scholarly journals Giambelli and Degeneracy Locus Formulas for Classical G/P spaces

2016 ◽  
Vol 16 (1) ◽  
pp. 125-177 ◽  
Author(s):  
Harry Tamvakis
Keyword(s):  
Degeneracy Locus

scholarly journals Diagrams and Essential Sets for Signed Permutations

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.

IRREDUCIBLE DECOMPOSITIONS OF DEGENERACY LOCI OF MATRICES

2008 ◽  
Vol 18 (02) ◽  
pp. 257-270
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Degeneracy Loci ◽  
Irreducible Components ◽  
Degeneracy Locus ◽  
Rank Conditions ◽  
Rank Functions

In this paper, we study the irreducible decompositions of the degeneracy loci of matrices which are defined by rank conditions on upper left submatrices. By introducing the concepts of standard and essential rank functions, we give an explicit classification of these degeneracy loci. Based on standard rank functions, we design an algorithm to write a degeneracy locus as a union of its irreducible components. This gives an answer to a problem raised by Sturmfels.

scholarly journals A BOGOMOLOV UNOBSTRUCTEDNESS THEOREM FOR LOG-SYMPLECTIC MANIFOLDS IN GENERAL POSITION

2018 ◽  
Vol 19 (5) ◽  
pp. 1509-1519
Author(s):  
Ziv Ran
Keyword(s):  
Poisson Structure ◽  
General Position ◽  
Tangent Space ◽  
Hodge Structure ◽  
Symplectic Manifolds ◽  
Mixed Hodge Structure ◽  
Deformation Space ◽  
Normal Crossing ◽  
Degeneracy Locus ◽  
Kählerian Manifolds

We consider compact Kählerian manifolds $X$ of even dimension 4 or more, endowed with a log-symplectic holomorphic Poisson structure $\unicode[STIX]{x1D6F1}$ which is sufficiently general, in a precise linear sense, with respect to its (normal-crossing) degeneracy divisor $D(\unicode[STIX]{x1D6F1})$. We prove that $(X,\unicode[STIX]{x1D6F1})$ has unobstructed deformations, that the tangent space to its deformation space can be identified in terms of the mixed Hodge structure on $H^{2}$ of the open symplectic manifold $X\setminus D(\unicode[STIX]{x1D6F1})$, and in fact coincides with this $H^{2}$ provided the Hodge number $h_{X}^{2,0}=0$, and finally that the degeneracy locus $D(\unicode[STIX]{x1D6F1})$ deforms locally trivially under deformations of $(X,\unicode[STIX]{x1D6F1})$.

scholarly journals Blow-ups and infinitesimal automorphisms of CR-manifolds

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1701-1724
Author(s):  
Boris Kruglikov
Keyword(s):  
Lie Group ◽  
Blow Up ◽  
Symmetry Algebra ◽  
Levi Form ◽  
Cr Manifolds ◽  
Parabolic Subalgebras ◽  
Degeneracy Locus ◽  
Real Analytic ◽  
Infinitesimal Automorphisms

Abstract For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$ n ⩾ 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$ s = s ( M ) : (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$ dim s = n 2 + 4 n + 3 and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$ dim s ⩽ n 2 + 2 n + 2 + δ 2 , n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

scholarly journals Schubert complexes and degeneracy loci

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Steven V Sam
Keyword(s):  
Euler Characteristic ◽  
Vector Bundles ◽  
Homology Class ◽  
Schur Polynomials ◽  
Degeneracy Loci ◽  
Schubert Polynomials ◽  
Conceptual Proof ◽  
Degeneracy Locus ◽  
Alternating Sums ◽  
International Audience

International audience The classical Thom―Porteous formula expresses the homology class of the degeneracy locus of a generic map between two vector bundles as an alternating sum of Schur polynomials. A proof of this formula was given by Pragacz by expressing this alternating sum as the Euler characteristic of a Schur complex, which gives an explanation for the signs. Fulton later generalized this formula to the situation of flags of vector bundles by using alternating sums of Schubert polynomials. Building on the Schubert functors of Kraśkiewicz and Pragacz, we introduce Schubert complexes and show that Fulton's alternating sum can be realized as the Euler characteristic of this complex, thereby providing a conceptual proof for why an alternating sum appears. \par La formule classique de Thom―Porteous exprime la classe d'homologie du locus de la dégénérescence d'une fonction générique entre deux fibrés vectoriels comme une somme alternée des polynômes de Schur. Un preuve de cette formule a été donnée par Pragacz en exprimant ce alternant somme comme la caractéristique d'Euler d'un complexe de Schur, ce qui donne une explication pour les signes. Fulton puis généralisée cette formule à la situation des drapeaux de fibrés vectoriels à l'aide alternant des sommes de polynômes de Schubert. S'appuyant sur le Schubert foncteurs de Kraśkiewicz et Pragacz, nous introduisons les complexes de Schubert et montrent que la somme alternée de Fulton peuvent être réalisées en tant que Euler caractéristique de ce complexe, fournissant ainsi une preuve conceptuelle pour lesquelles une somme alternée appara\^ıt.

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