Equivariant ChernSchwartzMacPherson classes in partial flag varieties: interpolation and formulae
Abstract.
Consider the natural torus action on a partial flag manifold . Let be an open Schubert variety, and let be its torus equivariant ChernSchwartzMacPherson class. We show a set of interpolation properties that uniquely determine , as well as a formula, of ‘localization type’, for . In fact, we proved similar results for a class — in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that .
1. Introduction
An interesting chapter of enumerative geometry is the theory of characteristic classes of singular varieties. One of the fundamental results about ChernSchwartzMacPherson (CSM) characteristic classes is their calculation for degeneracy loci in [PP] by Pragacz and Parusiński. In this short note—dedicated to the 60th birthday of P. Pragacz—we prove a result about CSM classes of Schubert cells in partial flag varieties.
Namely, we present a set of interpolation properties that uniquely determine the sought CSM class. Such interpolation characterisation was known before for the leading term of the CSM class, the fundamental class [FR]. A solution of these interpolation conditions is a weight function in the terminology of our earlier works. Weight functions were considered by Tarasov and Varchenko [TV1, TV2] in the context of qhypergeometric solutions of qKZ differential equations, and turn up in our recent works (with Tarasov, Gorbounov) in the context of geometric interpretations of Bethe algebras.
In fact, our results on the interpolation conditions and their solution are present in [RTV1, RTV2, RTV3] for certain cohomology classes —see also one of the main motivations [MO, Section 3.3.4]. In the present note we only prove that is equal to the CSM class of a Schubert cell . By giving a proof of this fact and by presenting an accessible description of the interpolation conditions and the weight functions (with appropriate convention changes) we hope to bring the attention of researchers in Schubert calculus and characteristic classes to the quantum group aspects of CSM classes.
In Section 8 we extend this result to the case of ChernSchwartzMacPherson classes of Schubert cells in where is a parabolic subgroup of a semisimple group .
The authors thank P. Aluffi, L. M. Fehér, and T. Ohmoto for useful discussions on the topic.
2. Weight functions
2.1. Definition of weight functions
Let us fix nonnegative integers and , as well as with . Consider tuples , where , , and for . The set of such tuples will be denoted by . For we will use the notations
and the variables
(1) 
For , , , define
with the convention that .
Let the group act on the set of variables in such a way that elements of permute the lower indexes of ’s. Let
Denote .
Definition 2.1.
Define the weight function
and the modified weight function
The weight functions are polynomials (despite the appearance of factors in the denominators), while the modified weight functions are rational functions.
Example 2.2.
For , we have
3. Combinatorial description of weight function
In this section we show a diagrammatic interpretation of the terms of the weight function. Let . Consider a table with rows and columns. Number the rows from top to bottom and number the columns from left to right. Certain boxes of this table will be distinguished, as follows. In the ’th column distinguish boxes in the ’th row if . This way all the boxes in the last column will be distinguished since .
Now we will define fillings of the tables by putting various variables in the distinguished boxes. First, put the variables into the last column from top to bottom. Now choose permutations . Put the variables in the distinguished boxes of the ’th column from top to bottom.
Each such filled table will define a rational function as follows. Let be a variable in the filled table in one of the columns . If is a variable in the next column, but above the position of then consider the factor (‘type1 factor’). If is a variable in the next column, but below the position of then consider the factor (‘type2 factor’). If is a variable in the same column, but below the position of then consider the factor (‘type3 factor’). The rule is illustrated in the following figure.
For each variable in the table consider all these factors and multiply them together. This is “the term associated with the filled table”.
One sees that is the sum of terms associated with the filled tables corresponding to all choices . For example, is the sum of two terms associated with the filled tables
The term corresponding to the first filled table is
and the term corresponding to the second filled table is
4. Partial flag manifold, equivariant cohomology, Schubert varieties
Let be the standard basis in . Consider the partial flag manifold parameterizing chains
where . The standard action of the torus on induces an action of on . The fixed points of this action are the points with .
Consider variables for , and let . The group acts on by permuting the variables with the same first index. The complex coefficient, equivariant cohomology ring of is presented as
(2) 
Here for are the Chern roots of the bundle whose fiber is , and are the Chern roots of the torus.
Let . For define the Schubert cell
The Schubert cell is an affine space of dimension
The point is a smooth point of the Schubert cell . The weights of the torus action on the tangent space are
The weights of the torus action on a invariant normal space to in are
Hence we have
for the tangent total Chern class and the normal Euler class of at .
5. Unique classes defined by interpolation
The restriction of a cohomology class to the fixed point will be denoted . In terms of variables this means the substitution
(3) 
For an symmetric function in variables as in (1) let denote the substitution
Theorem 5.1.
There are unique classes satisfying

is divisible by ;

;

has degree less than , if .
Moreover,
(4) 
Since is a rational function we need to explain what we mean by (4). This means that the substitution (3) of is (a polynomial, and is) equal to for all . It is remarkable that although can be represented by a polynomial in , variables (by definition, see (2)) but we represented it by a rational function.
6. ChernSchwartzMacPherson classes
Let the torus act on the algebraic manifold . For a equivariant constructible function on one can consider its ChernSchwartzMacPherson (CSM) class . For an invariant subset we write for where is the indicator function of .
The nonequivariant version of this notion was introduced by MacPherson [M], see also [Sch], and the equivariant version by Ohmoto [O1], see also [W, O2]. We refer the reader to these papers for definitions and main properties, namely natural normalization, functoriality, product, and localization properties. CSM classes of Schubert cells and varieties, their recursion and positivity properties, are studied in [AM1, H, AM2].
Remark 6.1.
It is customary to consider the CSM class of a variety in its (equivariant) homology —this version does not depend on an ambient manifold . In the present paper we will not consider this homology CSM class. We follow [O1] and [W] by mapping the homology CSM class first to by homology pushforward, then applying (equivariant) Poincaré duality for , and considering the resulting cohomology class living in .
The class is a refinement of the notion of (equivariant) fundamental class; namely higher degree terms, where is the (equivariant) fundamental class of the variety in .
Now we recall the properties of equivariant CSM classes we will need in the next section.

(Linearity.) For equivariant constructible functions and we have .

(Local model of CSM classes of varieties.) Suppose acts on a vector space , and let be an invariant smooth subvariety. Then
(5) where ’s are the tangent weights, and ’s are the normal weights of . Indeed, when is considered in the (co)homology of then it is the total Chern class of the tangent bundle of [O2, Thm. 3.10]. In our convention (i.e. when is pushed forward to the cohomology of the ambient space) we obtain (5).

Suppose acts on a vector space , and let be an invariant subvariety which contains an invariant smooth subvariety . Moreover assume that an invariant linear subspace is transversal to and . Then is divisible by . Indeed, in our convention Theorem 3.13 (2) of [O2] reads
which implies

Let be a subvariety of the manifold with isolated fixed points. Let be a fixed point of . Then [W, Thm 20]
A remarkable feature of this fact which we will use below is that the top degree part of does not depend on the variety as long as belongs to the variety.
7. CSM classes coincide with classes
Consider the torus action on and their Schubert cells as in Section 4. Also recall the classes defined axiomatically in Section 5.
Theorem 7.1.
We have
Proof.
We need to prove that the class satisfies the axioms defining .
Let us write the indicator function of the Schubert cell as a linear combination of the indicator functions of Schubert varieties:
(6) 
Here means and are integer coefficients, . By linearity of CSM classes
and hence
The behaviour of the varieties appearing on the RHS near are known in Schubert calculus: they all contain , is smooth at and the subspace required in (C) above exists. Therefore each term is divisible by . This proves axiom (I).
If then the right hand side is just the term which—according to the “local model” property above is . This proves axiom (II).
Now let . Using Theorem [W, Thm 20] as recalled above in (D) we obtain
where “l.d.t.”(lower degree terms) means terms of degree strictly less than . However, the sum vanishes due to substituting into the functional identity (6). We obtain that has degree strictly less than . This proves axiom (III). ∎
Corollary 7.2.
The equivariant CSM classes of Schubert cells of partial flag manifolds are defined by the axioms of Theorem 5.1. Moreover .
Remark 7.3.
The CSM classes satisfy obvious vanishing properties respecting the Bruhat order. Namely, if then . It is remarkable that one does not need to list this obvious vanishing property among the axioms, this property is a consequence of the axioms.
8. CSM classes of Schubert cells in generalized flag manifolds
Let be a semisimple algebraic group and let be a maximal torus, Borel subgroup, and a parabolic subgroup in . The orbits are called Schubert cells in . They are parameterized by certain elements of the Weyl group. Each orbit contains a fixed point . The first part of Corollary 7.2 generalizes to the general flag variety as follows.
Theorem 8.1.
The equivariant ChernSchwartzMacPherson classes are the unique classes in satisfying the properties

is divisible by ;

;

has degree less than , if .
Proof.
The existence and uniqueness of classes satisfying the itemized properties is proved in [MO]. (A guide to the reader: Maulik and Okounkov prove that certain classes are characterized by three axioms, see [MO, Section 3.3.4]. Their three axioms for are (II), (III) above, together with a global version of (I). The global versus local versions of (I) does not affect the arguments. Also, the MaulikOkounkov classes are homogeneous, depending on an extra parameter . Our nonhomogeneous versions are obtained by putting .)
The fact that the classes satisfy the properties follows the same way as in the special case detailed in Section 7. ∎
Remark 8.2.
The map sending fixed points to the classes is essentially the stable envelope map of MaulikOkounkov corresponding to the pair .
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