Fitness, Apprenticeship, and Polynomials

Sturmfels, Bernd

doi:10.1007/978-1-4939-7486-3_1

Cited by 14 publications

(1 citation statement)

References 54 publications

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“…We show that this is only the case when the two circles either meet in two points or are mutually tangent-in all other cases, the edge surface has higher degree and it is an irrational surface of degree eight when the circles are disjoint in CP 3 . This is related to Problem 3 on Convexity in [21], on the convex hull of three ellipsoids in R 3 . An algorithm was presented in [6] (see the video [7]), using projective duality.…”

Section: Introductionmentioning

confidence: 99%

The Convex Hull of Two Circles in ℝ 3 $$\mathbb{R}^{3}$$

Nash

Pir

Sottile

et al. 2017

Fields Institute Communications

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We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality.

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Section: Introductionmentioning

confidence: 99%

The Convex Hull of Two Circles in ℝ 3 $$\mathbb{R}^{3}$$

Nash

Pir

Sottile

et al. 2017

Fields Institute Communications

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Specht Polytopes and Specht Matroids

Wiltshire-Gordon¹,

Woo

Zajączkowska

2017

Fields Institute Communications

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The generators of the classical Specht module satisfy intricate relations. We introduce the Specht matroid, which keeps track of these relations, and the Specht polytope, which also keeps track of convexity relations. We establish basic facts about the Specht polytope, for example, that the symmetric group acts transitively on its vertices and irreducibly on its ambient real vector space. A similar construction builds a matroid and polytope for a tensor product of Specht modules, giving "Kronecker matroids" and "Kronecker polytopes" instead of the usual Kronecker coefficients. We dub this process of upgrading numbers to matroids and polytopes "matroidification," giving two more examples. In the course of describing these objects, we also give an elementary account of the construction of Specht modules different from the standard one. Finally, we provide code to compute with Specht matroids and their Chow rings.The irreducible representations of the symmetric group S n were worked out by Young and Specht in the early 20th century, and they remain omnipresent in algebraic combinatorics. The symmetric group S n has a unique irreducible representation for each partition of n. For example, S 4 has exactly five irreducible representations corresponding to the partitions (4) (

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