scholarly journals Poincaré polynomial of a class of signed complete graphic arrangements

2019 ◽  
Author(s):  
Guangfeng Jiang ◽  
Jianming Yu ◽  
Jianghua Zhang
Keyword(s):  
Poincaré Polynomial

scholarly journals Polynomial identities related to special Schubert varieties

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.

scholarly journals Poincaré polynomials of generic torus orbit closures in Schubert varieties

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.

On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of SLn

2006 ◽  
Vol 58 (5) ◽  
pp. 1000-1025 ◽  
Author(s):  
Ajneet Dhillon
Keyword(s):  
Vector Bundles ◽  
Hodge Structure ◽  
Betti Numbers ◽  
Mixed Hodge Structure ◽  
Projective Curve ◽  
Ground Field ◽  
Smooth Projective Curve ◽  
Poincaré Polynomial ◽  
Moduli Stack ◽  
Moduli Of Vector Bundles

AbstractWe compute some Hodge and Betti numbers of the moduli space of stable rank r, degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincaré polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one.

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

The Betti Numbers of the Simple Lie Groups

1958 ◽  
Vol 10 ◽  
pp. 349-356 ◽  
Author(s):  
A. J. Coleman
Keyword(s):  
Lie Groups ◽  
Orthogonal Group ◽  
Betti Numbers ◽  
Algebraic Methods ◽  
Classical Groups ◽  
Compact Lie Groups ◽  
Poincaré Polynomial ◽  
Unimodular Group ◽  
Simple Lie Groups

The purpose of the present paper1 is to simplify the calculation of the Betti numbers of the simple compact Lie groups. For the unimodular group and the orthogonal group on a space of odd dimension the form of the Poincaré polynomial was correctly guessed by E. Cartan in 1929 (5, p. 183). The proof of his conjecture and its extension to the four classes of classical groups was given by L. Pontrjagin (13) using topological arguments and then by R. Brauer (2) using algebraic methods.

scholarly journals Intersection Poincaré Polynomial for Nagaraj-Seshadri moduli space

2017 ◽  
Vol 291 (1) ◽  
pp. 7-23 ◽  
Author(s):  
Pabitra Barik ◽  
Arijit Dey ◽  
B. N. Suhas
Keyword(s):  
Moduli Space ◽  
Poincaré Polynomial
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