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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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The Klein Quadric

Combinatorics of Spreads and Parallelisms ◽

10.1201/ebk1439819463-54 ◽

2010 ◽

pp. 632-635

Keyword(s):

Klein Quadric

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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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