
Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all existence
proofs of obstructions only took into account the symmetry group of one of the
two polynomials (or tensors) that were to be separated. In our paper the
multiplicity obstruction is obtained by comparing the representation theoretic
decomposition coefficients of both symmetry groups.
Our proof uses a semi-explicit description of the coordinate ring of the
orbit closure of the power sum polynomial in terms of Young tableaux, which
enables its comparison to the coordinate ring of the orbit.Comment: 47 page
Explicit polynomial sequences with maximal spaces of partial derivatives and a question of K. Mulmuley
We answer a question of K. Mulmuley: In [Efremenko-Landsberg-Schenck-Weyman]
it was shown that the method of shifted partial derivatives cannot be used to
separate the padded permanent from the determinant. Mulmuley asked if this
"no-go" result could be extended to a model without padding. We prove this is
indeed the case using the iterated matrix multiplication polynomial. We also
provide several examples of polynomials with maximal space of partial
derivatives, including the complete symmetric polynomials. We apply Koszul
flattenings to these polynomials to have the first explicit sequence of
polynomials with symmetric border rank lower bounds higher than the bounds
attainable via partial derivatives.Comment: 18 pages - final version to appear in Theory of Computin
- …