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14
Gröbner geometry of Schubert polynomials
 Ann. Math
"... Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix S ..."
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Cited by 66 (13 self)
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Schubert polynomials, which a priori represent cohomology classes of Schubert varieties in the flag manifold, also represent torusequivariant cohomology classes of certain determinantal loci in the vector space of n ×n complex matrices. Our central result is that the minors defining these “matrix Schubert varieties” are Gröbner bases for any antidiagonal term order. The Schubert polynomials are therefore positive sums of monomials, each monomial representing the torusequivariant cohomology class of a component (a schemetheoretically reduced coordinate subspace) in the limit of the resulting Gröbner degeneration. Interpreting the Hilbert series of the flat limit in equivariant Ktheory, another corollary of the proof is that Grothendieck polynomials represent the classes of Schubert varieties in Ktheory of the flag manifold. An inductive procedure for listing the limit coordinate subspaces is provided by the proof of the Gröbner basis property, bypassing what has come to be known as Kohnert’s conjecture [Mac91]. The coordinate subspaces, which are
Four positive formulas for type A quiver polynomials, preprint
, 2003
"... Abstract. We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, ..."
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Cited by 20 (3 self)
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Abstract. We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rcgraphs). Three of our formulae are multiplicityfree and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torusinvariant scheme. The remaining (presently nongeometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled ” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82, Jos84, BB85, Ros89] via equivariant Chow groups [EG98]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds
Calulation of Thom polynomials and other cohomological obstructions for group actions
"... In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian in [Kaz97a]. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative alg ..."
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Cited by 16 (7 self)
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In this paper we propose a systematic study of Thom polynomials for group actions defined by M. Kazarian in [Kaz97a]. On one hand we show that Thom polynomials are first obstructions for the existence of a section and are connected to several problems of topology, global geometry and enumerative algebraic geometry. On the other hand
Thom polynomials for Lagrange, Legendre, and critical point function singularities
 Proc. London Math. Soc
, 2003
"... The classi®cation of isolated hypersurface singularities is known to be highly irregular and there is no hope of getting the complete classi®cation. Nevertheless one can make the following observation. Consider the hierarchy of singularities of ..."
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Cited by 7 (0 self)
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The classi®cation of isolated hypersurface singularities is known to be highly irregular and there is no hope of getting the complete classi®cation. Nevertheless one can make the following observation. Consider the hierarchy of singularities of
On the structure of Thom polynomials of singularities
 Bull. London Math. Soc
"... Abstract. Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a pr ..."
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Cited by 6 (2 self)
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Abstract. Thom polynomials of singularities express the cohomology classes dual to singularity submanifolds. A stabilization property of Thom polynomials is known classically, namely that trivial unfolding does not change the Thom polynomial. In this paper we show that this is a special case of a product rule. The product rule enables us to calculate the Thom polynomials of singularities if we know the Thom polynomial of the product singularity. As a special case of the product rule we define a formal power series (Thom series, TsQ) associated with a commutative, complex, finite dimensional local algebra Q, such that the Thom polynomial of every singularity with local algebra Q can be recovered from TsQ. 1.
Thom polynomials and Schur functions: towards the singularities Ai(−)
, 2008
"... We develop algebrocombinatorial tools for computing the Thom polynomials for the Morin singularities Ai(−) (i ≥ 0). The main tool is the function F (i) r defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the T ..."
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Cited by 4 (3 self)
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We develop algebrocombinatorial tools for computing the Thom polynomials for the Morin singularities Ai(−) (i ≥ 0). The main tool is the function F (i) r defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the Thom polynomial T Ai for the singularity Ai (any i) associated with maps (C •,0) → (C •+k,0), with any parameter k ≥ 0, under the assumption that Σ j = ∅ for all j ≥ 2, is given by F (i) k+1. Equivalently, this says that “the 1part ” of T Ai (i) equals F k+1. We investigate 2 examples when T Ai apart from its 1part consists also of the 2part being a single Schur function with some multiplicity. Our computations combine the characterization of Thom polynomials via the “method of restriction equations” of Rimányi et al. with the techniques of Schur functions.
Equivariant Chern classes of singular algebraic varieties with group actions
"... Abstract. We define the equivariant ChernSchwartzMacPherson class of a possibly singular algebraic Gvariety over the base field C, or more generally over a field of characteristic 0. In fact, we construct a natural transforfrom the Gequivariant constructible function functor FG to the mation CG ..."
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Cited by 4 (1 self)
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Abstract. We define the equivariant ChernSchwartzMacPherson class of a possibly singular algebraic Gvariety over the base field C, or more generally over a field of characteristic 0. In fact, we construct a natural transforfrom the Gequivariant constructible function functor FG to the mation CG ∗ Gequivariant homology functor HG ∗ or AG ∗ (in the sense of TotaroEdidinGraham). This CG ∗ may be regarded as MacPherson’s transformation for (certain) quotient stacks. We discuss on other type Chern classes and applications. The VerdierRiemannRoch formula takes a key role throughout. 1.
THE DEGREE OF THE DISCRIMINANT OF IRREDUCIBLE REPRESENTATIONS
, 2005
"... Abstract. We present a formula for the degree of the discriminant of irreducible representations of a Lie group, in terms of the roots of the group and the maximal weight of the representation. The proof uses equivariant cohomology techniques, namely, the theory of Thom polynomials, and a new method ..."
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Cited by 1 (1 self)
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Abstract. We present a formula for the degree of the discriminant of irreducible representations of a Lie group, in terms of the roots of the group and the maximal weight of the representation. The proof uses equivariant cohomology techniques, namely, the theory of Thom polynomials, and a new method for their computation. We study the combinatorics of our formulas in various special cases. 1.
Enumeration of unisingular algebraic hypersurfaces
, 2007
"... We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure ..."
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We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on an intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in detail the generalized Newtonnondegenerate singularities. We also give examples of enumeration in some other cases.