Journal of Algebraic Combinatorics

, Volume 42, Issue 3, pp 875–892 | Cite as

Schur polynomials and weighted Grassmannians

Article

Abstract

In this paper, we introduce a family of symmetric polynomials by specializing the factorial Schur polynomials. We show that it represents the weighted Schubert classes of the cohomology of the weighted Grassmannian introduced by Corti–Reid, and we regard it as an analogue of the Schur polynomials. Furthermore, we prove that these polynomials are the characters of certain representations, and hence, we give an interpretation of the Schubert structure constants of the weighted Grassmannians as the (rational) multiplicities of the tensor products of the representations. We also derive two determinantal formulas for the weighted Schubert classes: One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbi-bundles.

Keywords

Schur polynomial Weighted Grassmannian Orbifold  Schubert calculus Representation 

Mathematics Subject Classification

05E05 05E15 57R18 

1 Introduction

Let \({\mathcal P}({d})\) be the set of partitions with at most \({d}\) rows. For every \(\lambda \in {\mathcal P}({d})\), the Schur polynomial \(s_{\lambda }(x)\) is defined as a symmetric polynomial in the variables \((x_1,\ldots , x_{{d}})\). They form a \({\mathbb Z}\)-module basis of the algebra \({\mathbb Z}[x]^{{\mathfrak S}_{{d}}}\) of symmetric polynomials in x-variables with the coefficients in \({\mathbb Z}\). On the other hand, the Grassmannian \({\text {Gr}}({d},{n})\) of complex \({d}\)-planes in \({\mathbb C}^{{n}}\) has the distinguished subvarieties, called Schubert varieties, indexed by the set \({\mathcal P}({d},{n})\) of all partitions contained in the \({d}\times ({n}-{d})\) rectangle. Their associated cohomology classes \(S_{\lambda }, \lambda \in {\mathcal P}({d},{n})\) form a \({\mathbb Z}\)-module basis of the cohomology \(H^*({\text {Gr}}({d},{n});{\mathbb Z})\). The Schur polynomials \(s_{\lambda }(x)\) represent the Schubert classes \(S_{\lambda }\) of the Grassmannian in a sense that there is a surjective ring homomorphism
$$\begin{aligned} {\mathbb Z}[x]^{{\mathfrak S}_{{d}}} \rightarrow H^*({\text {Gr}}({d},{n});{\mathbb Z}) \end{aligned}$$
(1)
which sends \(s_{\lambda }(x)\) to \(S_{\lambda }\) if \(\lambda \in {\mathcal P}({d},{n})\), or 0 otherwise. The above map gives a representation theoretic interpretation to the structure constants with respect to Schubert classes, since we can regard \({\mathbb Z}[x]^{{\mathfrak S}_{{d}}}\) as the representation ring of the general linear group \(\text {GL}_{{d}}({\mathbb C})\) and the Schur polynomials \(s_{\lambda }(x)\) correspond to the irreducible representations.

The correspondence (1) has been generalized in several situations. For example, the equivariant Schubert classes for the Grassmannians are represented by the factorial Schur polynomials, cf. [9, 13, 14, 15]. This equivariant generalization of (1) will be the main tool in this paper. Other such examples include the (double/quantum) Schubert polynomials ([3, 11]) for the (equivariant/quantum) cohomology of the full flag varieties, and also the (factorial) Schur Q-polynomials ([4, 5, 6]) for the (equivariant) cohomology of the Lagrangian Grassmannians. One of the advantages of these correspondences is that we can study the structure constants by multiplying actual polynomials.

In this paper, we will introduce and study a twisting of the (factorial) Schur polynomials to generalize the above pictures to the (equivariant) cohomology of the weighted Grassmannians introduced by Corti–Reid [2]. Below we summarize only the non-equivariant results of this paper to avoid complexity, although we build the correspondence for the equivariant cohomology of the weighted Grassmannians first and then derive the non-equivariant one.

Let \(w_1,w_2,\ldots \) be an infinite sequence of nonnegative integers and u a positive integer. Let \({\text {wGr}}({d},{n})\) be the weighted Grassmannian introduced in [2]. Its rational cohomology has a distinguished \({\mathbb Q}\)-basis consisting of the weighted Schubert classes \({w\mathrm {S}}_{\lambda }\), \(\lambda \in {\mathcal P}({d},{n})\), and the corresponding structure constants with respect to this basis are computed in [1].

For each \(\lambda \in {\mathcal P}({d})\), let \(s_{\lambda }(x|a)\) be the factorial Schur polynomial defined by the formula (3) below. We consider the polynomial \(s^w_{\lambda }(x)\) obtained by specializing \(s_{\lambda }(x|a)\) at \(a_i=-(w_i/u)(x_1 + \cdots + x_{{d}})\) for all \(i=1,2,\ldots \). If \(w_1=w_2=\cdots =0\), this is nothing but the usual Schur polynomial \(s_{\lambda }(x)\). In Proposition 3, we show that these polynomials \(s^w_{\lambda }(x), \lambda \in {\mathcal P}({d})\) form a \({\mathbb Q}\)-basis of \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}}\) and represent the weighted Schubert classes.

Theorem A

(Theorem 2 below). The map \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}} \rightarrow H^*({\text {wGr}}({d},{n});{\mathbb Q})\) defined by sending \(s^w_{\lambda }(x)\) to \({w\mathrm {S}}_{\lambda }\) if \(\lambda \in {\mathcal P}({d},{n})\) and to 0 otherwise, is a surjective ring homomorphism.

To prove Theorem A, we first obtain its equivariant analogue (Proposition 1).

Furthermore, we prove the following from the definition of \(s^w_{\lambda }(x)\).

Theorem B

(Theorem 3 below). Suppose that \(u=1\). For any partition \(\lambda \in {\mathcal P}({d})\), the polynomial \(s^w_{\lambda }(x)\) is integral and Schur-positive, i.e., it is a linear combination of the Schur polynomials such that the coefficients are nonnegative integers. In particular, there exists a representation \(V^{\lambda }_w\) of \({{\text {GL}}}_{{d}}({\mathbb C})\) such that \(s^w_{\lambda }(x)\) is the character \(\mathrm{ch}(V^{\lambda }_w)\) of \(V^{\lambda }_w\).

We would like to stress that one obtains the nonnegative coefficients of \(s^w_{\lambda }(x)\) explicitly in (14) and (15).

This theorem allows us to interpret the weighted Schubert constants in terms of representations. Suppose that \(u=1\), and the weights are non-increasing, i.e.,\(w_1\ge w_2\ge \cdots \). Then, for each \(\lambda ,\mu \), and \(\nu \in {\mathcal P}({d})\), there exist nonnegative integers \(\ell _{\lambda \mu }\in {\mathbb Z}_{\ge 1}\) and \(\ell _{\lambda \mu }^{\nu }\in {\mathbb Z}_{\ge 0}\) such that
$$\begin{aligned} (V^{\lambda }_w \otimes V^{\mu }_w)^{\oplus \ell _{\lambda \mu }} = \bigoplus _{\nu \in {\mathcal P}({d})} (V^{\nu }_w)^{\oplus \ell _{\lambda \mu }^{\nu }} \quad \text { as representations of } \hbox {GL}_{{d}}({\mathbb C}). \end{aligned}$$
By Theorem A and Theorem B, we see that
$$\begin{aligned} {w\mathrm {S}}_{\lambda }\cdot {w\mathrm {S}}_{\mu } = \sum _{\nu \in {\mathcal P}({d},{n})} \Big (\frac{\ell _{\lambda \mu }^{\nu }}{\ell _{\lambda \mu }}\Big ) {w\mathrm {S}}_{\nu }. \end{aligned}$$
Therefore, we can think of the structure constants \(\frac{\ell _{\lambda \mu }^{\nu }}{\ell _{\lambda \mu }}\) as rational multiplicities of \(V^{\nu }_w\) in the tensor product \(V^{\lambda }_w \otimes V^{\mu }_w\).

The equivariant analogue of Theorem A (Proposition 1 below) provides algebraic proofs of the two determinantal formulas for the weighted Schubert classes \({w\mathrm {S}}_{\lambda }\): One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbifold vector bundles. Let \((k)\in {\mathcal P}({d})\) be the one row partition with k boxes. Let \(\mathcal {S}_w\hookrightarrow {\mathcal E}_w \twoheadrightarrow \mathcal {Q}_w\) be the sequence of the tautological orbifold vector bundles over the weighted Grassmannian defined in Sect. 7. We show

Theorem C

(Theorem 4 and 5 below) For each \(\lambda \in {\mathcal P}({d},{n})\),
$$\begin{aligned}&{w\mathrm {S}}_{\lambda }\\&\quad = \det \left[ \sum _{k=0}^{\lambda _i-i+j} \big (c_1({\mathcal S}_w)/u\big )^{k}h_{k}(w_{\lambda _i-i+{d}+1}, \ldots , w_{{n}}) c_{\lambda _i-i+j-k}({\mathcal Q}_w) \right] _{1\le i\le j\le {d}}\\&\quad = \det \left[ \sum _{r=0}^{j-1} h_r(w_{\lambda _i-i+1+{d}},\ldots ,w_{\lambda _i-i+j-r+{d}}) \left( \frac{{w\mathrm {S}}_{(1)}}{-w_{\bar{\emptyset }}}\right) ^{r} {w\mathrm {S}}_{(\lambda _i-i+j-r)}\right] _{1\le i,j\le {d}} \end{aligned}$$
where \(w_{\bar{\emptyset }}=w_1+\cdots +w_{{d}}+u\).

These two formulas coincide in the case of ordinary Grassmannians since the special Schubert classes are the Chern classes of the dual of the tautological bundle of Grassmannians. However, this is not the case for the weighted Grassmannians.

It follows from Theorem C that the cohomology of the weighted Grassmannian is generated by the Chern classes of the tautological orbifold bundles. Based on this fact, we give the quotient ring description of the cohomology with generators corresponding to the those Chern classes and their relations.

2 Preliminary

Let \({d}\) and \({n}\) be positive integers with \({d}< {n}\). Let \({\mathsf M}_{{n},{d}}({\mathbb C})^*\) be the space of \({n}\times {d}\) complex matrices of rank \({d}\). The general linear groups \({{\text {GL}}}_{{d}}({\mathbb C})\) and \({{\text {GL}}}_{{n}}({\mathbb C})\) naturally act on \({\mathsf M}_{{n},{d}}({\mathbb C})^*\) by the right and left multiplications, respectively. Let
$$\begin{aligned} {\text {aPl}}^{\times }({d}, {n}) := {\mathsf M}_{{n},{d}}({\mathbb C})^*/{\text{ SL }}_{{d}}({\mathbb C}). \end{aligned}$$
There is the residual action of \({{\text {Det}}}:= {{\text {GL}}}_{{d}}({\mathbb C})/{\text{ SL }}_{{d}}({\mathbb C})\) on \({\text {aPl}}^{\times }({d},{n})\) from the right since \({\text{ SL }}_{{d}}({\mathbb C})\) is a normal subgroup of \({{\text {GL}}}_{{d}}({\mathbb C})\). We identify \({{\text {Det}}}\) with \({\mathbb C}^{\times }\) by the determinant map. Let \({\mathsf T}:= ({\mathbb C}^{\times })^{{n}}\) be the diagonal torus in \({{\text {GL}}}_{{n}}({\mathbb C})\) embedded in the standard way. We consider the induced action of the \(({n}+1)\)-torus \({\mathsf K}:= {\mathsf T}\times {{\text {Det}}}\) on \({\text {aPl}}^{\times }({d},{n})\).

Definition 1

(Corti–Reid [2] ) Let \(w_1,\ldots , w_{{n}}\) be nonnegative integers and u a positive integer. Let \({\mathsf D}_w:={\mathbb C}^{\times }\) and consider the homomorphism
$$\begin{aligned} \rho _w:{\mathsf D}_{w} \rightarrow {\mathsf K}; \quad t \mapsto \left( t^{w_{n}},\ldots , t^{w_{1}}; t^u\right) . \end{aligned}$$
The weighted Grassmannian is the quotient variety \({\text {wGr}}({d},{n}):= {\text {aPl}}^{\times }({d},{n})/{\mathsf D}_w\) with the residual action of \({\mathsf T}_{w}:= {\mathsf K}/{\mathsf D}_w\). It is a projective variety with at worst orbifold singularities.

One obtains the ordinary Grassmannian \({\text {Gr}}({d},{n})\) by setting \(w_1=\cdots = w_{{n}}=0\) and \(u=1\). In this case, we can identify \({\mathsf D}_w\) with \({{\text {Det}}}\) and \({\mathsf T}_{w}\) with \({\mathsf T}\). Note that the weight \((w_1,\ldots ,w_{{n}})\) has the reversed order, compared with the one in [2] and [1].

It was shown in [1] that there exist the following ring isomorphisms among the rational equivariant cohomologieswhere \({\mathsf f}^*\) and \({\mathsf f}_w^*\) are the pullback of the natural maps between Borel constructionsThe isomorphisms \({\mathsf f}^*\) and \({\mathsf f}_w^*\) are actually the isomorphisms of algebras over \(H^*(B{\mathsf T})\) and \(H^*(B{\mathsf T}_{w})\), respectively. All the cohomologies treated in this paper are the singular cohomology with the rational coefficients.

Let \({\mathcal P}({d},{n})\) be the set of all partitions \(\lambda =(\lambda _1\ge \cdots \ge \lambda _{{d}})\) that fit inside of the \({d}\times ({n}-{d})\) rectangle. For each \(\lambda \in {\mathcal P}({d},{n})\), there is a \({\mathsf K}\)-invariant subvariety \(\text {a}\varOmega _{\lambda }\) in \({\text {aPl}}^{\times }({d},{n})\), which coincides with the pullback of the usual Schubert variety \(\varOmega _{\lambda }\) in \({\text {Gr}}({d},{n})\) by the quotient map \({\text {aPl}}^{\times }({d},{n})\rightarrow {\text {Gr}}({d},{n})\). We call the associated equivariant cohomology class \(\widetilde{aS}_{\lambda }:=[\text {a}\varOmega _{\lambda }]_{{\mathsf K}} \in H_{{\mathsf K}}^*({\text {aPl}}^{\times }({d},{n}))\) the \({\mathsf K}\)-equivariant Schubert class. This class coincides with the pullback of the usual \({\mathsf T}\)-equivariant Schubert class \(\widetilde{S}_{\lambda }:=[\varOmega _{\lambda }]_{{\mathsf T}} \in H_{{\mathsf T}}^*({\text {Gr}}({d},{n}))\) along \({\mathsf f}\). Let \(\widetilde{wS}_{\lambda } \in H_{{\mathsf T}_{w}}^*({\text {wGr}}({d},{n}))\) be the image of \(\widetilde{aS}_{\lambda }\) under the inverse of \({\mathsf f}_w^*\). It is shown in [1] that \(\widetilde{wS}_{\lambda }, \lambda \in {\mathcal P}({d},{n})\) form a basis of \(H_{{\mathsf T}_{w}}^*({\text {wGr}}({d},{n}))\) as an \(H^*(B{\mathsf T}_{w})\)-module.

Definition 2

Let \(a_l, l\in {\mathbb N}\) and \(x_i, i= 1,\ldots , {d}\) be indeterminates. Let \({\mathbb Q}[a]\) be the ring of polynomials in a’s and \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}}\) the ring of symmetric polynomials in x’s. Let \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}}:={\mathbb Q}[a]\otimes _{{\mathbb Q}} {\mathbb Q}[x]^{{\mathfrak S}_{{d}}}\). Let \({\mathcal P}({d})\) be the set of partitions with at most \({d}\) rows. For each \(\lambda =(\lambda _1,\ldots ,\lambda _{{d}}) \in {\mathcal P}({d})\), let
$$\begin{aligned} \bar{\lambda }_i:=\lambda _i + ({d}+ 1 -i), \quad \ \ i=1,\ldots , {d}\end{aligned}$$
so that the sequence \((\bar{\lambda }_1,\ldots , \bar{\lambda }_{{d}})\) is strictly decreasing. The factorial Schur polynomial\(s_{\lambda }(x|a)\) is defined as follows [12]:
$$\begin{aligned} s_{\lambda }(x|a) = \frac{\det \left[ \prod _{p=1}^{\bar{\lambda }_i -1}(x_j - a_p) \right] _{1\le i,j\le {d}}}{\prod _{1 \le i<j \le {d}} (x_i - x_j)}. \end{aligned}$$
(3)
It is well known that \(\{s_{\lambda }(x|a), \lambda \in {\mathcal P}({d})\}\) is a basis of \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}}\) as a \({\mathbb Q}[a]\)-module.

Let \(\{b_1,\ldots , b_{{n}}\}\) be the basis of the integral lattice \({\text {Lie}}({\mathsf T})_{{\mathbb Z}}^*\) such that as a character of \({\mathsf T}\), \(b_i\) sends an element \((t_1,\ldots ,t_{{n}}) \in {\mathsf T}\) to \(t_{{n}+ 1 -i}^{-1} \in {\mathbb C}^{\times }\). We identify \(H^*(B{\mathsf T})\) with \({\mathbb Q}[b_1,\ldots , b_{{n}}]\) as usual. Consider the projection \({\mathbb Q}[a] \rightarrow {\mathbb Q}[b_1,\ldots , b_{{n}}]\) by sending \(a_i\) to \(b_i\) if \(i=1,\ldots , {n}\) and 0 otherwise. In this way, we regard \(H_{{\mathsf T}}^*({\text {Gr}}({d},{n}))\) as a \({\mathbb Q}[a]\)-module. The following is well known.

Theorem 1

([9, 13, 14, 15]) There is a surjective homomorphism of algebras over \({\mathbb Q}[a]\)
$$\begin{aligned} \varPhi : {\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \rightarrow H_{{\mathsf T}}^*({\text {Gr}}({d},{n})) \end{aligned}$$
that sends \(s_{\lambda }(x|a)\) to \(\widetilde{S}_{\lambda }\) if \(\lambda \in {\mathcal P}({d},{n})\), and 0 if otherwise.

By composing with the isomorphisms in (2), we obtain

Corollary 1

There are surjective ring homomorphisms
$$\begin{aligned} \varPhi _a: {\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \rightarrow H^*_{{\mathsf K}}({\text {aPl}}^{\times }({d},{n})) \quad \mathrm{and } \quad \varPhi _w: {\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \rightarrow H^*_{{\mathsf T}_{w}}({\text {wGr}}({d},{n})) \end{aligned}$$
where \(\varPhi _a:=f^*\circ \varPhi \) and \(\varPhi _w:=(f_w^*)^{-1}\circ \varPhi _a\). The maps \(\varPhi _a\) and \(\varPhi _w\) send \(s_{\lambda }(x|a)\) to \(\widetilde{aS}_{\lambda }\) and \(\widetilde{wS}_{\lambda }\), respectively, if \(\lambda \in {\mathcal P}({d},{n})\), and 0 if otherwise.

3 Twisting the module structure

In this section, we will regard \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}}\) as an algebra over a certain polynomial ring in order to regard \(\varPhi _w\) as a homomorphism of rings over the polynomial ring.

Let \(\{z\}\) be the standard basis of \({\text {Lie}}({{\text {Det}}})_{{\mathbb Z}}^*\) and \(\{\gamma \}\) is the standard basis of \({\text {Lie}}({\mathsf D}_w)_{{\mathbb Z}}^*\). The induced map \(\rho _w^*: {\text {Lie}}({\mathsf K})^*_{{\mathbb Q}} \rightarrow {\text {Lie}}({\mathsf D}_w)^*_{{\mathbb Q}}\) is given by \(b_i \mapsto -w_i \gamma \) and \(z \mapsto u\gamma \), where \({\text {Lie}}({\mathsf K})^*_{{\mathbb Q}}={\text {Lie}}({\mathsf K})^*_{{\mathbb Z}}\otimes _{{\mathbb Z}}{\mathbb Q}\) and \({\text {Lie}}({\mathsf D}_w)^*_{{\mathbb Q}} = {\text {Lie}}({\mathsf D}_w)^*_{{\mathbb Z}} \otimes _{{\mathbb Z}}{\mathbb Q}\). We identify the kernel of \(\rho _w^*\) with \({\text {Lie}}({\mathsf T}_{w})_{{\mathbb Q}}^*\). Let
$$\begin{aligned} b^w_i:= b_i + (w_i/u) z, \quad i=1,\ldots ,{n}. \end{aligned}$$
(4)
Then \(\{b^w_1,\ldots ,b^w_{{n}}\}\) is a basis of \({\text {Lie}}({\mathsf T}_{w})_{{\mathbb Q}}^*\). Thus, we can identify \(H^*(B{\mathsf T}_{w})\) with \({\mathbb Q}[b^w_1,\ldots , b^w_{{n}}]\).
For each \(\lambda \in {\mathcal P}({d},{n})\), let
$$\begin{aligned} \varepsilon ^{\lambda }_i := {n}+ 1 - \bar{\lambda }_i, \quad \forall i =1,\ldots ,{d}. \end{aligned}$$
Let \(e_1,\ldots , e_n\) be the standard basis of \({\mathbb C}^{{d}}\). The set of all \({\mathsf T}\)-fixed points of \({\text {Gr}}({d},{n})\) is in bijection with \({\mathcal P}({d},{n})\) and for each \(\lambda \in {\mathcal P}({d},{n})\), the corresponding fixed point \(p_{\lambda }\) is given as the image of the matrix \(E_{\lambda }:=[e_{\varepsilon ^{\lambda }_1} \ldots e_{\varepsilon ^{\lambda }_{{d}}}] \in {{\text {Mat}}}_{{n},{d}}({\mathbb C})^*\). The \({\mathsf T}_{w}\)-fixed points in \({\text {wGr}}({d},{n})\) are also given by the images of the matrices \(E_{\lambda }\) in the quotient and denoted by \(p^{w}_{\lambda }\). The \({\text{ SL }}_{{d}}({\mathbb C})\)-orbit of \(E_{\lambda }\) in \({\text {aPl}}^{\times }({d},{n})\) is denoted by \(F_{\lambda }\).

Remark 1

In [1, 9], \(\varepsilon ^{\lambda }_i\) is denoted by \(\lambda _i\) and regarded as the location of 1’s in 01 strings. For example, for the maximum partition \(\lambda =(({n}-{d})^{{d}}) \in {\mathcal P}({d},{n})\), we have \(\bar{\lambda }=({n},{n}-1,\ldots , {n}-{d}+1)\) and \(\varepsilon ^{\lambda }=(1,2,\ldots ,{d})\). For the empty partition \(\emptyset =(0,\ldots , 0)\), we have \(\bar{\emptyset }=({d},{d}-1,\ldots , 1)\) and \(\varepsilon ^{\emptyset }=({n}+1-{d}, \ldots , {n}-1,{n})\).

We extend the diagram (2) with the restrictions to the fixed points. There is a commutative diagram of the pullback mapswhere we denote
$$\begin{aligned} b_{\bar{\lambda }}:= b_{\bar{\lambda }_1} + \cdots + b_{\bar{\lambda }_{{d}}}. \end{aligned}$$
The stabilizer \({\mathsf K}_{\lambda }\) of the points of \(F_{\lambda }\) is the kernel of the homomorphism
$$\begin{aligned} {\mathsf K}\rightarrow {\mathbb C}^{\times }; \ \ \ (t_1,\ldots , t_{{n}}, s) \mapsto s\cdot t_{\varepsilon ^{\lambda }_1} \ldots t_{\varepsilon ^{\lambda }_{{d}}}. \end{aligned}$$
Since \(H_{{\mathsf K}}^*(F_{\lambda }) = H^*(B{\mathsf K}_{\lambda })\), we can identify
$$\begin{aligned} H_{{\mathsf K}}^*(F_{\lambda })={\mathbb Q}[b_1,\ldots ,b_{{n}},z]/(-b_{\bar{\lambda }}+z) \end{aligned}$$
as in (5). The vertical maps on the right are also isomorphisms of rings since the kernel of the composition \({\mathsf K}_{\lambda } \rightarrow {\mathsf K}\rightarrow {\mathsf T}_{w}\) is finite for arbitrary weights w’s and u.

Lemma 1

Under the isomorphism \({\mathsf f}^*: H_{{\mathsf T}}^*({\text {Gr}}({d},{n})) \rightarrow H_{{\mathsf K}}^*({\text {aPl}}^{\times }({d},{n}))\), the preimage of \(z=z\cdot 1\) is given by
$$\begin{aligned} ({\mathsf f}^*)^{-1}(z) = \widetilde{S}_{(1)} + b_{\bar{\emptyset }} \end{aligned}$$
where (1) is the partition \((1,0,\ldots ,0)\).

Proof

If we restrict \(\widetilde{S}_{(1)} \in H_{{\mathsf T}}^*({\text {Gr}}({d},{n}))\) to \(p_{\lambda }\), we have \(\widetilde{S}_{(1)}|_{\lambda } = b_{\bar{\lambda }} -b_{\bar{\emptyset }}\) by Lemma 3 [9]. Therefore, if we restrict \(\widetilde{aS}_{(1)} \in H_{{\mathsf K}}^*({\text {aPl}}^{\times }({d},{n}))\) to \(F_{\lambda }\), we have \(\widetilde{aS}_{(1)}|_{\lambda } = b_{\bar{\lambda }} -b_{\bar{\emptyset }}= -b_{\bar{\emptyset }} + z\) by the commutative diagram (5). Thus, \(({\mathsf f}^*)^{-1}(z) = \widetilde{S}_{(1)} + b_{\bar{\emptyset }}\). \(\square \)

By definition, we have \(s_{(1)}(x|a)=x_{\bar{\emptyset }} - a_{\bar{\emptyset }}\). Therefore, we have \(\varPhi _a(x_{\bar{\emptyset }}) = \widetilde{aS}_{(1)} + b_{\bar{\emptyset }}=z\). The \({\mathbb Q}[a]\)-algebra \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}}\) has also a \({\mathbb Q}[x_{\bar{\emptyset }}, a_1, a_2, \ldots ]\)-algebra structure by the obvious multiplication. This extension of the coefficient ring makes \(\varPhi _a\) a module homomorphism as follows.

Corollary 2

The map
$$\begin{aligned} \varPhi _a: {\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \rightarrow H_{{\mathsf K}}^*({\text {aPl}}^{\times }({d},{n})) \end{aligned}$$
is a homomorphism of algebras over \({\mathbb Q}[x_{\bar{\emptyset }}, a_1, a_2, \ldots ]\) with respect to
$$\begin{aligned} {\mathbb Q}[x_{\bar{\emptyset }}, a_1, a_2, \ldots ] \rightarrow {\mathbb Q}[z, b_1,\ldots , b_{{n}}]; \quad x_{\bar{\emptyset }} \mapsto z, \quad \mathrm{and } \quad a_l \mapsto {\left\{ \begin{array}{ll} b_l &{} 1 \le l \le {n}\\ 0 &{} {n}< l \end{array}\right. }. \end{aligned}$$
Now we would like to find a subring of \({\mathbb Q}[x_{\bar{\emptyset }}, a_1, a_2, \ldots ]\) that corresponds to \({\mathbb Q}[b^w_1,\ldots , b^w_{{n}}]\) under \(\varPhi _w\). Choose an infinite sequence of nonnegative integers \(w_l, l\in {\mathbb N}\) where \(w_{1}, \ldots , w_{n}\) are the ones chosen to define the weighted Grassmannian. Let
$$\begin{aligned} a^w_l:= a_l + (w_l/u) x_{\bar{\emptyset }}, \quad l\in {\mathbb N}. \end{aligned}$$
(6)
Then apparently \(\{a^w_l, l\in {\mathbb N}\}\) is a set of algebraically independent variables so that \({\mathbb Q}[a^w]:= {\mathbb Q}[a^w_1,a^w_2,\ldots ]\) is a polynomial ring. Furthermore, it is easy to see that there is a canonical identification as rings \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \cong {\mathbb Q}[a^w]\otimes _{{\mathbb Q}} {\mathbb Q}[x]^{{\mathfrak S}_d}=:{\mathbb Q}[a^w][x]^{{\mathfrak S}_{{d}}}\) via \(x_i \mapsto x_i\) and \(a_l \mapsto a^w_l - (w_l/u)x_{\bar{\emptyset }}\). From the above corollary, we obtain

Proposition 1

The surjection
$$\begin{aligned} \varPhi _w: {\mathbb Q}[a^w][x]^{{\mathfrak S}_{{d}}} \rightarrow H^*_{{\mathsf T}_{w}}({\text {wGr}}({d},{n})) \end{aligned}$$
is a homomorphism of algebras over \({\mathbb Q}[a^w]\) with respect to
$$\begin{aligned} {\mathbb Q}\left[ a^w\right] \rightarrow {\mathbb Q}\left[ b^w_1,\ldots , b^w_{{n}}\right] ; \quad a^w_l \mapsto {\left\{ \begin{array}{ll} b^w_l &{} 1 \le l \le {n}\\ 0 &{} {n}< l. \end{array}\right. } \end{aligned}$$

Remark 2

The set \(\{s_{\lambda }(x|a), \lambda \in {\mathcal P}({d})\}\) of the factorial Schur polynomials is also a basis of \({\mathbb Q}[a^w][x]^{{\mathfrak S}_{{d}}}\) as a \({\mathbb Q}[a^w]\)-module. Indeed, any non-trivial \({\mathbb Q}[a^w]\)-linear relation among \(s_{\lambda }(x|a)\)’s will give a non-trivial \({\mathbb Q}[b^w]\)-linear relation among \(\widetilde{wS}_{\lambda }\)’s for a sufficiently large \({n}\). Therefore, the claim follows from the \({\mathbb Q}[b^w]\)-linear independency for \(\widetilde{wS}_{\lambda }\)’s. It should also be noted that Proposition 1 and Corollary 1 imply that the corresponding structure constants of \({\mathbb Q}[a^w][x]^{{\mathfrak S}_{{d}}}\) (over \({\mathbb Q}[a^w]\)) give the structure constants of \(H^*_{{\mathsf T}_{w}}({\text {wGr}}({d},{n}))\) with respect to the equivariant Schubert classes \(\widetilde{wS}_{\lambda }\)’s.

Remark 3

For \(M \in M^*_{{n},{d}}({\mathbb C})\), denote the subspace of \({\mathbb C}^{{n}}\) spanned by the columns of M by \({\langle }M{\rangle }\). The tautological vector bundle is defined by \({\mathcal S}= \{ ([M], v) \ |\ [M] \in {\text {Gr}}({d},{n}), v \in {\langle }M {\rangle }\} \subset {\text {Gr}}({d},{n}) \times {\mathbb C}^{{n}}\) and the action of \({\mathsf T}\) is defined by \(t \cdot ([M], v) = ([t\cdot M], t\cdot v)\) where \(t \in {\mathsf T}\) acts as a diagonal matrix in \({{\text {GL}}}_{{n}}({\mathbb C})\). The element \(\varPhi (x_{\bar{\emptyset }})=({\mathsf f}^*)^{-1}z\) in \(H_{{\mathsf T}}^*({\text {Gr}}({d},{n}))\) is \(-c^{{\mathsf T}}_1({\mathcal S})\) where \(c^{{\mathsf T}}_1({\mathcal S})\) is the equivariant first Chern class of \({\mathcal S}\).

4 Vanishing Lemma and restriction to the fixed points

In this section, we obtain the formula for the restriction of the factorial Schur polynomials and the equivariant Schubert classes to the fixed points in the weighted case. Let \(s^w_{\lambda }(x|a^w)\) be the image of \(s_{\lambda }(x|a)\) under the identification \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \cong {\mathbb Q}[a^{w}][x]^{{\mathfrak S}_{{d}}}\) defined by \(a_i=a^w_i - (w_i/u) x_{\bar{\emptyset }}\), i.e.,
$$\begin{aligned} s^w_{\lambda }(x|a^w) := s_{\lambda }\left( x | a^w_1 - (w_1/u) x_{\bar{\emptyset }}, a^w_2 - (w_2/u) x_{\bar{\emptyset }}, \ldots \right) . \end{aligned}$$
For each \(\mu \in {\mathcal P}({d})\), let
$$\begin{aligned} a^w_{\bar{\mu }} := a^w_{\bar{\mu }_1} + \cdots + a^w_{\bar{\mu }_{{d}}} \quad \text{ and } \ \ w_{\bar{\mu }}:=w_{\bar{\mu }_1} + \cdots + w_{\bar{\mu }_{{d}}}+u. \end{aligned}$$
Consider the map \(\psi _{\mu }^w : {\mathbb Q}[a^w][x]^{{\mathfrak S}_d} \rightarrow {\mathbb Q}[a^w]\) defined by the substitution
$$\begin{aligned} x_i \mapsto a^w_{\bar{\mu }_i} - (w_{\bar{\mu }_i}/w_{\bar{\mu }}) \cdot a^w_{\bar{\mu }} \quad \text { for all }\, i=1,\ldots , {d}. \end{aligned}$$

Lemma 2

Let \(\mu \in {\mathcal P}({d})\). We have
$$\begin{aligned} \psi _{\mu }^w\left( s_{\lambda }^w(x|a^w)\right) = s_{\lambda }\left( c^w_{\bar{\mu }_1},\ldots ,c^w_{\bar{\mu }_{{d}}}|c^w_1,c^w_2, \ldots \right) , \end{aligned}$$
where \(c^w_l:=a^w_l - (w_l/w_{\bar{\mu }})a^w_{\bar{\mu }}\) for each \(l \in {\mathbb N}\).

Proof

By a direct computation, we see that \(\psi _{\mu }^w(x_{\bar{\emptyset }}) = (u/w_{\bar{\mu }})a^w_{\bar{\mu }}\). Therefore,
$$\begin{aligned} \psi _{\mu }^w\left( s^w_{\lambda }\left( x|a^w\right) \right)= & {} \psi _{\mu }^w\left( s_{\lambda }\left( x | a^w_1 - (w_1/u) x_{\bar{\emptyset }}, a^w_2 - (w_2/u) x_{\bar{\emptyset }}, \ldots \right) \right) \\= & {} s_{\lambda }\left( c^w_{\bar{\mu }_1},\ldots ,c^w_{\bar{\mu }_{{d}}}|c^w_1,c^w_2, \ldots \right) . \end{aligned}$$
\(\square \)

Let \([\lambda ]_{-}:=\{\rho \in {\mathcal P}({d}) \mid \rho \subset \lambda , \ |\overline{\rho } \cap \overline{\lambda }|={d}-1\}\). The following proposition is the generalization of the Vanishing Theorem ([14, 15]).

Proposition 2

For each \(\lambda ,\mu \in {\mathcal P}({d})\), we have
$$\begin{aligned} \psi _{\mu }^w(s_{\lambda }^w(x|a^w))={\left\{ \begin{array}{ll} 0 &{}\quad \mathrm{if }\, \lambda \not \subset \mu \\ \prod _{\rho \in [\lambda ]_{-}} \left( (w_{\bar{\rho }}/w_{\bar{\lambda }})a^w_{\bar{\lambda }} -a^w_{\bar{\rho }} \right) &{}\quad \mathrm{if }\, \lambda =\mu . \end{array}\right. } \end{aligned}$$

Proof

In [14] and [15], it is shown that
$$\begin{aligned} s_{\lambda }(a_{\bar{\mu }}|a) = {\left\{ \begin{array}{ll} 0 &{}\quad \mathrm{if }\,\lambda \not \subset \mu \\ \prod _{\rho \in [\lambda ]_{-}} \left( a_{\bar{\lambda }} -a_{\bar{\rho }} \right) &{}\quad \mathrm{if }\,\lambda =\mu . \end{array}\right. } \end{aligned}$$
From this and Lemma 2, the first part of the case is obvious. The second claim follows from the following computation. Suppose \(\lambda =\mu \). We compute
$$\begin{aligned} c^w_{\bar{\lambda }} = c^w_{\bar{\lambda }_1} + \cdots + c^w_{\bar{\lambda }_{{d}}} =a^w_{\bar{\lambda }} - \frac{w_{\bar{\lambda }} - u}{w_{\bar{\lambda }}} a^w_{\bar{\lambda }} = \frac{u}{w_{\bar{\lambda }}} a^w_{\bar{\lambda }} \\ \end{aligned}$$
and
$$\begin{aligned} c^w_{\bar{\rho }} = c^w_{\bar{\rho }_1} + \cdots + c^w_{\bar{\rho }_{{d}}} = a^w_{\bar{\rho }} - \frac{w_{\bar{\rho }}-u}{w_{\bar{\lambda }}}a^w_{\bar{\lambda }} = a^w_{\bar{\rho }} - \frac{w_{\bar{\rho }}}{w_{\bar{\lambda }}}a^w_{\bar{\lambda }} + \frac{u}{w_{\bar{\lambda }}} a^w_{\bar{\lambda }} \\ \end{aligned}$$
Thus,
$$\begin{aligned} \psi _{\lambda }^w(s_{\lambda }^w(x|a^w))= & {} s_{\lambda }\left( c^w_{\bar{\lambda }_1},\ldots ,c^w_{\bar{\lambda }_{{d}}}|c^w_1,c^w_2, \ldots \right) \\= & {} \prod _{\rho \in [\lambda ]_{-}} \left( c^w_{\bar{\lambda }} -c^w_{\bar{\rho }} \right) =\prod _{\rho \in [\lambda ]_{-}} \left( \frac{w_{\bar{\rho }}}{w_{\bar{\lambda }}}a^w_{\bar{\lambda }} -a^w_{\bar{\rho }} \right) . \end{aligned}$$
\(\square \)

We also have the following generalization of [9, Section 6, Lemma].

Lemma 3

For each \(\lambda , \mu \in {\mathcal P}({d},{n})\), we have
$$\begin{aligned} \widetilde{wS}_{\lambda }|_{\mu }= \psi _{\mu }^w\left( s_{\lambda }^w\left( x|a^w\right) \right) \big |_{a^w_l=b^w_l, \forall l\in {\mathbb N}}. \end{aligned}$$
where \(b^w_l:=0\) for all \(l>{n}\).

Proof

In (5), the right vertical isomorphisms send \(b_i \in {\mathbb Q}[b_1,\ldots , b_{{n}}]\) to \(b^w_i\)\(- (w_i/w_{\bar{\mu }})b^w_{\bar{\mu }} \in {\mathbb Q}[b^w_1,\ldots , b^w_{{n}}]\). We know from Lemma 6 in [9] that
$$\begin{aligned} \widetilde{S}_{\lambda }|_{\mu }=s_{\lambda }( b_{\bar{\mu }_1}, \ldots , b_{\bar{\mu }_{{d}}} | b_1, \ldots , b_{{n}}, 0, \ldots ) \in {\mathbb Q}[b_1,\ldots ,b_{{n}}]. \end{aligned}$$
Therefore,
$$\begin{aligned} \widetilde{wS}_{\lambda }|_{\mu }= & {} \left. \left( \widetilde{S}_{\lambda }|_{\mu }\right) \right| _{b_i \mapsto b^w_i- (w_i/w_{\bar{\mu }})b^w_{\bar{\mu }}}\\= & {} s_{\lambda }( b_{\bar{\mu }_1}, \cdots , b_{\bar{\mu }_{{d}}} | b_1, \dots , b_{{n}}, 0, \dots )\big |_{b_i \mapsto b^w_i- (w_i/w_{\bar{\mu }})b^w_{\bar{\mu }}}\\= & {} \psi _{\mu }^w\left( s_{\lambda }^w\left( x|a^w\right) \right) \big |_{a^w_l=b^w_l, \forall l\in {\mathbb N}}, \end{aligned}$$
where the last equality follows from Lemma 2. \(\square \)

5 Twisting Schur polynomials

Recall that, for given nonnegative integers \(w_l, l \in {\mathbb N}\) and a positive integer u, we identify \({\mathbb Q}[a][x]^{{\mathfrak S}_{{d}}} \cong {\mathbb Q}[a^w][x]^{{\mathfrak S}_{{d}}}\) by sending \( a_l\) to \(a^w_l - (w_l/u)x_{\bar{\emptyset }}\). The polynomial \(s^w_{\lambda }(x | a^w)\) is nothing but the factorial Schur polynomial \(s_{\lambda }(x | a)\) after this parameter change.

Definition 3

For each \(\lambda \in {\mathcal P}({d})\), define
$$\begin{aligned} s^w_{\lambda }(x) := s^w_{\lambda }(x | 0) \ \ \ \in {\mathbb Q}[x]^{{\mathfrak S}_{{d}}}. \end{aligned}$$
Equivalently, we have \(s^w_{\lambda }(x) = s_{\lambda }(x | - (w_1/u)x_{\bar{\emptyset }}, - (w_2/u)x_{\bar{\emptyset }}, \ldots )\).

Example 1

Since \(s_{(1)}(x|a) = x_{\bar{\emptyset }} - a_{\bar{\emptyset }}\), by the substitution \(a^w_l - (w_l/u) x_{\bar{\emptyset }} = a_l\), we have
$$\begin{aligned} s^w_{(1)}(x|a^w) = (w_{\bar{\emptyset }}/u)x_{\bar{\emptyset }} - a^w_{\bar{\emptyset }}. \end{aligned}$$
By setting \(a^w=0\), we have
$$\begin{aligned} s^w_{(1)}(x) = (w_{\bar{\emptyset }}/u) x_{\bar{\emptyset }}. \end{aligned}$$
(7)
Since \(s_{(1,1)}(x|a) = \sum _{1\le i < j \le {d}} (x_i - a_i)(x_j - a_{j-1})\), by the substitution, we have
$$\begin{aligned} s_{(1,1)}^w(x|a^w) = \sum _{1\le i < j \le {d}} \left( x_i - a^w_i + (w_i/u) x_{\bar{\emptyset }}\right) \left( x_j - a^w_{j-1} + (w_{j-1}/u) x_{\bar{\emptyset }}\right) . \end{aligned}$$
By setting \(a^w=0\), we have
$$\begin{aligned} s_{(1,1)}^w(x) = \sum _{1\le i < j \le {d}} \left( x_i + (w_i/u) x_{\bar{\emptyset }}\right) \left( x_j + (w_{j-1}/u) x_{\bar{\emptyset }}\right) . \end{aligned}$$

Since \(s^w_{\lambda }(x | a^w), \lambda \in {\mathcal P}({d})\) form a basis of \({\mathbb Q}[a^w][x]^{{\mathfrak S}_{{d}}}\) as a \({\mathbb Q}[a^w]\)-module, the polynomials \(s^w_{\lambda }(x), \lambda \in {\mathcal P}({d})\) generate \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}}\) over \({\mathbb Q}\). For each homogeneous part of \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}}\), the numbers of \(s_{\lambda }(x)\) and \(s^w_{\lambda }(x)\) of the given degree are the same. Thus, we have the following proposition.

Proposition 3

The polynomials \(s^w_{\lambda }(x), \lambda \in {\mathcal P}({d})\) form a basis of \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}}\).

For each \(\lambda \in {\mathcal P}({d},{n})\), let \({w\mathrm {S}}_{\lambda }\) be the corresponding weighted Schubert class in \(H^*({\text {wGr}}({d},{n}))\). It is the image of \(\widetilde{wS}_{\lambda }\) under the natural map \(H_{{\mathsf T}_{w}}^*({\text {wGr}}({d},{n}))\rightarrow H^*({\text {wGr}}({d},{n}))\). Proposition 1 implies the following theorem.

Theorem 2

The map \({\mathbb Q}[x]^{{\mathfrak S}_{{d}}} \rightarrow H^*({\text {wGr}}({d},{n}))\) defined by sending \(s^w_{\lambda }(x)\) to \({w\mathrm {S}}_{\lambda }\) if \(\lambda \in {\mathcal P}({d},{n})\) and 0 otherwise is a surjective ring homomorphism.

We conclude this section with the Pieri rule for the twisted Schur polynomials \(s_{\lambda }^w(x)\). For partitions \(\lambda , \lambda ^{\prime }\in {\mathcal P}({d})\), we denote by \(\lambda ^{\prime }\rightarrow \lambda \) the condition that \(\lambda ^{\prime }\supset \lambda \) and \(|\lambda ^{\prime }/\lambda |=1\) where \(\lambda ^{\prime }/\lambda \) is the corresponding skew Young diagram and \(|\lambda ^{\prime }/\lambda |\) is the number of the boxes in \(\lambda ^{\prime }/\lambda \).

Proposition 4

(The Pieri rule)
$$\begin{aligned} s^w_{(1)}(x) \cdot s^w_{\lambda }(x)= & {} \frac{w_{\bar{\emptyset }}}{w_{{\bar{\lambda }}}} \sum _{\lambda ^{\prime }\rightarrow \lambda }s^w_{\lambda ^{\prime }}(x) \end{aligned}$$
(8)

Proof

It is well known (cf. [14, p.4434]) that
$$\begin{aligned} s_{(1)}(x|a) s_{\lambda }(x|a) = (a_{{\bar{\lambda }}} - a_{{\bar{\emptyset }}}) s_{\lambda }(x|a) + \sum _{\lambda ^{\prime } \rightarrow \lambda } s_{\lambda ^{\prime }}(x|a). \end{aligned}$$
By substituting \(a_i \mapsto -(w_i/u)x_{\bar{\emptyset }}\), we have
$$\begin{aligned} a_{{\bar{\lambda }}} - a_{{\bar{\emptyset }}} \mapsto (x_{\bar{\emptyset }}/u)(- w_{ {\bar{\lambda }}} + w_{ {\bar{\emptyset }}}). \end{aligned}$$
Thus, we obtain
$$\begin{aligned} \sum _{\lambda ^{\prime } \rightarrow \lambda } s^w_{\lambda ^{\prime }}(x)&= s^w_{(1)}(x) s^w_{\lambda }(x) - \frac{- w_{ {\bar{\lambda }}} + w_{ {\bar{\emptyset }}}}{u} x_{\bar{\emptyset }} s^w_{\lambda }(x) \\&= s^w_{(1)}(x) s^w_{\lambda }(x)- \frac{- w_{ {\bar{\lambda }}} + w_{ {\bar{\emptyset }}}}{w_{{\bar{\emptyset }}}} s^w_{(1)}(x) s^w_{\lambda }(x) = \frac{w_{{\bar{\lambda }}}}{w_{\bar{\emptyset }}} s^w_{(1)}(x) s^w_{\lambda }(x). \end{aligned}$$
where the second equality follows from (7). This proves the claim. \(\square \)

6 Representations associated with \(s^w_{\lambda }(x)\)

In this section, we will see that the twisted Schur polynomial \(s^w_{\lambda }(x)\) is a character of a representation of \(\text {GL}_{{d}}({\mathbb C})\) under the assumption that \(u=1\). It basically follows from [14].

Recall that the character \(\text {ch}(V^{\lambda })\) of the irreducible \(\text {GL}_{{d}}({\mathbb C})\)-representation \(V^{\lambda }\) with the highest weight \(\lambda =(\lambda _1,\ldots ,\lambda _{{d}})\) coincides with the Schur polynomial \(s_{\lambda }(x)\). This gives us the ring isomorphism
$$\begin{aligned} \text {ch}: R(\text {GL}_{{d}}({\mathbb C})) \mathop {\longrightarrow }\limits ^{\cong } {\mathbb Z}[x]^{{\mathfrak {S}}_{{d}}} \end{aligned}$$
(9)
where \(R(\text {GL}_{{d}}({\mathbb C}))\) is the polynomial representation ring.

Theorem 3

Suppose that \(u=1\). For any partition \(\lambda \in {\mathcal P}({d})\), the polynomial \(s^w_{\lambda }(x)\) is integral and Schur-positive, i.e., it is a linear combination of the Schur polynomials such that the coefficients are nonnegative integers. In particular, there exists a representation \(V^{\lambda }_w\) of \(\mathrm{GL}_{{d}}({\mathbb C})\) such that \(\mathrm{ch}(V^{\lambda }_w)=s^w_{\lambda }(x)\).

Proof

For each \(\lambda \in {\mathcal P}({d})\), we can find in [14, p.4433] the expression
$$\begin{aligned} s_{\lambda }(x|-a)=\sum _{\nu \subset \lambda }\bar{g}_{\lambda \nu }(a)s_{\nu }(x) \end{aligned}$$
(10)
where
$$\begin{aligned} \bar{g}_{\lambda \nu }(a) = \sum _{T\in {\mathcal {T}}(\lambda ,\nu )}\prod _{\begin{array}{c} \alpha \in \lambda \\ T(\alpha )\ \text {unbarred} \end{array}} a_{T(\alpha )+c(\alpha )} \end{aligned}$$
(11)
and \({\mathcal {T}}(\lambda ,\nu )\) is a subset of semi-standard tableaux of shape \(\lambda \) depending on \(\nu \). For the completeness, we recall the definition of \({\mathcal {T}}(\lambda ,\nu )\). Consider a sequence of partitions
$$\begin{aligned} R : \emptyset =\rho ^{(0)} \rightarrow \rho ^{(1)} \rightarrow \cdots \rightarrow \rho ^{(l)} = \nu . \end{aligned}$$
(12)
Let \(r_i\) be the row number of the box that one added to \(\rho ^{(i-1)}\) to obtain \(\rho ^{(i)}\). An element of \({\mathcal {T}}(\lambda ,R)\) is a semi-standard tableaux T of shape \(\lambda \) with entries in \(\{1,\ldots ,{d}\}\), together with a sequence of boxes \(\alpha _1,\ldots ,\alpha _{l}\) in \(\lambda \) such that the column order of \(\alpha _i\) are strictly increasing and \(T(\alpha _i)=r_i\). These entries \(T(\alpha _i), i=1,\dots ,l\) are called barred. Let \({\mathcal {T}}(\lambda ,\nu ):= \bigsqcup _{R}{\mathcal {T}}(\lambda ,R)\) where the union is taken over all sequence of the form (12).
By the definition (11), we see that \(\bar{g}_{\lambda \nu }(a)\) is a homogeneous polynomial in a’s of degree \(|\lambda |-|\nu |\) and with positive integral coefficients. By substituting \(a_i \mapsto -(w_i/u)x_{\bar{\emptyset }}\) (assume \(u=1\)) to the equation (10), we obtain
$$\begin{aligned} s^{w}_{\lambda } (x) = s_{\lambda }(x|(-w_1)x_{\bar{\emptyset }},(-w_2)x_{\bar{\emptyset }},\ldots ) = \sum _{\nu \subset \lambda } \bar{g}_{\lambda \nu }(w) s_{(1)}(x)^{|\lambda |-|\nu |} s_{\nu }(x). \end{aligned}$$
(13)
Define \(K_{\nu , k}^{\mu }\in {\mathbb Z}_{\ge 0}\) by the equation
$$\begin{aligned} s_{(1)}(x)^k s_{\nu }(x) = \sum _{\mu \in {\mathcal P}({d})} K_{\nu ,k}^{\mu }s_{\mu }(x), \quad k\in {\mathbb Z}_{\ge 0}. \end{aligned}$$
Note that \(K_{\nu ,k}^{\mu }=0\) unless \(\nu \subset \mu \) and \(|\mu |=|\nu |+k\). From (13), we have
$$\begin{aligned} s^{w}_{\lambda } (x) =\sum _{\mu \in {\mathcal P}({d})} \left( \sum _{\nu \subset \lambda } \bar{g}_{\lambda \nu }(w) K_{\nu , |\lambda |-|\nu |}^{\mu }\right) s_{\mu }(x). \end{aligned}$$
(14)
Observe that the coefficient of \(s_{\mu }(x)\)
$$\begin{aligned} m_{\lambda \mu }(w):=\sum _{\nu \subset \lambda } \bar{g}_{\lambda ,\nu }(w) K_{\nu , |\lambda |-|\nu |}^{\mu } \end{aligned}$$
(15)
is a nonnegative integer and vanishes unless \(|\mu |=|\lambda |\). Thus, \(s^{w}_{\lambda } (x) \in {\mathbb Z}[x]^{{\mathfrak {S}}_{{d}}}\) and it is the character of the representation
$$\begin{aligned} V^{\lambda }_w:= {\mathop {\mathop {\bigoplus } \limits _{\mu \in {\mathcal P}({d})}}\limits _{|\lambda |=|\mu |}} (V^{\mu })^{\oplus m_{\lambda \mu }(w)} \end{aligned}$$
where \(V^{\mu }\) is the irreducible representation of \(\text {GL}_{{d}}({\mathbb C})\) with the highest weight \(\mu =(\mu _1,\cdots ,\mu _{{d}})\). \(\square \)

Remark 4

Since \(s^w_{\lambda }(x)\) is a homogeneous polynomial, the representation \(V^{\lambda }_w\) is homogeneous. That is, there exists a representation \(S^{\lambda }_w\) of \({\mathfrak {S}}_{|\lambda |}\) such that
$$\begin{aligned} V^{\lambda }_w = ({\mathbb C}^{{d}})^{\otimes |\lambda |} \otimes _{{\mathbb C}[{\mathfrak {S}}_{|\lambda |}]} S^{\lambda }_w \end{aligned}$$
as representations of \(\text {GL}_{{d}}({\mathbb C})\) where \({{\text {GL}}}_{{d}}({\mathbb C})\) naturally acts on the first factor in the right hand side.

Example 2

Suppose that \(w_1=2, w_2=1, w_3=0\), and \(u=1\). Then
$$\begin{aligned} s^w_{(1)}(x)= & {} 4 s_{(1)}(x), \\ s^w_{(2)}(x)= & {} 6 s_{(2)}(x) + 5 s_{(1,1)}(x). \end{aligned}$$
Now we interpret the weighted Schubert structure constants in terms of representations. Suppose that \(u=1\) and the weights are non-increasing ; \(w_1\ge w_2\ge \cdots \). For all \({\mathsf N}>{d}\), Corollary 5.6 in [1] shows that, the structure constants with weights \((w_{{\mathsf N}},\ldots ,w_1)\) are nonnegative rational numbers. Therefore, by Theorem 2 and Theorem 3, we see that, for each \(\lambda ,\mu , \nu \in {\mathcal P}({d})\), there exist nonnegative integers \({ \ell _{\lambda \mu }}\in {\mathbb Z}_{\ge 1}\) and \({ \ell _{\lambda \mu }^{\nu }}\in {\mathbb Z}_{\ge 0}\) such that
$$\begin{aligned} (V^{\lambda }_w \otimes V^{\mu }_w)^{\oplus {\ell _{\lambda \mu }}} = \bigoplus _{\nu \in {\mathcal P}({d})} (V^{\nu }_w)^{\oplus {\ell _{\lambda \mu }^{\nu }}} \quad \text { as representations of } \hbox {GL}_{{d}}({\mathbb C}). \end{aligned}$$
We can think of the fraction \({\ell _{\lambda \mu }^{\nu }/\ell _{\lambda \mu }}\) as the rational multiplicity of \(V^{\nu }_w\) in the tensor product of \(V^{\lambda }_w \otimes V^{\mu }_w\), and they are nothing but the Schubert structure constants for the weighted Grassmannians.

7 Determinantal formulas

In this section, we give two determinantal formulas for the weighted Schubert classes \({w\mathrm {S}}_{\lambda } \). One is in terms of the special weighted Schubert classes, and the other is in terms of the Chern classes of the tautological orbifold vector bundles. Recall that the weight \((w_1,\ldots ,w_{{n}})\) has the reversed order, compared with the one in [2] and [1].

7.1 Known formulas

Let \({\mathcal S}\hookrightarrow {\mathcal E}\twoheadrightarrow {\mathcal Q}\) be the sequence of the tautological bundles of \({\text {Gr}}({d},{n})\) where \({\mathcal E}={\text {Gr}}({d},{n})\times {\mathbb C}^{{n}}\) is the trivial bundle. Let \({\mathcal F}^{\ell }\) be the subbundle of \({\mathcal E}\) defined by the coordinate plane generated by the last\(\ell \) coordinates. Then
$$\begin{aligned} \sum _{r\ge 0}c^{{\mathsf T}}_r({\mathcal F}^{\ell } - {\mathcal S}) = \frac{\prod _{i=1}^{\ell } (1 - b_i)}{\prod _{j=1}^{{d}}(1 - {\mathbf x }_j)} \end{aligned}$$
where \({\mathbf x }_1,\dots , {\mathbf x }_{{d}}\) are the \({\mathsf T}\)-equivariant Chern roots of the dual of \({\mathcal S}\). The following is known as Kempf–Laksov’s formula [8] (cf. [12, p.17]):
$$\begin{aligned} \widetilde{S}_{\lambda }=\det \big [c^{{\mathsf T}}_{\lambda _i-i+j}({\mathcal F}^{\lambda _i-i+{d}} - {\mathcal S}) \big ]_{1\le i\le j\le {d}}. \end{aligned}$$
The equality \(c^{{\mathsf T}}_r({\mathcal F}^{\ell } - {\mathcal S}) = \sum _{k=0}^{r} h_{k}(b_{\ell +1}, \dots , b_{{n}}) c^{{\mathsf T}}_{r-k}({\mathcal Q})\) follows from
$$\begin{aligned} \sum _{r\ge 0}c^{{\mathsf T}}_r({\mathcal F}^{\ell } - {\mathcal S}) = \frac{1}{\prod _{i=\ell +1}^{{n}}(1 - b_i)} \sum _{r\ge 0}c^{{\mathsf T}}_r({\mathcal E}- {\mathcal S}) = \frac{1}{\prod _{i=\ell +1}^{{n}}(1 - b_i)} \sum _{r\ge 0}c^{{\mathsf T}}_r({\mathcal Q}). \end{aligned}$$
Therefore, we have
$$\begin{aligned} \widetilde{S}_{\lambda }= \det \left[ \sum _{k=0}^{\lambda _i-i+j} h_{{k}}(b_{\lambda _i-i+{d}+1}, \ldots , b_{{n}}) c^{{\mathsf T}}_{{\lambda _i-i+j-k}}({\mathcal Q}) \right] _{1\le i\le j\le {d}}. \end{aligned}$$
(16)
On the other hand, it is shown in [10] that
$$\begin{aligned} \widetilde{S}_{\lambda }= \det \left[ \sum _{k=0}^{j-1}h_k(b_{\lambda _i-i+1+{d}},\ldots ,b_{\lambda _i-i+j-k+{d}}) \widetilde{S}_{(\lambda _i-i+j-k)} \right] _{1\le i\le j\le {d}}. \end{aligned}$$
(17)
When we specialize the above formulas to \(b_k=0\) for all \(k\in {\mathbb N}\), we obtain the same determinant formula for the Schubert class in the ordinary cohomology of the Grassmannian.

7.2 Determinantal formulas for the weighted Grassmannian

Let \({\mathcal S}_a \hookrightarrow {\mathcal E}_a \twoheadrightarrow {\mathcal Q}_a\) be the pullback of the sequence of tautological bundles along \({\text {aPl}}^{\times }({d},{n}) \rightarrow {\text {Gr}}({d},{n})\). By quotienting the sequence by \({\mathsf D}_w\), we obtain the tautological sequence \({\mathcal S}_w \hookrightarrow {\mathcal E}_w \twoheadrightarrow {\mathcal Q}_w\) of orbifold vector bundles over \({\text {wGr}}({d},{n})\). Under the isomorphismsthe equivariant Chern classes of those tautological bundles coincide. As we have seen, the equivariant Schubert classes also coincide. Therefore, the formulas (16) and (17) hold for both \(H_{{\mathsf K}}({\text {aPl}}^{\times }({d},{n}))\) and \(H_{{\mathsf T}_{w}}({\text {wGr}}({d},{n}))\). In particular, (16) gives the following in the non-equivariant setting.

Theorem 4

For each \(\lambda \in {\mathcal P}({d},{n})\),
$$\begin{aligned} {w\mathrm {S}}_{\lambda }\,=\, \det \, \left[ \sum _{k=0}^{\lambda _i-i+j} \big (c_1({\mathcal S}_w)/u\big )^{k}h_{k}(w_{\lambda _i-i+{d}+1}, \dots , w_{{n}}) c_{\lambda _i-i+j-k}({\mathcal Q}_w) \right] _{1\le i\le j\le {d}}. \end{aligned}$$

Proof

From (16), we have
$$\begin{aligned} \widetilde{wS}_{\lambda }\,=\,\det \left[ \sum _{k=0}^{\lambda _i-i+j} \tilde{h}_{k,i}(b^w) c^{{\mathsf T}_{w}}_{\lambda _i-i+j-k}({\mathcal Q}_w) \right] _{1\le i\le j\le {d}} \end{aligned}$$
where
$$\begin{aligned} \tilde{h}_{k,i}(b^w)=h_{k}\left( b_{\lambda _i-i+{d}+1}^w + (w_{\lambda _i-i+{d}+1}/u)c_1^{{\mathsf T}_{w}}({\mathcal S}_w) , \dots , b_{{n}}^w + (w_{{n}}/u)c_1^{{\mathsf T}_{w}}({\mathcal S}_w)\right) . \end{aligned}$$
Then we have \(\tilde{h}_{k,i}(0) = \big (c_1({\mathcal S}_w)/u\big )^{k}h_{k}\left( w_{\lambda _i-i+{d}+1}, \ldots , w_{{n}}\right) \). \(\square \)

Similarly, by substituting \(b_i \mapsto -(w_i/w_{\bar{\emptyset }}){w\mathrm {S}}_{(1)}\) to (17), one obtains

Theorem 5

For each \(\lambda \in {\mathcal P}({d},{n})\), we have
$$\begin{aligned}&{w\mathrm {S}}_{\lambda }\\&\quad = \det \,\left[ \sum _{k=0}^{j-1} h_k\left( w_{\lambda _i-i+1+{d}},\ldots ,w_{\lambda _i-i+j-k+{d}}\right) \left( \frac{{w\mathrm {S}}_{(1)}}{-w_{{\bar{\emptyset }}}}\right) ^{k} {w\mathrm {S}}_{(\lambda _i-i+j-k)}\right] _{1\le i,j\le {d}}. \end{aligned}$$

Example 3

Let \(({d},{n})=(2,4)\) and \(\lambda =(1,1)\). Assume \(u=1\). Then we have
$$\begin{aligned}&{w\mathrm {S}}_{(1,1)}\\&\quad =\left[ \begin{matrix} c_{1}({\mathcal Q}_w) + c_1({\mathcal S}_w)h_{1}(w_3, w_4)&{} c_{2}({\mathcal Q}_w)+ c_1({\mathcal S}_w)h_{1}(w_3, w_4) c_{1}({\mathcal Q}_w) + c_1({\mathcal S}_w)^{2}h_{2}(w_3, w_4)\\ 1 &{} c_{1}({\mathcal Q}_w) + c_1({\mathcal S}_w)h_{1}(w_2, w_3, w_4) \end{matrix}\right] . \end{aligned}$$
If \((w_1,w_2,w_3,w_4)=(0,2,1,0)\), then
$$\begin{aligned} {w\mathrm {S}}_{(1,1)}= \det \left[ \begin{matrix} c_1(\mathcal {Q}_w) + c_1(\mathcal {S}_w) &{} c_2(\mathcal {Q}_w) + c_1(\mathcal {Q}_w)c_1(\mathcal {S}_w) +c_1(\mathcal {S}_w)^2 \\ 1 &{} c_1(\mathcal {Q}_w) + 3c_1(\mathcal {S}_w) \end{matrix}\right] . \end{aligned}$$

Example 4

Suppose \(\lambda =(1,1) \in {\mathcal P}(2,4)\).
$$\begin{aligned} {w\mathrm {S}}_{\lambda } =\det \left[ \begin{matrix} {w\mathrm {S}}_{(1)}&{} {w\mathrm {S}}_{(2)}+h_1(w_{3}) \left( -{w\mathrm {S}}_{(1)}/w_{{\bar{\emptyset }}}\right) {w\mathrm {S}}_{(1)}\\ 1&{} {w\mathrm {S}}_{(1)} +h_1(w_2) \left( -{w\mathrm {S}}_{(1)}/w_{{\bar{\emptyset }}}\right) \end{matrix} \right] . \end{aligned}$$
For \((w_1,w_2,w_3,w_4)=(0, 2, 1,0)\) and \(u=1\), we have
$$\begin{aligned} {w\mathrm {S}}_{(1,1)} = \det \left[ \begin{matrix} {w\mathrm {S}}_{(1)} &{} {w\mathrm {S}}_{(2)} - \frac{1}{3}{w\mathrm {S}}_{(1)}^2 \\ 1 &{} \frac{1}{3}{w\mathrm {S}}_{(1)} \end{matrix}\right] \end{aligned}$$
The reader can also check this equality directly by the Pieri rule (8).

Remark 5

In the ordinary case, the above two propositions coincide since we have \(S_{(r)} = c_r({\mathcal Q})\). In general, they are different formulas, reflecting the fact that \({w\mathrm {S}}_{(r)} \not = c_r({\mathcal Q}_w)\).

7.3 A presentation of \(H^*({\text {Gr}}({d},{n}))\)

We conclude by giving a presentation of the ordinary cohomology ring of the weighted Grassmannians \({\text {wGr}}({d},{n})\) over \({\mathbb Q}\), in terms of Chern classes. It is a well-known fact that \(H_{{\mathsf T}}^*({\text {Gr}}({d},{n}))\) is generated by \(c_i^{{\mathsf T}}({\mathcal S})\) and \(c_j^{{\mathsf T}}({\mathcal Q})\) as a \({\mathbb Q}[b]\)-algebra and the relations are given by
$$\begin{aligned} c^{{\mathsf T}}({\mathcal S}) c^{{\mathsf T}}({\mathcal Q}) = c^T({\mathcal E}), \quad \textit{i.e.,} \ \sum _{i=0}^rc^{{\mathsf T}}_i({\mathcal S})c^{{\mathsf T}}_{r-i}({\mathcal Q}) = (-1)^re_r(b_1,\ldots , b_n). \end{aligned}$$
Therefore, by
$$\begin{aligned} {\mathsf f}^*(b_i)={\mathsf f}_w^*\left( b_i^w + (w_i/u)c_1^{{\mathsf T}_{w}}({\mathcal S}_w)\right) , \end{aligned}$$
we have
$$\begin{aligned}&\sum _{i=0}^rc^{{\mathsf T}_{w}}_i({\mathcal S}_w)c^{{\mathsf T}_{w}}_{r-i}({\mathcal Q}_w) \\&\quad = c_r^{{\mathsf T}_{w}}({\mathcal E}_w)\\&\quad = (-1)^re_r\left( b_1^w + (w_1/u)c_1^{{\mathsf T}_{w}}({\mathcal S}_w), \dots , b_{{n}}^w + (w_{{n}}/u)c_1^{{\mathsf T}_{w}}({\mathcal S}_w) \right) . \end{aligned}$$
By setting \(b^w_i=0\), we find that
$$\begin{aligned} \sum _{i=0}^rc_i({\mathcal S}_w)c_{r-i}({\mathcal Q}_w) = c_r({\mathcal E}_w) = (-1)^r(c_1({\mathcal S}_w)/u)^r e_r(w_1, \dots , w_{{n}}). \end{aligned}$$
(18)
Thus, \(H^*({\text {wGr}}({d},{n}))\) is generated by \(c_i({\mathcal S}_w)\) and \(c_j({\mathcal Q}_w)\) with the relation (18).

Proposition 5

We have the following graded ring isomorphism
$$\begin{aligned} H^*({\text {wGr}}({d},{n})) \cong \frac{{\mathbb Q}\left[ c_1,\ldots , c_{{d}}, \bar{c}_1,\ldots ,\bar{c}_{{{n}-{d}}}\right] }{\big (\sum _{i=0}^rc_i \bar{c}_{r-i} \,=\, (-1)^r(c_1/u)^r e_r(w_1, \ldots , w_{{n}}), \ {r=1,\ldots ,{n}} \big )} \end{aligned}$$
where \(c_0=\bar{c}_0 =1\), \(c_i=0\) if \(i>{d}\) and \(\bar{c}_j=0\) if \(j > {n}-{d}\).
When \(d=1\) (i.e., when \({\text {wGr}}({d},{n})\) is a weighted projective space), it is straightforward to check that our presentation turns out to be
$$\begin{aligned} H^*({\text {wGr}}(1,{n})) \cong \frac{{\mathbb Q}[c_1]}{\big ( c_1^n \big )} \end{aligned}$$
which is isomorphic to the cohomology ring of the ordinary projective space \({\mathbb C}{\mathbb P}^n\) (cf. Kawasaki [7]).

Notes

Acknowledgments

The authors would like to thank the organizers of “MSJ Seasonal Institute 2012 Schubert calculus” for providing us an excellent environment for the discussions on the topic of this paper. The authors also would like to show their gratitude to Takeshi Ikeda and Tatsuya Horiguchi for many useful discussions. The first author is particularly grateful to Takashi Otofuji for many helpful comments. The first author was supported by JSPS Research Fellowships for Young Scientists. The second author was supported by the National Research Foundation of Korea (NRF) Grants funded by the Korea government (MEST) (No. 2012-0000795, 2011-0001181). He also would like to express his gratitude to the Algebraic Structure and its Application Research Institute at KAIST for providing him an excellent research environment in 2011–2012.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Osaka City University Advanced Mathematical InstituteOsakaJapan
  2. 2.Department of Applied MathematicsOkayama University of ScienceOkayama-shiJapan

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