Equivariant Schubert calculus

c 2009 by Institut Mittag-Leffler. All rights reserved

Equivariant Schubert calculus

Letterio Gatto and Ta´ıse Santiago

Abstract. We describeT-equivariant Schubert calculus onG(k, n),Tbeing ann-dimensio- nal torus, through derivations on the exterior algebra of a freeA-module of rankn, whereAis theT-equivariant cohomology of a point. In particular, T-equivariant Pieri’s formulas will be determined, answering a question raised by Lakshmibai, Raghavan and Sankaran (Equivariant Gi- ambelli and determinantal restriction formulas for the Grassmannian,Pure Appl. Math. Quart.2 (2006), 699–717).

1. Introduction

In this paper we deal with G(k, n), the complex Grassmannian variety para- meterizing k-dimensional vector subspaces of Cⁿ, equipped with a certain linear action of an n-dimensional torus T (either (S¹)ⁿ or (C^∗)ⁿ) such that the fixed point locus is isolated. It turns out that H_T^∗(G(k, n)), the integral T-equivariant cohomology ring of G(k, n), is a finite free module over the ring A:=H_T^∗(pt), the T-equivariant cohomology of a point. Our main result (Theorem 2.2) shows that the multiplicative structure of the A-algebra H_T^∗(G(k, n)), for all 1≤k≤n, can be described through derivations on the Grassmann algebra of a freeA-module of rank n, in the same spirit as [2], [5], [6] and [7]. We stress that the description of the product structure ofH_T^∗(G(k, n)) heavily depends on the particular choice of anA- basis for it. Ifσ:=(σI) is any suchA-basis, whereIruns over a certain finite index set, the product structure ofH_T^∗(G(k, n)) concerns the structure constantsC_IJ^K(σ) in the product expansion σIσJ=

KC_IJ^K(σ)σK. The notation reflects the depen- dence of the structure constants on the basisσ. As the consideredT-action on the Grassmannian has only isolated fixed points, it turns out thatG(k, n) can be decom- posed into the disjoint union ofT-invariant complex cells{B_I}. A natural A-basis

Work partially sponsored by PRIN “Geometria sulle Variet`a Algebriche” (Coordinatore Alessandro Verra), INDAM-GNSAGA, Scuola di Dottorato (ScuDo) del Politecnico di Torino, FAPESB proc. n. 8057/2006 and CNPq proc. n. 350259/2006-2.

Y:={YI} forH_T^∗(G(k, n)) is then obtained by letting YI be the equivariant coho- mology class of the closure ofBI. Such a basisY is used in the important work [9], by Knutson and Tao, which sheds light on the deep relationship between the beau- tiful combinatorics of puzzles and the product structure of the ring H_T^∗(G(k, n)).

The main result of [9], as the authors put it, is that “puzzles compute equivariant Schubert calculus”, i.e., more precisely, the constant structuresC_IJ^K(Y)∈A. Alter- natively we may phrase our main result by saying thatderivations on a Grassmann algebra compute (equivariant)Schubert calculus. Still using the basis Y, Mihalcea achieves, in a couple of important papers ([16] and [17]), a detailed description of the equivariant small quantum deformation ofH_T^∗(G(k, n)). In particular (see [17]) he produces a quantum equivariant Giambelli’s formula, exploiting the combinatorics of the factorial Schur functions. In [10], instead, the authors aim to a formulation ofT-equivariant Schubert calculus closer to the classical one, based on “equivariant versions” of Giambelli’s and Pieri’s formulas. Indeed they prove a determinantal formula for the restriction to a torus fixed point of each elementY_I of the basis Y. An elegant, new and illuminating approach is that pursued by Laksov ([12]; see also [11]): based on a previous joint work with Thorup [13], [14], he shows that the T-equivariant Schubert calculus on G(k, n) is recovered by the unique symmetric structure on thekth exterior power ofH_T^∗(Pⁿ⁻¹), the quotient of the ringA[X] by a certain monic polynomial of degreen.

Our work is mostly related to Laksov’s, but we put the emphasis on the exterior algebra ofA[X] rather than on a single exterior power. In our formulation, Schubert calculus takes place in a subring A^∗(

M) of the ring of endomorphisms of the exterior algebra of a freeA-moduleM of rankn, as in [7], generated by derivations of

M. The (equivariant) Schubert calculus onG(k, n) is obtained by considering theA^∗(

M)-module structure ofk

M, the degree-kdirect summand of M. The paper is organized as follows. We put our main result at the end of Section2, whose first part lists the basics we believe are necessary to keep the paper as self-contained as possible. In Section3we write down a full set of “equivariant”

Pieri’s formulas (proposing a solution, analogous to Laksov’s one, for the question raised in [10]) recovering, in particular, the equivariant Pieri’s rule as in [9], p. 237.

As most expositions on (equivariant) Schubert calculus do, including [9] and [16], we give a quick look, at the end of the paper, to the example of G(2,4), the first Grassmannian which is not a projective space. We revisit the example at p. 231 of [9], depicting the basis of the equivariant cohomology of G(2,4), using solely Leibniz’s rule and a bit of integration by parts (if one wants to avoid the general Giambelli’s formula due to Laksov and Thorup [13], Theorem 0.1 (2)).

Acknowledgements. Part of this work is contained in the Doctoral thesis [18]

of the second author, who wishes to thank all the supporting institutions, especially the Doctoral School (ScuDo) of Politecnico di Torino and the Universidade Estadual de Feira de Santana (UEFS) for support during the preparation of this work. We also thank the valuable comments and suggestions of the referee, who helped us to improve the shape of this paper. Finally, we are very deeply indebted to D. Laksov and A. Thorup for sharing many ideas with us. The former, especially, helped us not only with substantial remarks but also with constant encouragement.

2. Preliminaries and notation 2.1. T-equivariant cohomology

Let T be an n-dimensional torus, either T=(S¹)ⁿ or T=(C^∗)ⁿ. The ring A:=H_T^∗(pt), the integral equivariant cohomology of a point, coincides with the polynomial ring Z[y1, ..., yn]. Let Ai:=H_Tⁱ(pt). Then, for eachi≥0,A2i+1=0 and A2i is the Z-submodule of A generated by the monomials of degree i. Thus A=

i≥0A_i. We apply the theory exposed in [7] to the T-equivariant cohomology of G(k, n), in the case that theT-action is induced by the diagonal linear action

(t₁, ..., t_n)(z₁, ..., z_n) = (t₁z₁, ..., t_nz_n)

of T on Cⁿ. If F is an equivariant vector bundle over G(k, n), denote byc^T(F) its equivariant Chern polynomial

i≥0c^T_i (F)tⁱ, where, for each i≥0,c^T_i(F) is the T-equivariantith Chern class of F. All the bundles occurring in the tautological exact sequence overG(k, n), 0→Sk→Fk→Qk→0, are equivariant and then it makes sense to speak of their equivariant Chern classes. Setc^T(Qk−Fk)=c^T(Qk)/c^T(Fk), the ratio taken into the ring of formal power series with coefficients inH_T^∗(G(k, n)).

The equality

c^T(Qk−Fk) =

i≥0

ci(Qk−Fk)tⁱ definesci(Qk−Fk) (see [4] for further foundational details).

As is well known, H_T^∗(Pⁿ⁻¹):=H_T^∗(G(1, n)) is a free A-module of rank n, spanned by 1, ξ, ..., ξⁿ⁻¹, where ξ:=c^T₁(O_Pⁿ−1(1)), the first T-equivariant Chern class of the hyperplane bundle. Letpbe the minimal polynomial of the “multiplying- by-ξ” endomorphism ofH_T^∗(Pⁿ⁻¹), i.e. p∈A[X] is the monic polynomial of minimal degree such thatp(ξ)=0 inH_T^∗(Pⁿ⁻¹). Then, obviously, the mapA[X]→H_T^∗(Pⁿ⁻¹), sendingX→ξ, is an epimorphism with kernel p, inducing an isomorphism

ι₁:A[X]/p−→H_T^∗ Pⁿ⁻¹

.

2.2. Schubert calculus on a Grassmann algebra

LetM:=XA[X] and M(p)=M/pM. The quotient M(p) is a free A-module generated byεⁱ:=Xⁱ+pM,i≥1: in particular anA-basis ofM(p) isε:=(ε¹, ..., εⁿ).

Furthermore, M(p) is a free A[X]/pmodule of rank 1 generated by ε¹. For each h≥0 define anA-endomorphisms_hof M(p) through the equality

shε^j=

X^h+p X^j+p

=X^h+j+p=ε^i+j,

for each j≥1 (in the equalities aboveX^h+pis seen as an element ofA[X]/p). Let k be a positive integer and letIn^k be the set of all strictly increasing sequences of positive integers not bigger thann. Thenk

M(p), thekth exterior power ofM(p), is freely generated overAbyk

ε:={I

ε|I∈In^k}where, ifI:=(i1, ..., ik)∈In^k, the notationI

εstands forεⁱ¹∧...∧ε^ik. Let

M(p):=

k

k

M(p) be the exterior algebra of M(p). Identify M(p) with the submodule 1

M(p) of

M(p). By [7], Proposition 2.7, there exists a unique sequence (D0, D1, ...) of A-endomorphisms of

M(p) such that, for each α, β∈

M(p), eachh≥0 andi≥1, (thehth order) Leibniz’s rule

(1) D_h(α∧β) =

h1 +h2 =h h1,h2≥0

D_h1α∧D_h2β

holds and the initial conditionsD_h|

M(p)=s_hare satisfied. If one defines Dt:=

i≥0

Ditⁱ:

M(p)−→

M(p)[[t]],

equation (1) can be more compactly phrased by saying that Dt is an A-algebra homomorphism ([5], [6], [7], [18]), i.e., more explicitly, that

Dt(α∧β) =Dtα∧Dtβ.

2.3. The Poincar´e isomorphism

IfA[T]:=A[T1, T2, ...] is the polynomial ring in infinitely many indeterminates, let A^∗(

M(p)) be its image in EndA(

M(p)) via the natural evaluation map T_i→D_i. It is a commutativeA-subalgebra of End_A(

M(p)). Let

ev_ε1∧...∧ε^k:A^∗ M(p) −→

k

M(p)

be the A-module homomorphism given by P(D)→P(D)ε¹∧...∧ε^k, P∈A[T].

By [7], the map ev_ε1∧...∧ε^k is surjective. This can be seen either by integration by parts(as in [7]) or by noticing, as in [13], that Giambelli’s formula

(2) I

ε= Δ_I(D)·ε¹∧...∧ε^k

holds for eachI∈In^k, where ifI=(i₁, ..., i_k) then Δ_I(D):=det(D_ij−i)∈A^∗( M(p)).

There is anA-module isomorphism (Poincar´e isomorphism) Πk:A^∗

k

M(p)

−→ k

M(p), whereA^∗(k

M(p)):=A^∗(

M(p))/ker ev_ε1∧...∧ε^k. For eachA-basisν=(ν¹, ..., νⁿ) ofM(p) and eachI∈In^k, letG^ν_I(D)=Π⁻_k¹(I

ν). ThenG^ν(D):={G^ν_I(D)|I∈In^k}is anA-basis of the ringA^∗(k

M(p)). A representative inA^∗(

M(p)) ofG^ν_I(D) can be explicitly written via the general determinantal formula by Laksov and Thorup ([13], Theorem 0.1 (2)). In particular, ifν=ε, one may takeG^ε_I(D) as the class of ΔI(D) modulo ker ev_ε1∧...∧ε^k.

2.4. Schubert calculus on an exterior power

LetC_IJ^K(ν)∈Abe defined by the equalityG^ν_I(D)G^ν_J(D)=

K∈In^kC_IJ^K(ν)G^ν_K(D) for each triple I, J, K∈In^k. Since each G^ν_I(D) is a polynomial expression in (D₁, D2, ...), it turns out that {C_IJ^K(ν)} can be computed once one knows a ν-Pieri’s formula, i.e. a rule to expand the product DhG^ν_I(D) as a linear combination of elements ofG^ν(D). Notice that ifDhG^ν_I(D)=

JC_h,I^J (ν)G^ν_J(D), then Dh

I

ν

=DhG^ν_I(D)ν¹∧...∧ν^k=

J

C_hI^J (ν)G^ν_J(D)ν¹∧...∧ν^k

=

J

C_hI^J(ν)·J

ν

i.e. aν-Pieri’s formula in the ringA^∗(k

M(p)) amounts to knowing the expansion ofD_h(I

ν) as a linear combination of the elements of the basisk

ν. IfI, H∈N^k, denote byI+H their componentwise sum. When ν=ε, Pieri’s formula looks the same as that holding in the classical (i.e. non-equivariant) case (see e.g. [5]).

Theorem 2.1.(ε-Pieri’s formula ([5] and [13])) For each h≥0 and eachI:=

(i1, ..., ik)∈In^k,

(3) D_hI

ε=

H∈P(I,h)

I+H

ε,

where P(I, h) denotes the set of all H:=(h1, ..., hk)∈N^k such that h1+...+hk=h and

1≤i1≤i1+h1< i2≤i2+h2< ...≤ik−1+hk−1< ik. Abusing notation, denote by the same symbol both ΔI(D)∈A^∗(

M(p)) and its class modulo ker(ε¹∧...∧ε^k) (cf. Section 2.3). Then, by (2), {ΔI(D)|I∈In^k} is an A-basis of A^∗(k

M(p)) and the A-linear extension of the map sending Δ_I(D)→Δ_I(c^T(Q_k−F_k)) defines anA-module homomorphism

(4) ιk:A^∗

k

M(p)

−→H_T^∗(G(k, n)).

Our main result, below, relates derivations on a Grassmann algebra to the equivari- ant cohomology ofG(k, n).

Theorem 2.2. The A-module homomorphism (4) is an A-algebra isomor- phism.

Proof. LetδI=ΔI(c^T(Qk−Fk)). Using well-known approximations of the equi- variant cohomology of G(k, n) by the ordinary cohomology of finite-dimensional spaces [1], [4], one easily shows that δI is indeed an A-basis of H_T^∗(G(k, n)) and that the same kind of Pieri’s rule as displayed in [3], p. 266 (by replacing partitions of length≤k with elements ofIn^k) holds in the equivariant case:

c^T_h(Qk−Fk)∪δI=

H∈P(I,h)

δI+H.

We are hence left to prove that ιk(ΔI(D)·ΔJ(D))=ιk(ΔI(D))∪Tιk(ΔJ(D)) (∪T denotes theT-equivariant cup product). It suffices to check the equality when the first factor is of the formDh(corresponding toI:=(1, ..., k−1, k+h)). One has

ιk(DhΔI(D)) =ιk

H∈P(I,h)

ΔI+H(D)

=

H∈P(I,h)

δ_I+H=c^T_h(Q_k−F_k)∪Tδ_I=ι_k(D_h)∪Tι_k(Δ_I(D)),

and the proof is complete.

3. Equivariant Pieri’s formulas 3.1. Equivariant cohomology ofPⁿ⁻¹

LetA:=H_T^∗(pt)=Z[y1, ..., yn] be theT-equivariant cohomology ring of a point.

IfP is aT-fixed point ofPⁿ⁻¹, let{P}→Pⁿ⁻¹be the canonical inclusion. It induces an injective map ([9], p. 232)

(5) H_T^∗

Pⁿ⁻¹

−→

T-fixed points ofPⁿ⁻¹

H_T^∗(P).

Let A^⊕n be the direct sum of n copies of A, a ring with respect to the natural componentwise product structure. By [9], p. 230, H_T^∗(Pⁿ⁻¹) can be identified with the subring ofA^⊕n satisfying certain Goresky–Kottwitz–MacPherson (GKM) con- ditions [8]. For each 1≤i≤nletY_i:=y_i−y₁ (in particularY₁=0), and define monic polynomialspi∈A[X] by settingp0=1 andpi=i

j=1(X−Yj), if 1≤i≤n. The poly- nomialpn will simply be denoted byp. Let:

S1=

⎛

⎜⎝ Y₁

...

Yn

⎞

⎟⎠∈A^⊕n

andSi=pi(S1). An easy check shows thatS0=1_A⊕n,S1, ...,Sn−1 are A-linearly independentn-tuples of polynomials. All of them satisfy the GKM conditions (as in [9], p. 230). SinceH_T^∗(G(1, n)) is free of rankn overA, they spanH_T^∗(G(1, n)), identified with the image of the map (5). Furthermore H_T^∗(G(1, n)) is generated byS1 as anA-algebra, and becausepis theA-minimal polynomial ofS1, one has indeed the isomorphismH_T^∗(G(1, n))∼=A[S₁]/p. Let M(p) be as at the beginning of Section 2.2. Then ι1:A^∗(M(p))−→H_T^∗(G(1, n)), defined by D1→S1, is a ring isomorphism. It follows thatμ:=(μ¹, ..., μⁿ) defined by (notation as in Sections2.3 and2.4)

(6) μⁱ:= Π₁¨ι⁻¹(S_i−1) =X·pi−1+pM∈M(p)

is anA-basis ofM(p). Our next task is to explicitly write down the (equivariant) Pieri’s formulas with respect to the basisμ(cf. Section 2.4). To this purpose, first notice that

(7) D₁μ^j=X(Xp_j−1)+pM= (X−Y_j)Xp_j−1+Y_jXp_j−1+pM=μ^j+1+Y_jμ^j. We also need the following result.

Lemma 3.1. For all i>0,

(8) Di(μ^j) =D₁ⁱ(μ^j) = i l=0

hi−l(Yj, ..., Yj+l)μ^j+l, 1≤j≤n,

whereh₀=1and,for eachm≥1,h_m(X₁, ..., X_k)is the complete homogeneous sym- metric polynomial in X1, ..., Xk of degreem [15].

Proof. The proof is by induction over the integer i. Ifi=1, one has, by (7) (9) D₁μ^j=μ^j+1+Y_jμ^j=μ^j+1+h₁(Y_j)μ^j.

Suppose that the formula holds fori≥1. SinceDiμ^j=D1Dⁱ₁⁻¹μ^j, by the inductive hypothesis

(10) D_i(μ^j) =D₁ i−1

l=0

h_i−l−1(Y_j, ..., Y_j+l)μ^j+l

=

i−1

l=0

h_i−l−1(Y_j, ..., Y_j+l)D₁μ^j+l. Using (9), the last member of (10) is thence equal to

i−1

l=0

hi−l−1(Yj, ..., Yj+l)(μ^j+1+h1(Yj)μ^j)

=

i−1

l=0

hi−l−1(Yj, ..., Yj+l)μ^j+l+1+

i−1

l=0

hi−l−1(Yj, ..., Yj+l)h1(Yj+l)μ^j+l

= i l=1

hi−l(Yj, ..., Yj+l−1)μ^j+l+

i−1

l=0

hi−l−1(Yj, ..., Yj+l)h1(Yj+l)μ^j+l

=h0(Yj, ..., Yj+i−1)μ^j+i+

i−1

l=1

hi−l(Yj, ..., Yj+l−1)μ^j+l

+hi−1(Yj)h1(Yj)μ^j+

i−1

l=1

hi−l−1(Yj, ..., Yj+l)h1(Yj+l)μ^j+l

=h0(Yj, ..., Yj+i−1)μ^j+i+hi(Yj)μ^j +

i−1

l=1

(h_i−l(Y_j, ..., Y_j+l−1)+h_i−l−1(Y_j, ..., Y_j+l)h₁(Y_j+l))μ^j+l

=h0(Yj, ..., Yj+i−1)μ^j+i+hi(Yj)μ^j+

i−1

l=1

hi−l(Yj, ..., Yj+l)μ^j+l,

i.e.Di(μ^j)=i

l=0hi−l(Yj, ..., Yj+l)μ^j+l, 1≤j≤n, as desired.

Let I∈In^k, I

μ∈k

M(p) and l≥0. Our next task is to get an expression for Dl(I

μ). It will be an A-linear combination of elements of the basis k

μ of k

M(p). Predicting which terms cancel because of the skew-symmetry of the wedge product, will give us aμ-Pieri’s formula (see Theorem3.2below). Leibniz’s rule forDl (cf. (1)) and an easy induction onk≥1 gives

Dl I

μ

=

l1 +...+lk=l li≥0

Dl1μ^l1∧...∧Dlkμ^lk.

Using (8),

l1 +...+lk=l li≥0

Dl1μ^l1∧...∧Dlkμ^lk

=

l1 +...+lk=l li≥0

l₁ m1=0

h_l1−m1(Y_i1, Y_i1+1, ..., Y_i1+m1)μ^i1+m1

∧...∧ lk

m_k=0

h_lk−mk(Y_ik, Y_ik+1, ..., Y_ik+mk)μ^ik+mk

.

Expanding the wedge products, the last member can be written as

=

l1 +...+lk=l li≥0

l1+...+l _k

m1 +...+mk=0 mi≥0

k j=1

hl_j−m_j(Yi_j, Yi_j+1, ..., Yi_j+m_j)μ^i1+m1∧...∧μ^ik+mk

=

l

m1 +...+mk=0

mi≥0 l1 +...+lk=l li≥0

k j=1

hl_j−m_j(Yi_j, Yi_j+1, ..., Yi_j+m_j)

μ^i1+m1∧...∧μ^ik+mk

=

l

m1 +...+mk=0 mi≥0

h_l

−k

j=1m_j(Yi₁, ..., Yi₁+m₁, ..., Yi_k, ..., Yi_k+m_k)μ^i1+m1∧...∧μ^ik+mk. The last equality uses basic properties of the complete symmetric polynomials (see e.g. [15]). Puttingu=l−k

j=1mj, one finally has D_l(μⁱ¹∧...∧μ^ik) =

l u=0

m1 +...+mk+u=l mi≥0

h_u(Y_i1, ..., Y_i1+m1, ..., Y_ik, ..., Y_ik+mk) (11)

×μ^i1+m1∧...∧μ^ik+mk.

Our aim is now to rewrite the right-hand side of (11) keeping into account the cancellations due to the alternating feature of the∧-product.

Theorem 3.2. (See also [12].) The following T-equivariant Pieri’s formula holds:

(12) Dl I

μ

= l u=0

M∈P(I,l−u)

hu(Yi₁, ..., Yi₁+h₁, ..., Yi_k, ..., Yi_k+h_k)·I+M

μ,

whereM:=(m1, ..., mk)∈P(I, l−u)is as in Theorem 2.1.

Proof. (Cf. [5], Theorem 2.4.) The proof is by induction over the integerk. For k=1, formula (12) is trivially true. The first case where the alternating behavior of the wedge product comes into play, causing cancellations in the expansion (11), is k=2. This case we analyze directly. For eachl≥0, let us split the sum (12) as

Dl(μⁱ¹∧μⁱ²) = l u=0

m1 +m2 =l−u mi≥0

hu(Yi1, ..., Yi1+m1, Yi1, ..., Yi2+m2)μ^i1+m1∧μ^i2+m2

=U+U , where

U= l u=0

m1 +m2 =l−u mi≥0 i1 +m1<i2

hu(Yi₁, ..., Yi₁+m₁, Yi₂, ..., Yi₂+m₂)μ^i1+m1∧μ^i2+m2

and U=

l u=0

m1 +m2 =l−u mi≥0 i1 +m1<i2

hu(Yi₁, ..., Yi₁+m₁, Yi₂, ..., Yi₂+m₂)μ^i1+m1∧μ^i2+m2.

We contend thatU vanishes. In fact, on the finite set of all integersi2−i1≤a≤l−u, define the bijectionγ(a)=i₂−i₁+l−u−a. Then

2U= l u=0

l−u

m₁=i₂−i₁

h_u(Y_i1, ..., Y_i1+m1, Y_i2, ..., Y_{i2+l−u−m1})μ^i1+m1∧μ^{i2+l−u−m1}

+ l u=0

l−u

m1=i2−i1

hu(Yi₁, ..., Y_i1+γ(m1), Yi₂, ..., Y_{i2+l−u−γ(m1)})

×μ^i1+γ(m1)∧μ^{i2+l−u−γ(m1)}

=− l u=0

l−u

m₁=i₂−i₁

hu(Yi₁, ..., Yi₁+m₁, Yi₂, ..., Yi₂+l−u−m₁)μ^{i2+l−u−m1}∧μ^i2+m1

+ l u=0

l−u

m1=i2−i1

hu(Yi₁, ..., Yi₁+m₁, Yi₂, ..., Yi₂+l−u−m₁)μ^{i2+l−u−m1}∧μ^i1+m1

= 0.

HenceU=0 and (12) holds fork=2. Suppose now that (12) holds for all 1≤k≤k−1.

Then, for eachl≥0,

(13) Dl(μⁱ¹∧μⁱ²∧...∧μ^ik) =

l_k+lk=l

D_l

k(μⁱ¹∧...∧μ^ik−1)∧Dl_kμ^ik. By the inductive hypothesis, the right-hand side of (13) is equal to (14)

l_k

u=0

Hk−1(u)∧

l_k

m_k=0

hl_k−m_k(Yi_k, Yi_k+1, ..., Yi_k+m_k)μ^ik+mk,

where for notational brevity we setI:=(i₁, ..., i_k−1), M=(m₁, ..., m_k−1) and Hk−1(u) =

M∈P(I,l_k−u)

h_u(Y_i1, ..., Y_i1+m1, ..., Y_ik−1, ..., Y_{ik−1+mk−1})I+M

μ.

But now (14) can be equivalently written as (15)

l−l u=0

Hk−2(u)∧D_l(μ^ik−1∧μ^ik), where

Hk−2(u) =

M∈P(I,l−l”−u)

hu(Yi₁, ..., Yi₁+m₁, ..., Yi_k−2, ..., Yi_k−2+m_k−2)

×μ^i1+m1∧...∧μ^{ik−2+mk−2},

withM=(m₁, ..., m_k−2). The inductive hypothesis implies that Dl(μ^ik−1∧μ^ik)

=

l

u=0

mk−1 +mk=l−u ik−1 +mk−1<ik

hu(Yi_k−1, ..., Yi_k−1+m_k−1, Yi_k, ..., Yi_k+m_k)μ^{ik−1+mk−1}∧μ^ik+mk,

which, substituted into (15), yields precisely (12).

Example3.3. (i) The coefficient ofμ²∧μ³∧μ⁷, in the expansion of D3(μ²∧μ³∧μ⁵),

is

h1(Y2, Y3, Y5, Y6, Y7) =Y2+Y3+Y5+Y6+Y7=y2+y3+y5+y6+y7−5y1. (ii) Forl=1, Pieri’s formula (12) reads as

D1(μⁱ¹∧...∧μ^ik) = k j=1

μⁱ¹∧...∧μ^ij−1∧μ^ij+1∧μ^ij+1∧...∧μ^ik (16)

+ (Y_i1+...+Y_ik)μⁱ¹∧...∧μ^ik,

that is the coefficient of μⁱ¹∧...∧μ^ik in the expansion of D1(μⁱ¹∧...∧μ^ik) is Yi₁+ ...+Yi_k.

3.2. The “equivariant Pieri’s rule”

Theequivariant Pieri’s rule stated in [9], p. 237, can be inferred as follows.

Let 1≤k≤n be a fixed integer and _n

k

the set of all functions a_i1...ik:{1, ..., n}→

{0,1}, 1≤i1<...<ik≤n, such that ai1...i_k(j)=0 if j∈{i1, ..., ik} and ai1...i_k(j)=1 otherwise. Eachλ∈_n

k

can also be represented by a stringλ(1)λ(2)...λ(n) of zeros and ones only (as in [9]). Ifλ=a_i1...ik, writeS_λ:=G^μ_(i

1,...,ik)(D)∈A^∗(k

M(p)) and S_div:=S_a12...k−1,k+1=S_0...010. It is easily seen that S_div=D₁−k

r=1Y_r and then, using (7) and (16),

SdivSλμ¹∧...∧μ^k

=S_divμⁱ¹∧...∧μ^ik

=

D1− k r=1

Yr

μⁱ¹∧...∧μ^ik

=D1μⁱ¹∧...∧μ^ik− k r=1

Yrμⁱ¹∧...∧μ^ik

= k j=1

μⁱ¹∧...∧μ^ij−1∧μ^ij+1∧μ^ij+1∧...∧μ^ik+ k

r=1

(Yi_r−Yr)

μⁱ¹∧...∧μ^ik

= k j=1

μⁱ¹∧...∧μ^ij−1∧μ^ij+1∧μ^ij+1∧...∧μ^ik+ k

r=1

(yi_r−yr)

μⁱ¹∧...∧μ^ik.