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Ketan D. Mulmuley

Publications and source records attributed to Ketan D. Mulmuley.

13 recordsLinked to original sources
arXiv2017

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 $(λ, μ, π)$ such that the Kronecker coefficient $k^λ_{μ, π} = 0$ but the Kronecker coefficient $k^{l λ}_{l μ, l π} > 0$ for some integer $l>1$. 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 $0<ε\leq1$, there exists $0<a<1$ such that, for all $m$, there exist $Ω(2^{m^a})$ partition triples $(λ,μ,μ)$ in the Kronecker cone such that: (a) the Kronecker coefficient $k^λ_{μ,μ}$ is zero, (b) the height of $μ$ is $m$, (c) the height of $λ$ is $\le m^ε$, and (d) $|λ|=|μ| \le m^3$. The proof of the last result illustrates the effectiveness of the explicit proof strategy of GCT.

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arXiv2017

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.

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arXiv2016

Geometric Complexity Theory V: Efficient algorithms for Noether Normalization

We study a basic algorithmic problem in algebraic geometry, which we call NNL, of constructing a normalizing map as per Noether's Normalization Lemma. For general explicit varieties, as formally defined in this paper, we give a randomized polynomial-time Monte Carlo algorithm for this problem. For some interesting cases of explicit varieties, we give deterministic quasi-polynomial time algorithms. These may be contrasted with the standard EXPSPACE-algorithms for these problems in computational algebraic geometry. In particular, we show that: (1) The categorical quotient for any finite dimensional representation $V$ of $SL_m$, with constant $m$, is explicit in characteristic zero. (2) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in the dimension of $V$. (3) The categorical quotient of the space of $r$-tuples of $m \times m$ matrices by the simultaneous conjugation action of $SL_m$ is explicit in any characteristic. (4) NNL for this categorical quotient can be solved deterministically in time quasi-polynomial in $m$ and $r$ in any characteristic $p$ not in $[2,\ m/2]$. (5) NNL for every explicit variety in zero or large enough characteristic can be solved deterministically in quasi-polynomial time, assuming the hardness hypothesis for the permanent in geometric complexity theory. The last result leads to a geometric complexity theory approach to put NNL for every explicit variety in P.

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arXiv2016

Boundaries of VP and VNP

One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = $\overline{\textrm{VP}}$, where VP is the class of families of polynomials that are of polynomial degree and can be computed by arithmetic circuits of polynomial size, and $\overline{\textrm{VP}}$ is the class of families of polynomials that are of polynomial degree and can be approximated infinitesimally closely by arithmetic circuits of polynomial size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that $\overline{\textrm{VP}}$ is not contained in VP. Towards that end, we introduce three degenerations of VP (i.e., sets of points in $\overline{\textrm{VP}}$), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP*. We also introduce analogous degenerations of VNP. We show that Stable-VP $\subseteq$ Newton-VP $\subseteq$ VP* $\subseteq$ VNP, and Stable-VNP = Newton-VNP = VNP* = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating $\overline{\textrm{VP}}$ from VP. Although we do not yet construct explicit candidates for the polynomial families in $\overline{\textrm{VP}}\setminus$VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP $\setminus$ VP based on semi-invariants of quivers would have to be non-generic by showing that, for many finite quivers (including some wild ones), any Newton degeneration of a generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits.

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arXiv2013

Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

The Kronecker coefficient g_{λμν} is the multiplicity of the GL(V)\times GL(W)-irreducible V_λ\otimes W_μin the restriction of the GL(X)-irreducible X_νvia the natural map GL(V)\times GL(W) \to GL(V \otimes W), where V, W are \mathbb{C}-vector spaces and X = V \otimes W. A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality. Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam to construct, in the case dim(V) = dim(W) =2, a representation \check{X}_νof the nonstandard quantum group that specializes to Res_{GL(V) \times GL(W)} X_νat q=1. We then define a global crystal basis +HNSTC(ν) of \check{X}_νthat solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ,μ) is the Kronecker coefficient g_{λμν}. We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U_q(\sl_2) graphical calculus, and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki. As a byproduct of the approach, we also obtain a rule for the decomposition of Res_{GL_2 \times GL_2 \rtimes §_2} X_νinto irreducibles.

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arXiv2009

On P vs. NP, Geometric Complexity Theory, and the Riemann Hypothesis

Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems. A high level overview of this research plan and the results obtained so far was presented in a series of three lectures in the Institute of Advanced study, Princeton, Feb 9-11, 2009. This article contains the material covered in those lectures after some revision, and gives a mathematical overview of GCT. No background in algebraic geometry, representation theory or quantum groups is assumed.

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arXiv2009

Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry

This article belongs to a series on geometric complexity theory (GCT), an approach to the P vs. NP and related problems through algebraic geometry and representation theory. The basic principle behind this approach is called the flip. In essence, it reduces the negative hypothesis in complexity theory (the lower bound problems), such as the P vs. NP problem in characteristic zero, to the positive hypothesis in complexity theory (the upper bound problems): specifically, to showing that the problems of deciding nonvanishing of the fundamental structural constants in representation theory and algebraic geometry, such as the well known plethysm constants--or rather certain relaxed forms of these decision probelms--belong to the complexity class P. In this article, we suggest a plan for implementing the flip, i.e., for showing that these relaxed decision problems belong to P. This is based on the reduction of the preceding complexity-theoretic positive hypotheses to mathematical positivity hypotheses: specifically, to showing that there exist positive formulae--i.e. formulae with nonnegative coefficients--for the structural constants under consideration and certain functions associated with them. These turn out be intimately related to the similar positivity properties of the Kazhdan-Lusztig polynomials and the multiplicative structural constants of the canonical (global crystal) bases in the theory of Drinfeld-Jimbo quantum groups. The known proofs of these positivity properties depend on the Riemann hypothesis over finite fields and the related results. Thus the reduction here, in conjunction with the flip, in essence, says that the validity of the P vs. NP conjecture in characteristic zero is intimately linked to the Riemann hypothesis over finite fields and related problems.

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arXiv2008

Geometric Complexity Theory VII: Nonstandard quantum group for the plethysm problem

This article describes a {\em nonstandard} quantum group that may be used to derive a positive formula for the plethysm problem, just as the standard (Drinfeld-Jimbo) quantum group can be used to derive the positive Littlewood-Richardson rule for arbitrary complex semisimple Lie groups. The sequel \cite{GCT8} gives conjecturally correct algorithms to construct canonical bases of the coordinate rings of these nonstandard quantum groups and canonical bases of the dually paired nonstandard deformations of the symmetric group algebra. A positive $#P$-formula for the plethysm constant follows from the conjectural properties of these canonical bases and the duality and reciprocity conjectures herein.

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arXiv2008

Geometric Complexity Theory VIII: On canonical bases for the nonstandard quantum groups

This article gives conjecturally correct algorithms to construct canonical bases of the irreducible polynomial representations and the matrix coordinate rings of the nonstandard quantum groups in GCT4 and GCT7, and canonical bases of the dually paired nonstandard deformations of the symmetric group algebra therein. These are generalizations of the canonical bases of the irreducible polynomial representations and the matrix coordinate ring of the standard quantum group, as constructed by Kashiwara and Lusztig, and the Kazhdan-Lusztig basis of the Hecke algebra. A positive ($#P$-) formula for the well-known plethysm constants follows from their conjectural properties and the duality and reciprocity conjectures in \cite{GCT7}.

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arXiv2007

Geometric Complexity Theory: Introduction

These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the second author in the spring quarter, 2003. It gives introduction to invariant theory with a view towards GCT. No background in algebraic geometry or representation theory is assumed. These lecture notes in conjunction with the article \cite{GCTflip1}, which describes in detail the basic plan of GCT based on the principle called the flip, should provide a high level picture of GCT assuming familiarity with only basic notions of algebra, such as groups, rings, fields etc.

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arXiv2007

On P vs. NP, Geometric Complexity Theory, and the Flip I: a high level view

Geometric complexity theory (GCT) is an approach to the $P$ vs. $NP$ and related problems through algebraic geometry and representation theory. This article gives a high-level exposition of the basic plan of GCT based on the principle, called the flip, without assuming any background in algebraic geometry or representation theory.

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arXiv2005

Geometric Complexity III: on deciding positivity of Littlewood-Richardson coefficients

We point out that the remarkable Knutson and Tao Saturation Theorem and polynomial time algorithms for LP have together an important and immediate consequence in Geometric Complexity Theory. The problem of deciding positivity of Littlewood-Richardson coefficients for GLn(C) belongs to P. Furthermore, the algorithm is strongly polynomial. The main goal of this article is to explain the significance of this result in the context of Geometric Complexity Theory. Furthermore, it is also conjectured that an analogous result holds for arbitrary symmetrizable Kac-Moody algebras.

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