Torus actions and tensor products of intersection cohomology

AbstractEvery tropical ideal in the sense of Maclagan–Rincón has an associated tropical variety, a finite polyhedral complex equipped with positive integral weights on its maximal cells. This leads to the realisability question, ubiquitous in tropical geometry, of which weighted polyhedral complexes arise in this manner. Using work of Las Vergnas on the non-existence of tensor products of matroids, we prove that there is no tropical ideal whose variety is the Bergman fan of the direct sum of the Vámos matroid and the uniform matroid of rank two on three elements and in which all maximal cones have weight one.

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Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

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Normal singularities with torus actions

Tohoku Mathematical Journal ◽

10.2748/tmj/1365452628 ◽

2013 ◽

Vol 65 (1) ◽

pp. 105-130 ◽

Cited By ~ 12

Author(s):

Alvaro Liendo ◽

Hendrik Süss

Keyword(s):

Torus Actions

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The dual of the space of bounded operators on a Banach space

Concrete Operators ◽

10.1515/conop-2020-0109 ◽

2021 ◽

Vol 8 (1) ◽

pp. 48-59

Author(s):

Fernanda Botelho ◽

Richard J. Fleming

Keyword(s):

Banach Space ◽

Tensor Product ◽

Banach Spaces ◽

Dual Space ◽

Tensor Products ◽

Bounded Operators ◽

Projective Tensor Product ◽

Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

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