The singular locus of a Schubert variety

Abstract We derive explicit formulae to compute the a and c central charges of four dimensional $$ \mathcal{N} $$ N = 2 superconformal field theories (SCFTs) directly from Coulomb branch related quantities. The formulae apply at arbitrary rank. We also discover general properties of the low-energy limit behavior of the flavor symmetry of $$ \mathcal{N} $$ N = 2 SCFTs which culminate with our $$ \mathcal{N} $$ N = 2 UV-IR simple flavor condition. This is done by determining precisely the relation between the integrand of the partition function of the topologically twisted version of the 4d $$ \mathcal{N} $$ N = 2 SCFTs and the singular locus of their Coulomb branches. The techniques developed here are extensively applied to many rank-2 SCFTs, including new ones, in a companion paper.This manuscript is dedicated to the memory of Rayshard Brooks, George Floyd, Breonna Taylor and the countless black lives taken by US police forces and still awaiting justice. Our hearts are with our colleagues of color who suffer daily the consequences of this racist world.

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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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Integrity and normality. The singular locus

Lecture Notes in Mathematics - Determinantal Rings ◽

10.1007/bfb0080384 ◽

1988 ◽

pp. 64-72

Author(s):

Winfried Bruns ◽

Udo Vetter

Keyword(s):

Singular Locus

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Nil-Hecke Ring and the Singular Locus of X(w)

Singular Loci of Schubert Varieties ◽

10.1007/978-1-4612-1324-6_7 ◽

2000 ◽

pp. 91-102

Author(s):

Sara Billey ◽

V. Lakshmibai

Keyword(s):

Singular Locus

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Poincaré polynomials of generic torus orbit closures in Schubert varieties

Topology, Geometry, and Dynamics - Contemporary Mathematics ◽

10.1090/conm/772/15490 ◽

2021 ◽

pp. 189-208

Author(s):

Eunjeong Lee ◽

Mikiya Masuda ◽

Seonjeong Park ◽

Jongbaek Song

Keyword(s):

Schubert Variety ◽

Flag Variety ◽

Schubert Varieties ◽

Orbit Closure ◽

Poincaré Polynomial ◽

Content Type ◽

Eulerian Polynomial ◽

Orbit Closures

The closure of a generic torus orbit in the flag variety G / B G/B of type A A is known to be a permutohedral variety, and its Poincaré polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincaré polynomial of a generic torus orbit closure in a Schubert variety in G / B G/B . When the generic torus orbit closure in a Schubert variety is smooth, its Poincaré polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.

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Diagrams and Essential Sets for Signed Permutations

The Electronic Journal of Combinatorics ◽

10.37236/8106 ◽

2018 ◽

Vol 25 (3) ◽

Author(s):

David Anderson

Keyword(s):

Schubert Variety ◽

Signed Permutation ◽

Signed Permutations ◽

Degeneracy Locus ◽

Essential Set ◽

Diagrammatic Method ◽

Essential Sets ◽

Rank Conditions ◽

Theoretic Version

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations. In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation. Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

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