1 Introduction

Let G be a semisimple algebraic group over ℂ with a Borel subgroup B and maximal torus T. Let Gr G =G(ℂ((t)))/G(ℂ[[t]]) be the affine Grassmannian of G. The T-equivariant homology H T (Gr G ) and cohomology H T(Gr G ) are dual Hopf algebras over S=H T(pt) with Pontryagin and cup products, respectively. Let \(W_{\mathrm{af}}^{0}\) be the minimal length cosets in W af/W where W af and W are the affine and finite Weyl groups. Let \(\{\xi_{w}\mid w\in W_{\mathrm{af}}^{0}\}\) be the Schubert basis of H T (Gr G ). Define the equivariant Schubert homology structure constants \(d_{uv}^{w}\in S\) by

(1)

where \(u,v\in W_{\mathrm{af}}^{0}\). One interest in the polynomials \(d^{w}_{uv}\) is the fact that they are precisely the Schubert structure constants for the T-equivariant quantum cohomology rings QH T(G/B) [9, 13]. Due to a result of Mihalcea [12], they have the positivity property

(2)

Our first main result (Theorem 6) is an “equivariant homology Chevalley formula”, which describes \(d_{r_{0},v}^{w}\) for an arbitrary affine Grassmannian. Our second main result (Theorem 20) is an “equivariant homology Pieri formula” for G=SL n , which is a manifestly positive formula for \(d_{\sigma_{m},v}^{w}\) where the homology classes \(\{\xi_{\sigma_{m}}\mid1\le m\le n-1\}\) are the special classes that generate \(H_{T}(\mathrm{Gr}_{SL_{n}})\). In a separate work [10] we use this Pieri formula to define new symmetric functions, called k-double Schur functions and affine double Schur functions, which represent the equivariant Schubert homology and cohomology classes for \(\mathrm{Gr}_{SL_{n}}\).

2 The equivariant homology of Gr G

We recall Peterson’s construction [13] of the equivariant Schubert basis \(\{j_{w}\mid w\in W_{\mathrm{af}}^{0}\}\) of H T (Gr G ) using the level-zero variant of the Kostant and Kumar (graded) nilHecke ring [6]. We also describe the equivariant localizations of Schubert cohomology classes for the affine flag ind-scheme in terms of the nilHecke ring; these are an important ingredient in our equivariant Chevalley and Pieri rules.

2.1 Peterson’s level-zero affine nilHecke ring

Let I and I af=I∪{0} be the finite and affine Dynkin node sets and (a ij ∣i,j∈I af) the affine Cartan matrix.

Let \(P_{\mathrm{af}}= \mathbb{Z}\delta\oplus\bigoplus_{i\in I_{\mathrm{af}}} \mathbb{Z}\varLambda_{i}\) be the affine weight lattice, with δ the null root and Λ i the affine fundamental weight. The dual lattice \(P_{\mathrm{af}}^{*}=\mathrm{Hom}_{\mathbb{Z}}(P_{\mathrm{af}},\mathbb{Z})\) has dual basis \(\{d\} \cup\{\alpha_{i}^{\vee}\mid i\in I_{\mathrm{af}}\}\) where d is the degree generator and \(\alpha_{i}^{\vee}\) is a simple coroot. The simple roots {α i ∣i∈I af}⊂P af are defined by \(\alpha_{j} = \delta_{j0} \delta+ \sum_{i\in I_{\mathrm{af}}} a_{ij}\varLambda_{i}\) for j∈I af where (a ij ∣i,j∈I af) is the affine Cartan matrix. Then \(a_{ij}=\langle\alpha_{i}^{\vee},\alpha_{j}\rangle\) for all i,j∈I af. Let (a i ∣i∈I af) (resp. \((a_{i}^{\vee}\mid i\in I_{\mathrm{af}})\)) be the tuple of relatively prime positive integers giving a relation among the columns (resp. rows) of the affine Cartan matrix. Then \(\delta= \sum_{i\in I_{\mathrm {af}}} a_{i}\alpha_{i}\). Let \(c=\sum_{i\in I_{\mathrm{af}}} a_{i}^{\vee}\alpha_{i}^{\vee}\in P_{\mathrm{af}}^{*}\) be the canonical central element. The level of a weight λ∈P af is defined by 〈c,λ〉.

There is a canonical projection P af→P where P is the finite weight lattice, with kernel ℤδ⊕ℤΛ 0. There is a section P→P af of this projection whose image lies in the sublattice of \(\bigoplus_{i\in I_{\mathrm{af}}} \mathbb {Z}\varLambda_{i}\) consisting of level-zero weights. We regard P⊂P af via this section.

Let W and W af denote the finite and affine Weyl groups. Denote by {r i ∣i∈I af} the simple generators of W af. W af acts on P af by \(r_{i} \cdot\lambda=\lambda- \langle\alpha_{i}^{\vee},\lambda \rangle \alpha_{i}\) for i∈I af and λ∈P af. W af acts on \(P_{\mathrm{af}}^{*}\) by \(r_{i}\cdot\mu= \mu- \langle\mu,\alpha_{i}\rangle \alpha_{i}^{\vee}\) for i∈I af and \(\mu\in P_{\mathrm{af}}^{*}\). There is an isomorphism W af≅W⋉Q ∨ where \(Q^{\vee}= \bigoplus_{i\in I} \mathbb{Z}\alpha_{i}^{\vee}\subset P_{\mathrm{af}}^{*}\) is the finite coroot lattice. The embedding Q ∨→W af is denoted μ↦t μ . The set of real affine roots is W af⋅{α i ∣i∈I af}. For a real affine root α=w⋅α i , the associated coroot is well-defined by \(\alpha^{\vee}= w\cdot\alpha_{i}^{\vee}\).

Let S=Sym(P) be the symmetric algebra, and Q=Frac(S) the fraction field. W af≅W⋉Q ∨ acts on P (and therefore on S and on Q) by the level-zero action:

(3)

Since \(t_{-\theta^{\vee}} = r_{\theta}r_{0}\) we have

(4)

Finally, we have δ=α 0+θ where θ∈P is the highest root. So under the projection P af→P, α 0↦−θ.

Let \(Q_{W_{\mathrm{af}}} = \bigoplus_{w\in W_{\mathrm{af}}} Q w\) be the skew group ring, the Q-vector space Q⊗ℚℚ[W af] with Q-basis W af and product (p⊗v)(q⊗w)=p(v⋅q)⊗vw for p,q∈Q and v,w∈W af. \(Q_{W_{\mathrm{af}}}\) acts on Q: q∈Q acts by left multiplication and W af acts as above.

For i∈I af define the element \(A_{i}\in Q_{W_{\mathrm {af}}}\) by

(5)

A i acts on S since

(6)
(7)

The A i satisfy \(A_{i}^{2}=0\) and

where

For w∈W af we define A w by

(8)
(9)

The level-zero graded affine nilHecke ring \(\mathbb{A}\) (Peterson’s [13] variant of the nilHecke ring of Kostant and Kumar [6] for an affine root system) is the subring of \(Q_{W_{\mathrm{af}}}\) generated by S and {A i ∣i∈I af}. In \(\mathbb{A}\) we have the commutation relation

(10)

In particular

(11)

2.2 Localizations of equivariant cohomology classes

Using the relation

(12)

w∈W af may be regarded as an element of \(\mathbb{A}\). For v,w∈W af define the elements ξ v(w)∈S by

(13)

Using a reduced decomposition (9) for w and substituting (12) for its simple reflections, one obtains the formula [1] [2]

(14)

where the sum runs over b such that \(\prod_{b_{j}=1} r_{i_{j}} = v\) is reduced and the product over j is an ordered left-to-right product of operators. Each b encodes a way to obtain a reduced word for v as an embedded subword of the given reduced word of w: if b j =1 then the reflection \(r_{i_{j}}\) is included in the reduced word for v. Given a fixed b and an index j such that b j =1, the root associated to the reflection \(r_{i_{j}}\) is by definition \(r_{i_{1}}r_{i_{2}}\dotsm r_{i_{j-1}} \cdot\alpha_{i_{j}}\). The summand for b is the product of the roots associated to reflections in the given embedded subword.

It is immediate that

(15)
(16)

The element ξ v(w)∈S has the following geometric interpretation. Let X af=G af/B af be the Kac–Moody flag ind-variety of affine type [7]. For every v∈W af there is a T-equivariant cohomology class [X v ]∈H T(X af) and for each w∈W af there is an associated T-fixed point (denoted w) in X af and a localization map \(i_{w}^{*}:H^{T}(X_{\mathrm{af}})\to H^{T}(w) \simeq H^{T}(\mathrm{pt})\) [7]. Then \(\xi^{v}(w) = i_{w}^{*}([X_{v}])\). Moreover, the map H T(X af)→H T(W af)≅Fun(W af,S) given by restriction of a class to the T-fixed subset W af⊂X af, is an injective S-algebra homomorphism where Fun(W af,S) is the S-algebra of functions W af→S with pointwise product. The function ξ v∈Fun(W af,S) is the image of [X v ]. The image Φ of H T(X af) in Fun(W af,S) satisfies the GKM condition [3] [6]: For f∈Φ we haveFootnote 1

(17)

Lemma 1

Suppose u,v∈W af with ℓ(uv)=ℓ(u)+ℓ(v). Then

(18)

Lemma 2

Suppose v,w∈W af. Then

(19)

2.3 Peterson subalgebra and Schubert homology basis

Let K⊂G denote the maximal compact subgroup of G. The homotopy equivalence between Gr G and the based loop space ΩK endows the equivariant homology H T (Gr G ) and cohomology H T(Gr G ) with the structure of dual Hopf algebras. The Pontryagin multiplication in the homology H T (Gr G ) is induced by the group structure of ΩK. We let {ξ w } denote the equivariant Schubert basis of H T (Gr G ), dual (via the cap product) to the basis {ξ w} of H T(Gr G ).

The Peterson subalgebra of \(\mathbb{A}\) is the centralizer subalgebra \(\mathbb{P}=Z_{\mathbb{A}}(S)\) of S in \(\mathbb{A}\).

Theorem 3

[13] There is an isomorphism H T (Gr G )→ℙ of commutative Hopf algebras over S. For \(w\in W_{\mathrm{af}}^{0}\) let j w denote the image of ξ w in ℙ. Then j w is the unique element of ℙ with the property that \(j_{w}^{w}=1\) and \(j_{w}^{x}=0\) for any \(x\in W_{\mathrm{af}}^{0} \setminus\{w\}\) where \(j_{w}^{x}\in S\) are defined by

(20)

Moreover, if \(j^{x}_{w}\ne0\) then ℓ(x)≥ℓ(w) and \(j^{x}_{w}\) is a polynomial of degree ℓ(x)−ℓ(w).

The Schubert structure constants for H T (Gr G ) are obtained as coefficients of the elements j w .

Proposition 4

([13]) Let \(u,v,w\in W_{\mathrm {af}}^{0}\). Then

(21)

Due to the fact [9, 13] that the collections of Schubert structure constants for H T (Gr G ) and QH T(G/B) are the same and Mihalcea’s positivity theorem for equivariant quantum Schubert structure constants, we have the positivity property

Proposition 5

\(j_{w}^{x} \in\mathbb{Z}_{\ge0}[\alpha_{i}\mid i\in I]\) for all \(w\in W_{\mathrm{af}}^{0}\) and x∈W af.

Given \(u\in W_{\mathrm{af}}^{0}\) let t u=t λ where λ∈Q ∨ is such that t λ W=uW.

Since the translation elements act trivially on S and \(W_{\mathrm{af}}\subset\mathbb{A}\) via (12), we have t λ ∈ℙ for all λ∈Q ∨, so that \(t_{\lambda}\in \bigoplus_{v\in W_{\mathrm{af}}^{0}} S j_{v}\). For any \(w\in W_{\mathrm{af}}^{0}\), we have

by the definitions and Lemma 1.

Define the \(W_{\mathrm{af}}^{0} \times W_{\mathrm{af}}^{0}\)-matrices

(22)
(23)

The matrix A is lower triangular by (15) and has nonzero diagonal terms, and is hence invertible over Q=Frac(S). We have

Taking the coefficient of A x for x∈W af, we have

(24)

Note that if \(\varOmega\subset W_{\mathrm{af}}^{0}\) is any order ideal (downwardly closed subset) then the restriction A| Ω×Ω is invertible. In the sequel we choose certain such order ideals and find a formula for the inverse of this submatrix. Since the values of ξ x are given by (14) we obtain an explicit formula for \(j_{v}^{x}\) for v∈Ω and all x∈W af.

3 Equivariant homology Chevalley rule

Theorem 6

For every x∈W af∖{id}, \(\xi^{x^{-1}}(r_{\theta})\in\theta S\) and

(25)

Proof

For x≠id, the GKM condition (17) and (15) implies that \(\xi^{x^{-1}}(r_{\theta})\in\theta S\). \(\varOmega= \{\mathrm{id}, r_{0}\}\subset W_{\mathrm{af}}^{0}\) is an order ideal. The matrix A| Ω×Ω and its inverse are given by

Since id=t id and \(t_{\theta^{\vee}} = t^{r_{0}}\) (as \(t_{\theta ^{\vee}}=r_{0}r_{\theta}\)), we have

By the length condition in Theorem 3 we have

By (15) \(j_{r_{0}}^{y} = 0\) unless \(y\le t_{\theta^{\vee}} =r_{0}r_{\theta}\). So assume this.

Suppose r 0 y<y. Write y=r 0 x. Then

If r 0 y>y then we write y=x≤r θ and

as required. □

The formula (14) shows that \(\xi^{x^{-1}}(r_{\theta}) \in \mathbb{Z}_{\ge0}[\alpha_{i}\mid i\in I]\). The same holds for \(\theta^{-1} \xi^{x^{-1}}(r_{\theta})\). Indeed,

Lemma 7

α −1 ξ x(r α )∈ℤ≥0[α i ∣i∈I] for any positive root α.

Proof

The reflection r α has a reduced word i=i 1 i 2⋯i r−1 i r i r−1⋯i 1 which is symmetric. Consider the different embeddings j of reduced words of x into i, as in (14). If j uses the letter i r , then the corresponding term in (14) has θ as a factor. Otherwise, j uses i s but not i s+1,…,i r , for some s. But then there is another embedding of j′ of the same reduced word of x into i, which uses the other copy of the letter i s in i. The two terms in (14) which correspond to j and j′ contribute A(β−r α ⋅β)=A(〈α ∨,β〉α) where A∈ℤ≥0[α i ∣i∈I], and β is an inversion of r α . It follows that 〈α ∨,β〉>0. The lemma follows. □

Remark 8

The polynomials \(\xi^{x^{-1}}(r_{\theta})\) appearing in (25) may be computed entirely in the finite Weyl group and finite weight lattice.

Remark 9

In [8, Proposition 2.17], we gave an expression for the non-equivariant part of \(j_{r_{0}}\), consisting of the terms \(j_{r_{0}}^{x}A_{x}\) where ℓ(x)=1=ℓ(r 0). This follows easily from Theorem 6 and the fact [6] that \(\xi^{r_{i}}(w) = \omega_{i}- w\cdot\omega_{i}\), where ω i is the ith fundamental weight.

3.1 Application to quantum cohomology

The equivariant homology Chevalley rule (Theorem 6) may be used to obtain a new formula for some Gromov–Witten invariants for QH T(G/P) where P⊆̷G is a parabolic subgroup.Footnote 2

For this subsection we adopt the notation of [9], some of which we recall presently. Our goal is Proposition 10, which is the equivariant generalization of [9, Prop. 11.2].

Consider the Levi factor of P. It has Dynkin node subset I P ⊂I, Weyl group W P ⊂W, coroot lattice \(Q_{P}^{\vee}\subset Q^{\vee}\), root system R P ⊂R and positive roots \(R_{P}^{+}\). Denote the natural projection Q af→Q by \(\beta\mapsto\overline{\beta}\). Define

Every element w∈W af has a unique expression w=w 1 w 2 with w 1∈(W P)af and w 2∈(W P )af; denote by π P :W af↦(W P)af the map that sends w↦w 1.

Recall that the ring H T (Gr G ) has an S-basis \(\{\xi_{x}\mid x\in W_{\mathrm{af}}^{-}\}\). It has an ideal

The set \(\mathcal{T} = \{\xi_{\pi_{P}(t_{\lambda})} \mid\lambda\in \tilde{Q}\}\) is multiplicatively closed, where \(\tilde{Q}=\{\lambda\in Q^{\vee}\mid\langle\lambda,\alpha_{i}\rangle \le0 \text{for all $i\in I$}\}\) is the set of antidominant elements of Q ∨. Finally let \(\eta_{P}:Q^{\vee}\to Q^{\vee}/Q_{P}^{\vee}\) be the natural projection. Then by [9, Thm. 10.16] there is an isomorphism

$$\varPsi_P: \bigl(H_T(\mathrm{Gr}_G)/J_P\bigr) \bigl[\xi_{\pi_P(t_\lambda)}^{-1} \mid \lambda \in\tilde{Q} \bigr]\cong QH^T(G/P)_{(q)}$$

where (q) denotes localization at the quantum parameters. For \(x\in W_{\mathrm{af}}^{-}\cap(W^{P})_{\mathrm{af}}\) with x=wt λ for w∈W and λ∈Q ∨, we have w∈W P and \(\lambda\in\tilde{Q}\). Then \(\varPsi_{P}(\xi_{x}) = q_{\eta_{P}(\lambda)} \sigma_{P}^{w}\) where \(\sigma_{P}^{w}\) is the quantum Schubert class in QH T(G/P) associated with w∈W P.

Proposition 10

Let w∈W P. Then

Proof

Choose λ∈Q ∨ such that 〈λ,α i 〉=0 for i∈I P and 〈λ,α i 〉≪0 for i∈I∖I P . Then 〈λ,α〉=0 for α∈R P and 〈λ,α〉≪0 for \(\alpha\in R^{+}\setminus R_{P}^{+}\).

We have \(x=wt_{\lambda}\in W_{\mathrm{af}}^{-} \cap(W^{P})_{\mathrm{af}}\) by [9, Lemmata 3.3, 10.1]. Define the set

(26)

Using the characterization of the Schubert basis in Theorem 3, for \(z\in W_{\mathrm{af}}^{-}\) the coefficient of j z in \(j_{r_{0}} j_{x}\) is given by the coefficient of A z in \(j_{r_{0}} A_{x}\). We obtain

(27)

where χ(true)=1 and χ(false)=0. We shall apply the map Ψ P to the above expression. First it is desirable to factor out the dependence of the right hand side on λ.

Suppose u∈W (which holds for u≤r θ ∈W). We claim that \(u\in\mathcal{A}_{x}\) if and only if ℓ(uw)=ℓ(w)−ℓ(u). Suppose \(u\in\mathcal{A}_{x}\). Since \(ux\in W_{\mathrm{af}}^{-}\) we have ℓ(ux)=ℓ(uwt λ )=ℓ(t λ )−ℓ(uw) and ℓ(u)+ℓ(x)=ℓ(u)+ℓ(t λ )−ℓ(w). Since ℓ(ux)=ℓ(u)+ℓ(x) it follows that ℓ(uw)=ℓ(w)−ℓ(u). Conversely suppose ℓ(uw)=ℓ(w)−ℓ(u). Since w∈W P it follows that uw∈W P. In particular \(uw t_{\lambda}\in W_{\mathrm{af}}^{-}\). Therefore ℓ(ux)=ℓ(u)+ℓ(x) and \(u\in \mathcal{A}_{x}\).

Let us fix the assumption that u∈W and ℓ(uw)=ℓ(w)−ℓ(u). Then \(u\in\mathcal{A}_{x}\) and ux∈(W P)af since uw∈W P. One may show that:

  1. (1)

    r 0 ux>ux if and only if (uw)−1⋅θ∈R + and (ux)−1⋅α 0∈ℤ>0 δ−(uw)−1⋅θ.

  2. (2)

    r 0 ux∉(W P)af if and only if \((uw)^{-1}\cdot \theta \in R_{P}^{+}\).

  3. (3)

    \(r_{0} ux \notin W_{\mathrm{af}}^{-}\) if and only if uxα i =α 0 for some i∈I.

It follows that under the assumption on u, \((uw)^{-1}\theta\in R^{+}\setminus R_{P}^{+}\) if and only if r 0 ux>ux, \(r_{0}ux \in W_{\mathrm{af}}^{-}\), and r 0 ux∈(W P)af.

We now apply the map Ψ P . By [9, Remark 10.1] \(r_{0}\in W_{\mathrm{af}}^{-}\cap (W^{P})_{\mathrm{af}}\). Since \(r_{0}= r_{\theta}t_{-\theta^{\vee}}\) we have \(\varPsi_{P}(\xi_{r_{0}}) = q_{\eta_{P}(-\theta^{\vee})} \sigma_{P}^{\pi _{P}(r_{\theta})}\).

By [9, Prop. 10.5, 10.8] π P (w)=w, π P (t λ )=t λ and π P (x)=x. Therefore \(\varPsi_{P}(\xi_{x}) = q_{\eta_{P}(\lambda)}\sigma_{P}^{w}\).

Let 1≠u≤r θ and \(u\in\mathcal{A}_{x}\). It follows that uw∈W P and ux=uwt λ ∈(W P)af. Then \(\varPsi_{P}(\xi_{ux}) = q_{\eta_{P}(\lambda)} \sigma_{P}^{uw}\).

Finally let 1≠u≤r θ be such that \(u\in\mathcal{A}_{x}\), \(r_{0}\in \mathcal{A}_{ux}\), and r 0 ux∈(W P)af. We have \(r_{0} ux = r_{\theta}t_{-\theta^{\vee}} u w t_{\lambda}= r_{\theta}uwt_{\lambda- (uw)^{-1}\theta^{\vee}}\). Therefore \(\varPsi_{P}(r_{0} ux) = q_{\eta_{P}(\lambda-(uw)^{-1}\theta^{\vee})} \sigma_{P}^{\pi_{P}(r_{\theta}uw)}\). Applying Ψ P to (27) yields the required equation. □

4 Alternating equivariant Pieri rule in classical types

We first establish some notation for G=SL n , Sp 2n , and SO 2n+1. Our root system conventions follow [5].

4.1 Special classes

We give explicit generating classes for H T (Gr G ).

4.1.1 \(H_{T}(\mathrm{Gr}_{SL_{n}})\)

Define the elements

(28)
(29)

So \(\ell(\hat{\sigma}_{p})=p-1\) and ℓ(σ p )=p. These elements have associated translations

(30)

4.1.2 \(H_{T}(\mathrm{Gr}_{Sp_{2n}})\)

For 1≤p≤2n−1 we define the elements \(\hat{\sigma}_{p}\in W\) by

For 1≤p≤2n−1 define \(\sigma_{p}\in W_{\mathrm{af}}^{0}\) and t p−1∈W af by

(31)
(32)

4.1.3 \(H_{T}(\mathrm{Gr}_{SO_{2n+1}})\)

For 1≤p≤2n−1 we define the elements \(\hat{\sigma}_{p}\in W_{\mathrm{af}}^{0}\) by

For 1≤p≤2n−1 define \(\sigma_{p}\in W_{\mathrm{af}}^{0}\) by

(33)

For 1≤p≤2n−2 define t p−1∈W af by

(34)

For 1≤p≤2n−1 let \(\sigma'_{p}\) be σ p but with every r 0 replaced by r 1. Then define

Then we conjecture that

(35)

where B is defined in (23). The sign is − for q≤2n−2 and + for q=2n−1.

4.1.4 Special classes generate

Let k′=n−1 for G=SL n and k′=2n−1 for G=Sp 2n or G=SO 2n+1. Let \(\hat{\mathbb{P}}:=S[[j_{\sigma_{m}}\mid1\le m\le k']]\) be the completion of ℙ≅H T (Gr G ) generated over S by series in the special classes. It inherits the Hopf structure from ℙ. The Hopf structure on ℙ is determined by the coproduct on the special classes.

Proposition 11

For G=SL n ,Sp 2n ,SO 2n+1, \(\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{P}\subset\mathbb{Q}\otimes_{\mathbb{Z}}\hat{\mathbb{P}}\).

Proof

It is known that the special classes generate the homology H ∗(Gr G ) non-equivariantly for G=SL n ,Sp 2n ,SO 2n+1 see [11, 14]. Furthermore, the equivariant homology Schubert structure constant \(d_{uv}^{w}\) is a polynomial in the simple roots of degree ℓ(w)−ℓ(u)−ℓ(v), and when ℓ(w)=ℓ(u)+ℓ(v), it is equal to the non-equivariant homology Schubert structure constant. It follows easily from this that each equivariant Schubert class can be expressed as a formal power series in the equivariant special classes. □

Remark 12

For G=SL n and G=Sp 2n the special classes generate H ∗(Gr G ) over ℤ.

4.2 The alternating equivariant affine Pieri rule

Let k=n−1 for G=SL n , k=2n−1 for G=Sp 2n , and k=2n−2 for G=SO 2n+1. Our goal is to compute \(j_{\sigma_{m}}^{x}\) for 1≤m≤k; note that for G=SO 2n+1, the element σ 2n−1 has been treated in (35). For this purpose consider the Bruhat order ideal \(\varOmega= \{ \mathrm{id}= \sigma_{0},\sigma_{1},\dotsc,\sigma_{k} \}\) in \(W_{\mathrm{af}}^{0}\). Since j 0=id, to compute \(j_{\sigma_{p}}^{x}\) for p≥1 we may assume x≠id by length considerations. It suffices to invert the matrix A given in (22) over Ω∖{id}×Ω∖{id}.

Define the matrices \(M_{pm}= (-1)^{m} \xi^{\sigma_{m}}(\sigma_{p})\) for 1≤p,m≤k, \(N_{mq}=\xi^{\hat{\sigma}_{m} r_{\theta}}(\hat{\sigma}_{q} r_{\theta})\) for 1≤m,q≤k, and the diagonal matrix \(D_{pq} = \delta_{pq} \,\xi^{t_{p-1}}(t_{p-1})\) for 1≤p,q≤k.

Conjecture 13

(36)

Conjecture 14

For 1≤m≤k and x≠id we have

(37)

In particular \(j_{\sigma_{m}}^{x}=0\) unless ℓ(x)≥m and x≤t q for some 0≤q≤m−1.

Conjecture 14 follows immediately from Conjecture 13: we have M −1=ND −1, and (37) follows from (24).

Theorem 15

Conjecture 14 holds for G=SL n .

The proof appears in Appendix A. Examples of (36) appear in Appendix B.

5 Effective Pieri rule for \(H_{T}(\mathrm{Gr}_{SL_{n}})\)

The goal of this section is to prove a formula for \(j_{\sigma_{m}}^{x}\) that is manifestly positive. In this section we work with G=SL n , W=S n , and \(W_{\mathrm{af}}=\tilde{S}_{n}\). We first establish some notation. For a≤b write

(38)
(39)
(40)

for upward and downward sequences of reflections and for sums of consecutive roots. In particular we have \(\theta= \alpha_{1}+\alpha_{2}+\dotsm+\alpha_{n-1} =\alpha_{1}^{n-1}\).

5.1 V’s and Λ’s

The support Supp(b) of a word b is the set of letters appearing in the word. For a permutation w, Supp(w) is the support of any reduced word of w. A V is a reduced word (for some permutation) that decreases to a minimum and increases thereafter. Special cases of V’s include the empty word, any increasing word and any decreasing word. A Λ is a reduced word that increases to a maximum and decreases thereafter. A (reverse) N is a reduced word consisting of a V followed by a Λ, such that the support of the V is contained in the support of the Λ. For example, the words 32012, 23521, and 32012453 are a V, Λ, and N, respectively.

By abuse of language, we say a permutation is a V if it admits a reduced word that is a V. We use similar terminology for Λ’s and N’s.

A permutation is connected if its support is connected (that is, is a subinterval of the integers). The following basic facts are left as an exercise.

Lemma 16

A permutation that is a V, admits a unique reduced word that is a V. Similarly for a connected Λ or a connected N.

Lemma 17

A connected permutation is a V if and only if it is a Λ, if and only if it is an N.

5.2 t q -factorizations

For 0≤q≤n−2, we call

(41)

the standard reduced word for t q . Since this word is an N it follows that any x≤t q is an N. We call the subwords \(q(q-1)\dotsm1\), \(12\dotsm(n-2)\) and \((n-2) \dotsm (q+1)\) the left, middle, and right branches.

Lemma 18

If \(x\in\tilde{S}_{n}\) admits a reduced word in which i+1 precedes i for some i∈ℤ/nℤ then \(x\not\le t_{i}\).

Proof

Suppose x≤t i . Since the standard reduced word of t i has all occurrences of i preceding all occurrences of i+1, it follows that x has a reduced word with that property. But this property is invariant under the braid relation and the commuting relation, which connect all reduced words of x. □

Let c(x) denote the number of connected components of Supp(x). If J and J′ are subsets of integers then we write J<J′−1 if max(J)<min(J′)−1. The following result follows easily from the definitions.

Lemma 19

Suppose x≤t q . Then x has a unique factorization \(x = v_{1}\cdots v_{r} y_{1}\* y_{2} \cdots y_{s}\), called the q-factorization, where each v i ,y i has connected support such that

  1. (1)

    Supp(v i )<Supp(v i+1)−1 and Supp(y i )<Supp(y i+1)−1

  2. (2)

    Supp(v 1⋯v r )⊂[0,q]

  3. (3)

    Supp(y 1⋯y s )⊂[q+1,n−1]

  4. (4)

    Each v i is a V

  5. (5)

    Each y i is a Λ.

We say that v r and y 1 touch if q∈Supp(v r ) and q+1∈Supp(y 1). We denote

(42)

Note that ϵ(x,q) depends only on Supp(x) and q.

Each k in the q-factorization of x≤t q , is (S1) in the left branch of some v i , or (S2) in the right branch of some v i , or (S3) at the bottom of a v i , or (S1′) in the left branch of some y i , or (S2′) in the right branch of some y i , or (S3′) at the top of a y i . We call these sets S1, S2, S3, S1′, S2′, and S3′. Note that k can belong to both S1 and S2, or both S1′ and S2′.

For each x and each q such that x≤t q , we define the polynomials

We also define \(R(x,q,m)= \prod_{k\in S2'\cap[m,n-1]}(-\alpha_{q+1}^{k})\).

5.3 The equivariant Pieri rule

Let

(43)

and

(44)

be the root associated with the reflection \(r_{\beta_{i}}\) that exchanges the numbers 1+q i and 1+q i+1. For a root β and f∈S define

Theorem 20

We have

$$j_{\sigma_m}^x = (-1)^{\ell(x)-m+p-1} M(x,q_1)\partial_{\beta_{p-1}} \dotsm\partial_{\beta_2} \partial_{\beta_1}Y(x,m)$$

where \(Y(x,m) = (\alpha_{0}^{q_{1}})^{c(x)-1}R(x,q_{1},m)\).

The proof of Theorem 20 is given in Sect. 6.

5.4 Positive formula

Define \(\tilde{S2}'=S2'\cap[m,n-1]\), and let \(K = \tilde{S2}' \cup\{n-1,\ldots,n-1\} = \{k_{1} \geq k_{2} \geq\cdots\geq k_{d}\}\) be the multiset where the element (n−1) is added to \(\tilde{S2}'\) (c(x)−1) times.

Theorem 21

(45)

where s(i,R)=#{r∈R∣i<r}+1.

The proof of Theorem 21 is given in Sect. 6.

Example 22

Let n=8, m=4, and x=r 0 r 4 r 5 r 7 r 4 r 2 r 1. The components of Supp(x) are [0,2], [4,5], and [7] so that c(x)=3. We have p=3 with (q 1,q 2,q 3)=(0,2,3), v 1=r 0, y 1=r 2 r 1, y 2=r 4 r 5 r 4, y 3=r 7, ϵ(x,q 1)=1, S1=S2=∅, S3={0}, S1′={4}, S2′={1,4}, S3′={2,5,7}, S2′∩[m,n−1]={4}. Thus K={7,7,4}. Then writing \(\alpha_{a}^{b} = x_{a}-x_{b+1}\), and noting that \(\alpha_{0}^{n-1} = 0\), Theorem 20 yields

agreeing with Theorem 21.

6 Proof of Theorems 20 and 21

6.1 Simplifying (37)

Let 0≤q≤m−1. By (14) and Lemma 2 we have

We also have

Define

$$ D(q,m) = \xi^{\sigma_{q+1}}(\sigma_{q+1})\xi^{u_{q+1}^{m-1}} \bigl(u_{q+1}^{m-1} \bigr).$$
(46)

so that by Theorem 15,

(47)

Explicitly we have

(48)
(49)

6.2 Evaluation at t q

Proposition 23

If x≤t q , then

(50)

Proof

We compute ξ x(t q ) using (14) by computing all embeddings of reduced words of x into the standard reduced word (41) of t q . We refer to the q-factorization of x. Each k∈S1 must embed into the left branch of the N, and has associated root \(\alpha_{k}^{q}\). Each k∈S2 embeds into the middle branch of the N and has associated root \(\alpha_{0}^{k-1}\). Each k∈S1′ embeds into the middle branch of the N and has associated root \(\alpha_{0}^{k}\). Each k∈S2′ embeds into the right branch of the N and has associated root \(-\alpha_{q+1}^{k}\). Each k∈S3 is either 0 and has associated root \(\alpha_{0}^{q}\), or can be embedded into the left or middle branch of the N, and the sum of the two associated roots for these positions is \(\alpha_{k}^{q}+\alpha_{0}^{k-1}=\alpha_{0}^{q}\). Each k∈S3′ is either n−1, which has associated root \(-\alpha_{q+1}^{n-1} = \alpha_{0}^{q}\), or can be embedded into the middle or right branch of the N, and the sum of associated roots is \(\alpha_{0}^{k}-\alpha_{q+1}^{k}=\alpha_{0}^{q}\). Since all the various choices for embeddings of elements of S3 and S3′ can be varied independently, the value of ξ x(t q ) is the product of the above contributions. Each minimum of a v i and maximum of a y j contributes \(\alpha_{0}^{q}\). If there is a component of x which contains both q and q+1 (that is, if v r and y 1 touch) then it is unique and contributes two copies of \(\alpha_{0}^{q}\). All this yields (50). □

6.3 Rotations

We now relate ξ x(t q ) with ξ x(t q′). Let r p,q denote the transposition that exchanges the integers p and q.

Proposition 24

Let x≤t q and consider the q-factorization of x. Let a be such that this reduced word of x contains the decreasing subword \((q+a)(q+a-1)\dotsm(q+1)\) but not \((q+a+1)(q+a)\dotsm(q+1)\). If q+1∉Supp(x), then set a=1. Then

(51)

and

(52)

Let y ↑ denote y with every r i changed to r i+1. The following lemma follows easily by induction.

Lemma 25

Let y be increasing with support in [b,a−1]. Then

$$y d^a_b = d^a_by^\uparrow $$

Proof of Proposition 24

We assume that q+1∈Supp(x), for otherwise the claim is easy.

By Lemma 18 we have \(x\not\le t_{q+i}\) for 1≤i≤a−1. Equation (51) follows from (15). We now prove (52). The first goal is to compute the q+a-factorization of x. Since x≤t q we may consider the q-factorization of x. The decreasing word \((q+a-1)\dotsm(q+2)(q+1)\) must embed into the right hand branch, that is, [q+1,q+a−1]⊂S2′. The hypotheses imply that \(q+a\not\in S2'\). There are two cases: either q+a∈S1′ or q+a∈S3′ (so that \(q+a+1\not\in\mathrm{Supp}(x)\)). We treat the former case, as the latter is similar: the two cases correspond to the touching and nontouching cases for the q+a-factorization of x, whose existence we now demonstrate.

Suppose q+a∈S1′. Then there is a \(y_{1}'\) with \(\mathrm{Supp}(y_{1}')\subset[q+a+1,n-1]\) and a y with an increasing reduced word such that Supp(y)⊂[q+1,q+a−1] and \(y_{1} = y r_{q+a} y_{1}' d^{q+a-1}_{q+1}= y d^{q+a}_{q+1} y_{1}'\). Suppose v r and y 1 touch. Then \(v_{r}':=v_{r} y d^{q+a}_{q+1}\) is an N and therefore a V. Moreover x≤t q+a since x has a q+a-factorization given by the q-factorization of x but with v r and y 1 replaced by \(v_{r}'\) and \(y_{1}'\), respectively. To verify that \(v_{r}'\) is a V, by the touching assumption, q∈Supp(v r ) and we have \(v_{r}' = v_{r} y d^{q+a}_{q+1} = v_{r} d^{q+a}_{q+1} y^{\uparrow}=d^{q+a}_{q+2} v_{r} r_{q+1} y^{\uparrow}\) which expresses \(v_{r}'\) in a V.

Suppose v r and y 1 do not touch, that is, q∉Supp(v r ). We have the V given by \(v'_{r+1} = y d^{q+a}_{q+1} = d^{q+a}_{q+1}y^{\uparrow}\). Then x≤t q+a , as x has the q+a factorization given by the q-factorization of x except that there is a new V, namely, \(v'_{r+1}\) and the first y is \(y_{1}'\) instead of y 1.

In every case we calculate that

The calculation for L and R follows from the fact that [q+2,q+a]⊂S1 q+a , but \([q+1,q+a-1]\subset S2'_{q}\). The calculation for M follows from the fact that Supp(y)⊂S2 q and Supp(y ↑)⊂S2 q+a , together with the following boundary cases:

If q+a+1∈Supp(x) then \(q+a \in S1_{q+a}\cap S1'_{q}\). Thus q+a contributes a factor of \(\alpha_{0}^{q+a}\) to M(x,q). This factor appears in M(x,q+a) as the factor \((\alpha_{0}^{q+a})^{\epsilon(x,q+a)}\), since ϵ(x,q+a)=1.

If q∈Supp(x) one has ϵ(x,q)=1 and q+1∈S2 q+a contributes a factor of \(\alpha_{0}^{q}\) to M(x,q+a). This factor appears in M(x,q) as the factor \((\alpha_{0}^{q})^{\epsilon(x,q)} = \alpha_{0}^{q}\).

Using that \(d^{q+a}_{q+1} \alpha_{0}^{q} = \alpha_{0}^{q+a}\), \(d^{q+a}_{q+1}(-\alpha_{q+1}^{q+a})=\alpha_{q+a}\), and \(r_{1+q,1+q+a}\alpha_{q+1}^{q+a} = -\alpha_{q+1}^{q+a}\), the above relations between M(x,q), L(x,q), R(x,q) and their counterparts for q+a, together with Proposition 23, yield

$$\xi^x(t_{q+a})= \bigl(\alpha_{q+1}^{q+a}\bigr)^{-1} M(x,q)\, d^{q+a}_{q+1} \, \bigl(-\alpha_{q+1}^{q+a} \bigr) \bigl(\alpha_0^q\bigr)^{c(x)}\; L(x,q)\;R(x,q).$$

To obtain (52), since \(r_{1+q,1+q+a} = d^{q+a}_{q+1}u_{q+2}^{q+a}\), it suffices to show that

However, it is clear that \(\alpha_{0}^{q}\) and L(x,q) are invariant, and the only part of R(x,q) that must be checked is the product ∏ k∈S2′∩[q+1,q+a](−α q+1,k ). However, we have S2′∩[q+1,q+a]=[q+1,q+a−1], and indeed the product \(\prod_{k=q+1}^{q+a} (-\alpha_{q+1}^{k})\) is invariant under \(u_{q+2}^{q+a}\), as required. □

Recall the definition of q j from (43). In light of the proof of Proposition 24, we write

(53)

Recall the definition of β i from (44). For i≤j we also define

Let

(54)

so that \(Y_{i}(x,m) = r_{\beta_{i-1}} Y_{i-1}(x,m)\).

Recall the definitions of D(q,m) and Y i (x,m) from (46).

Lemma 26

$$(-1)^{m-1-q_j-p+j} \frac{\xi^x(t_{q_j})}{D(q_j,m)}= \frac{M(x)Y_j(x,m)}{(\beta_1^{j-1}\beta_2^{j-1}\dotsm\beta_{j-1}^{j-1})(\beta_j^j\beta_j^{j+1}\dotsm\beta_j^{p-1})}.$$

Proof

The proof proceeds by induction on j. Let D j be the denominator of the right hand side. Suppose first that j=1. Consider the embedding of x into \(t_{q_{1}}\). By the definition of q 1, it follows that \(L(x,q_{1}) \alpha_{0}^{q_{1}} = \xi^{\sigma _{q_{1}+1}}(\sigma_{q_{1}+1})\). By the definition of the q j , we also have \(S2' \cap[q_{1}+1,m-1] =[q_{1}+1,m-1] \setminus\{q_{2},q_{3},\dotsc,q_{p}\}\). These considerations and Proposition 23 imply that

This proves the result for j=1. Suppose the result holds for 1≤j≤p−1. We show it holds for j+1. By induction we have

Proposition 24 yields

It remains to show

We have \(D(q_{j},m)=\prod_{k=0}^{q_{j}} \alpha_{k}^{q_{j}} \prod_{k=q_{j}+1}^{m-1} \alpha_{q_{j}+1}^{k}\). For k∈[0,q j ] we have \(r_{\beta_{j}} \alpha_{k}^{q_{j}} = \alpha_{k}^{q_{j+1}}\). For k∈[q j +1,q j+1−1] we have \(r_{\beta_{j}} \alpha_{q_{j}+1}^{k} = -\alpha_{k+1}^{q_{j+1}}\), \(r_{\beta_{j}} \alpha_{q_{j}+1}^{q_{j+1}} = -\alpha_{q_{j}+1}^{q_{j+1}}\), and for k∈[q j+1+1,m−1] we have \(r_{\beta_{j}} \alpha_{q_{j}+1}^{k} =\alpha_{q_{j+1}+1}^{k}\). Therefore

We also have \(r_{\beta_{j}} \beta_{j-1}^{i} = \beta_{j}^{i}\) for 1≤i≤j−1 and \(r_{\beta_{j}} \beta_{j}^{i} = \beta_{j+1}^{i}\) for j+1≤i≤p−1. Therefore

 □

The following result is immediate from the definitions.

Lemma 27

\(r_{\beta_{j}} Y_{i}(x,m) = Y_{i}(x,m)\) for j≥i+2.

6.4 Proof of Theorem 20

Note that if \(r_{\beta_{j+1}} Y = Y\) and i≤j then

$$\frac{1}{\beta_{j+1}}(1-r_{\beta_{j+1}})\frac{Y}{\beta_i^j} = \frac {Y}{\beta_i^j\beta_i^{j+1}}.$$

So using Lemma 27 we have

Thus

by (47), as required. □

6.5 Proof of Theorem 21

We first count the gratuitous negative signs in M(x)=M(x,q 1) and Y(x,m). Letting q=q 1, using the q 1-factorization of x, and recalling that \(\tilde{S2}'=S2'\cap[m,n-1]\), this number is

Therefore all signs cancel and we have

(55)

Let x i be the standard basis of the finite weight lattice ℤn with α i =x i −x i+1. Then \(r_{\beta_{j}}\) acts by exchanging \(x_{q_{j}+1}\) and \(x_{q_{j+1}+1}\). Let us write

where \(n-1 \ge k_{1} \ge k_{2} \ge\dotsm\ge k_{d} \ge m\). Note that q j +1≤q p +1≤m. Since

$$\partial_i \cdot(fg) = (\partial_i\cdot f)g +(r_i\cdot f) (\partial_i \cdot g),$$

and since ∂ i 1=0, we have

So \(\partial_{\beta_{1}}\) can act on any factor (giving the answer 1 and thus effectively removing the factor), and to the left each variable \(x_{q_{1}+1}\) is reflected to \(x_{q_{2}+1}\). Next we apply \(\partial_{\beta_{2}}\). It kills any factor \(x_{q_{1}+1}-x_{k_{i}+1}\). Therefore we may assume it acts on a factor of the form \(x_{q_{2}+1}-x_{k_{i}+1}\) which is to the left of the factor removed by \(\partial_{\beta_{1}}\). Continuing in this manner we see that \(\partial_{\beta_{p-1}}\dotsm \partial_{\beta_{1}} Z\) is the sum of products of positive roots, where a given summand corresponds to the selection of p−1 of the factors, which are removed, and between the rth and r+1th removed factor from the right, an original factor \(x_{q_{1}+1}-x_{k_{i}+1}\) is changed to \(x_{q_{r+1}+1}-x_{k_{i}+1}\).

It follows that Theorem 20 yields Theorem 21. □