Geometric complexity theory III: on deciding nonvanishing of a Littlewood–Richardson coefficient

Mulmuley, Ketan; Narayanan, Hariharan; Sohoni, Milind

doi:10.1007/s10801-011-0325-1

Cited by 47 publications

(35 citation statements)

References 26 publications

(92 reference statements)

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“…The Kronecker problem. This is a continuation of the series of articles [47,48,44] on geometric complexity theory (GCT), an approach to P vs. NP and related problems using geometry and representation theory. A basic philosophy of this approach is called the flip; see [46,42,43] for its detailed exposition.…”

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confidence: 98%

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

2015

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Abstract. The Kronecker coefficient g λµν is the multiplicity of the GL(V ) × GL(W )-irreducible V λ ⊗ W µ in the restriction of the GL(X)-irreducible X ν via the natural map, where V, W are C-vector spaces and X = V ⊗ 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 [1] to construct, in the case dim(V ) = dim(W ) = 2, a representatioň X ν of the nonstandard quantum group that specializes to Res GL(V )×GL(W ) X ν at q = 1. We then define a global crystal basis +HNSTC(ν) ofX ν that 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 from [19], and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, positive formula for two-row Kronecker coefficients, generalizing a formula in [15]. As a byproduct of the approach, we also obtain a rule for the decomposition of Res GL2×GL2⋊S2 X ν into irreducibles.

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confidence: 98%

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

2015

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“…This article belongs to a series [71,73,70,6] of articles on geometric complexity theory. See [67,66] for an overview of the earlier articles in this series, and [14] for an overview of the mathematical issues therein.…”

Section: Geometric Complexity Theory Approach To the Basic Algorithmimentioning

confidence: 99%

Geometric complexity theory V: Efficient algorithms for Noether normalization

Mulmuley

2016

J. Amer. Math. Soc.

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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 quasipolynomial in the dimension of V .(3) The categorical quotient of the space of r-tuples of m×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 quasipolynomial in m and r in any characteristic p ∈ [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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“…Note that even the case of G = SL(n, C) is extremely interesting in view of possible applications to P = NP, see [MS1], [MS2]. ) by induction on the rank: This procedure was first conjectured by R. Horn in the 1960s; it was proven in a combination of works by A. Klyachko [Kly1] and A. Knutson and T. Tao [KT].…”

Section: Other Developmentsmentioning

confidence: 99%

Generalized triangle inequalities and their applications

Kapovich¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. We present a survey of our recent work on generalized triangle inequalities in infinitesimal symmetric spaces, nonpositively curved symmetric spaces and Euclidean buildings. We also explain how these results can be used to analyze some basic problems of algebraic group theory including the problem of decomposition of tensor products of irreducible representations of complex reductive Lie groups. Among the applications is a generalization of the Saturation Theorem of Knutson and Tao to Lie groups other than SL(n, C).

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Geometric complexity theory III: on deciding nonvanishing of a Littlewood–Richardson coefficient

Cited by 47 publications

References 26 publications

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

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

Geometric complexity theory V: Efficient algorithms for Noether normalization

Generalized triangle inequalities and their applications

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