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A Chevalley-Monk and Giambelli's formula for Peterson varieties of all Lie types.

1 Introduction

Classical Schubert calculus computes the cohomology rings of Grassmannians and flag varieties. Each cohomology ring of a Grassmannian or flag variety has a basis of Schubert classes indexed by the elements of the corresponding Weyl group. This indexing set turns geometric and topological questions into combinatorial questions.

This paper will do "Schubert calculus" in the equivariant cohomology rings of Peterson varieties. Peterson varieties were introduced by D. Peterson in the 1990s. Peterson constructed the small quantum cohomology of partial flag varieties from what are now called Peterson varieties. Since then Kostant used Peterson varieties to describe the quantum cohomology of the flag manifold [13]. Rietsch described the totally non-negative part of type A Peterson varieties in 2006 using mirror symmetry constructions [16]. In 2012 Insko-Yong explicitly described the singular locus of type A Peterson varieties and intersected them with Schubert varieties [12].

Named for Goresky, Kottwitz, and MacPherson, GKM theory expresses the T-equivariant cohomology of certain spaces in terms of polynomials corresponding to T-fixed points [7]. The flag variety with the action of a maximal torus T [subset or equal to] B [subset or equal to] G is such a space. Peterson varieties are not GKM spaces; they lack many of the nice structures of G/B. However by using a one-dimensional torus [S.sup.1] several of those structures can be built for Peterson varieties anyway. Harada-Tymoczko gave the [S.sup.1]-fixed points of the Peterson variety explicitly [9]. In this paper we expand the GKM-like properties of Peterson varieties as illustrated in Figure 1.

Using work by Harada-Tymoczko [10] and Precup [15], we construct a basis for the [S.sup.1]-equivariant cohomology of Peterson varieties in all Lie types. This construction gives a basis of Peterson Schubert classes. A Peterson Schubert class is thus named because is the image of a Schubert classes under a certain ring homomorphism, not because the Peterson Schubert classes satisfy the standard properties of Schubert classes. In fact our first result is to give a subset of the Peterson Schubert classes which satisfy as many of the Schubert class properties as possible. Crucially this subset is a module basis for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Once we have a basis of Peterson Schubert classes, we ask classical Schubert calculus questions about how to multiply them. We give a Chevalley-Monk formula for multiplying a ring generator and a module generator, and a Giambelli's formula for expressing any Peterson Schubert class in terms of the ring generators. In type A the equivariant cohomology of the Peterson variety was studied by Harada-Tymoczko who gave a basis and a Monk's rule for the equivariant cohomology ring [9]. A type A Giambelli's formula was given by Bayegan-Harada [1]. This paper extends those results to all Lie types.

1.1 Multiplication rules for Peterson Schubert classes in all Lie types

Where Schubert classes in the flag variety are indexed by permutations in the associated Weyl group, Peterson Schubert classes are indexed by subsets of the set of simple roots. To each subset K of simple roots we associate a Peterson Schubert class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the flag variety case, the Schubert classes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] corresponding to words of length one generate the cohomology ring. For Grassmannians and partial flag varieties we need more complicated sets of Schubert classes to generate the cohomology ring. The Peterson case is analogous to the flag variety: classes indexed by words of length one generate the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For each simple root [[alpha].sub.i] [member of] [DELTA] we have a ring generator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A Chevalley-Monk rule is an explicit formula for multiplying an arbitrary module basis element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by a ring generator class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For the Peterson variety, a Chevalley- Monk formula gives a set of constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Graham showed that flag varieties have a Chevalley-Monk formula with non-negative integer coefficients [8, Theorem 3.1]. Giving these coefficients explicitly has been the work of many including Buch [5], Bergeron-Sanchez-Ortega-Zabrocki [2], and Lam-Shimozono [14]. In the Peterson case, we give the following formula.

Theorem 1.1 The Peterson Schubert classes satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our Chevalley-Monk rule is uniform across Lie types. This formula is similar in complexity to the rule for G/B and has positive, although occasionally non-integral, coefficients. On the other hand Giambelli's formula provides explicit structure constants and is surprisingly simple and uniform across Lie types.

Theorem 1.2 Giambelli's formula for Peterson Schubert classes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

1.2 Proof Techniques

Multiplication rules in Schubert calculus frequently require new combinatorial machines to prove explicit formulas. Billey's formula evaluates an equivariant Schubert class at a fixed point [3]. This combinatorial formula gives a polynomial in the simple roots for each pair of a Schubert class with a fixed point. Ikeda-Naruse use excited Young diagrams to encode and compute Billey's formula [11]. We modify both Ikeda-Naruse's excited Young diagrams and Billey's formula in order to evaluate Peterson Schubert classes at [S.sup.1]-fixed points of Peterson varieties.

While the uniform description of the result suggests that a geometric or topological proof exists, we proved the theorem in cases. The modified excited Young diagrams are used to prove Giambelli's formula in each of the classical Lie types. The exceptional Lie types do not have nice corresponding excited Young diagrams. In these cases we prove Giambelli's formula through direct computation. For types [F.sub.4] and [G.sub.2] the computation is relatively small and straightforward. Type E requires a computer assisted proof using Sage.

2 The [S.sup.1] Action on Peterson Varieties

Fix a complex reductive linear algebraic group G, a Borel subgroup B, and a maximal torus T [subset or equal to] B [subset or equal to] G. This choice gives

* a root system [PHI]

* positive roots [[PHI].sup.+] [subset or equal to] [PHI]

* simple roots [DELTA] [subset] [[PHI].sup.+]

* an associated Weyl group W

* associated Lie algebras t [subset or equal to] b [subset or equal to] g

* root spaces [g.sub.[alpha]] [subset] g for each root [alpha] [member of] [PHI].

We also choose a basis element [E.sub.[alpha]] [member of] [g.sub.[alpha]] for each of the root spaces. Some of our constructions rely on a specific ordering of the roots [[alpha].sub.1], [[alpha].sub.2], ..., [[alpha].sub.[absolute value of [DELTA]]] [member of] [DELTA]. This ordering is expressed in the Dynkin diagrams in Figure 2.

For any Lie type the Peterson subspace in g is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We fix a regular nilpotent operator [N.sub.0] [member of] g is

[N.sub.0] = [summation over ([alpha][member of][DELTA])][E.sub.[alpha]].

Definition 2.1 The Peterson variety Pet is a subvariety of the flag variety defined by

Pet = {gB [member of] G/B: Ad([g.sup.-1])([N.sub.0]) [member of] [H.sub.pet]}.

Peterson varieties are a type of regular nilpotent Hessenberg variety. They are irreducible and generally not smooth [12].

2.1 Billey's formula

With GKM theory the equivariant cohomology ring can be viewed as a subring of a product of polynomial rings: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where each ring [H.sup.*.sub.T](pt) [congruent to] C[[[alpha].sub.i]: [[alpha].sub.i] [member of] [DELTA]]. From this perspective each Schubert class [[sigma].sub.v] is actually a collection of polynomials [[sigma].sub.v](w) for w in the Weyl group W.

Billey gave an explicit combinatorial formula for the polynomial [[sigma].sub.v](w) [3]. Fix a reduced word of w, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

Proposition 2.2 (Billey [3]) The polynomial [[sigma].sub.v](w) has several useful properties:

* The polynomial [[sigma].sub.v](w) is homogeneous of degree l(v).

* The polynomial [[sigma].sub.v](w) is non-zero if and only if v [less than or equal to] w in the Bruhat order.

* The coefficients of [[sigma].sub.v](w) are non-negative integers.

* The polynomial [[sigma].sub.v](w) does not depend on the choice of reduced word for w.

When v and w are words of relatively short length it is simple to calculate [[sigma].sub.v](w) by hand.

Example 2.3 Let G/B have Weyl group W = [A.sub.2] and let w = [s.sub.1][s.sub.2][s.sub.1] and v = [s.sub.1]. The word v is found as a subword of [s.sub.1][s.sub.2][s.sub.1] in two places, [s.sub.1][s.sub.2][s.sub.1] and [s.sub.1][s.sub.2][s.sub.1].

[[sigma].sub.v](w) = r(1, [s.sub.1][s.sub.2][s.sub.1]) + r(3, [s.sub.1][s.sub.2][s.sub.1]) = [[alpha].sub.1] + [s.sub.1][s.sub.2]([[alpha].sub.1]) = [[alpha].sub.1] + [[alpha].sub.2]

2.2 GKM Theory and Peterson Varieties

The torus T does not preserve Pet but a one-dimensional subtorus [S.sup.1] [subset or equal to] T does. The equivariant cohomology of the Peterson variety can be defined with respect to [S.sup.1] and will still inject into the [S.sup.1]-fixed points of the Peterson variety.

Definition 2.4 [10, Lemma 5.1] The characters [[alpha].sub.1], ... [[alpha].sub.n] [member of] [t.sup.*] are a maximal Z-linearly independent set. Let [phi]: T [right arrow] [([C.sup.*]).sup.n] be the isomorphism of linear algebraic groups t [right arrow] ([[alpha].sub.1](t), [[alpha].sub.2](t), ..., [[alpha].sub.n](t)). Then define a one-dimensional torus [S.sup.1] by

[S.sup.1] = [[phi].sup.-1]({(c, c, ..., c) | c [member of] [C.sup.*]}).

Proposition 2.5 [10, Lemma 5.1] The torus [S.sup.1] acts on the Peterson variety.

Any point in Pet fixed by T will also be fixed by [S.sup.1]. In fact these are the only points in the Peterson variety fixed by [S.sup.1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Harada-Tymoczko gave the [S.sup.1]-fixed points of Pet explicitly. Let K [subset or equal to] [DELTA] be a subset of the simple roots. Define [W.sub.K] [subset or equal to] W as the parabolic subgroup generated by K and let [w.sub.K] be the longest element of [W.sub.K].

Proposition 2.6 [10, Proposition 5.8] An element w [member of] W is an [S.sup.1]-fixed point of Pet if and only if w = [w.sub.K] for some set K [subset or equal to] A.

3 Peterson Schubert classes as a basis of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first structure we want for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a basis. We use a projection from [H.sup.*.sub.T](G/B) to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

A priori [H.sup.*.sub.T](G/B) is a module over C[[[alpha].sub.i]: [[alpha].sub.i] [member of] [DELTA]]. The ring homomorphism [[pi].sub.1] takes simple roots [[alpha].sub.i] [member of] A to the variable t. The map [[pi].sub.2] forgets the T-fixed points of G/B that are not in the Peterson variety.

3.1 Peterson Schubert classes

The image of a Schubert class [[sigma].sub.v] [member of] [H.sup.*.sub.T](G/B) in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is denoted [p.sub.v] and called a Peterson Schubert class. The class [p.sub.v] has one polynomial for each [S.sup.1]-fixed point of Pet so a Peterson Schubert class can be thought of as a [2.sup.[absolute value of [DELTA]]]-tuple of polynomials in C[t]. Below is an example in type [A.sub.2].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 3.1 Because the longest word [w.sub.0] is a fixed point of the Peterson variety and the polynomial [[sigma].sub.v]([w.sub.0]) is non-zero by Proposition 2.2, the image [p.sub.v] of Schubert class [[sigma].sub.v] is always non-zero.

Theorem 3.2 The poset pinball machinery given by Harada-Tymoczko [10, Theorem 5.4] holds for Peterson varieties of all Lie types.

As consequence of this theorem we can use the maps [[pi].sub.1] and [[pi].sub.2] to study [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3.2 A basis of Peterson Schubert classes

The set of Peterson Schubert classes {[p.sub.v]: v [memebr of] W} is the image of the set of Schubert classes. The set does not satisfy the standard properties of Schubert classes. We now identify a subset that satisfies many of those properties. This subset is linearly independent and spans [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 3.3 A subset of simple roots K [subset or equal to] [DELTA] is called connected if the induced Dynkin diagram of K is a connected subgraph of the Dynkin diagram of [DELTA].

Any subset K [subset or equal to] [DELTA] can be written as K = [K.sub.1] x ... x [K.sub.m] where each [K.sub.i] is a maximally connected subset. Each connected subset corresponds to its own Lie type.

Definition 3.4 Let K [subset or equal to] [DELTA] be a connected subset. We define [v.sub.K] [member of] [W.sub.K] to be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [Root.sub.K](i) is the index of the corresponding root in a root system of the same Lie type as K, ordered as in Figure 2. If K = [K.sub.1] x ... x [K.sub.m] we write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example 3.5 Let [DELTA] = {[[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3], [[alpha].sub.4], [[alpha].sub.5], [[alpha].sub.6]} be a the set of simple roots of a type [E.sub.6] root system and let K = [DELTA]\{[[alpha].sub.6]}. The subset K [subset or equal to] [DELTA] is drawn as a marked set of vertices in the Dynkin diagram which is compared to the Dynkin diagram for [D.sub.5]. The word [v.sub.K] is [s.sub.1][s.sub.3][s.sub.4][s.sub.5][s.sub.2].

Lemma 3.6 For any set of simple roots A and any subsets J,K [subset or equal to] [DELTA] the Peterson Schubert class satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] unless J [subset or equal to] K. Since K contains itself, the polynomial [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is non-zero.

The following theorem is a version of Harada-Tymoczko's Theorem 5.9 [10] which, using Precup's work, we extend to all Peterson varieties.

Theorem 3.7 The Peterson Schubert classes {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} are a basis of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: Impose a partial order on the sets {K [subset or equal to] [DELTA]} by inclusion. Use that partial order to order the classes {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} and the [S.sup.1]-fixed points [w.sub.K] [member of] Pet. Lemma 3.6 implies that the collection {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} is lower-triangular and has full rank. Thus {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} is a linearly independent set.

By the properties of Billey's formula, the polynomial degree of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is [absolute value of K] and its cohomology degree is 2[absolute value of K]. As there are ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) subsets of [DELTA] with size [absolute value of K], there are exactly ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) Peterson Schubert varieties with cohomology degree 2[absolute value of K]. Precup's paving by affines reveals that the dimensions of the corresponding pavings are also ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) [15, Corollary 4.13]. As a linearly independent set with the correct number of elements of each degree, the set {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]} is a module basis of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [9, Proposition A.1].

Example 3.8 Below we give the Peterson Schubert classes which form a basis of the [S.sup.1]-equivariant cohomology of Pet in Lie type [C.sub.3]. The classes and fixed points are indexed by the subsets K [subset or equal to] [DELTA].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

4 Chevalley-Monk Formula

Now that we have a basis for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in terms of Peterson Schubert classes, we can examine the structure of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] through its multiplication rules. Recall that the Peterson Schubert classes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] generate the ring [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. A Chevalley-Monk rule is an explicit formula for multiplying an arbitrary module generator class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by a ring generator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For the Peterson variety, it gives a set of constants [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Lemma 4.1 If K [not subset or equal to] J then [c.sup.J.sub.i,K] = 0. Moreover if [absolute value of J] > [absolute value of K] + 1 then [c.sup.J.sub.i,K] = 0.

Having determined which coefficients are always zero, we can give our Chevalley-Monk formula for Peterson varieties. Our coefficients are complex polynomials in t. We say such a polynomial is nonnegative and rational if it is contained in [Q.sub.[greater than or equal to]0][t].

Theorem 4.2 (Chevalley-Monk formula for Peterson varieties) The Peterson Schubert classes satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the coefficients [c.sup.J.sub.i,K] are non-negative rational numbers given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Conjecture 4.3 In classical Schubert calculus the structure constants are generally non-negative integers. Frequently they are in bijection with dimensions of irreducible representations. However, structure constants for the Peterson variety are not necessarily integers. For example in type [D.sub.5] if we let K = {[[alpha].sub.1], [[alpha].sub.2], [[alpha].sub.3], [[alpha].sub.4]} and J = [DELTA] then

[c.sup.J.sub.5,K] = 5/2.

We conjecture that in this basis, non-integral structure constants only occur in Lie types D and E.

5 Giambelli's Formula

The last piece to understanding [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is Giambelli's formula. It expresses an arbitrary module basis element in terms of the ring generators. For the basis of [H.sup.*.sub.T](G/B) consisting of Schubert classes it looks like

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[sigma].sub.[lambda]] is the Schubert class corresponding to the partition [lambda] = ([[lambda].sub.1], ..., [[lambda].sub.r]) [6]. While easy to write down, this formula is hard to compute for a given Schubert class. Astonishingly, for Peterson varieties the Giambelli's formula for module basis elements of simplifies to a single product. A Giambelli's formula for the Peterson Schubert classes not in our basis is not yet known.

Lemma 5.1 For a Peterson Schubert class [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] there is an integer constant C satisfying

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

Proof: If [absolute value of K] = m let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Define subsets 0 = [K.sub.0] [subset or not equal to] [K.sub.1] [subset or not equal to] [K.sub.2] [subset or not equal to] ... [subset or not equal to] [K.sub.m] = K by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

From the Chevalley-Monk formula for Peterson varieties

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 4.2 says [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the polynomial [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is zero. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also zero. Thus if J [not equal to] [K.sub.i+1] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now the Chevalley-Monk rule reduces to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Solving for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] gives

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By induction on i we see

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This gives that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Remark 5.2 Like the determinantal formula for flag varieties, this formula holds in both the equivariant and ordinary cohomology of the Peterson variety.

Knowing that Giambelli's formula is a single product rather than a determinantal formula, we want to give the constant C explicitly. To find this C we consider the simplest non-trivial Peterson Schubert classes, those that are connected.

Theorem 5.3 If J,K [subset] [DELTA] are disjoint subsets such that no root in J is adjacent to any root of K, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

It follows that if K = [K.sub.1] x ... x [K.sub.m] then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: We show that equality holds when Equation (5) is evaluated at any [S.sup.1]-fixed point [w.sub.L]. If L does not contain J [union] K we can suppose without loss of generality that J [not subset or equal to] L. Then both [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are zero.

If L contains J [union] K then a subword b of a fixed reduced word [[??].sub.L] is a reduced word for [v.sub.J[union]K] if and only if b = [b.sub.J] [union] [b.sub.K] for subwords [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] reduced words for [v.sub.J] and [v.sub.K]. Billey's formula in Equation (1) is a sum over such subwords. We use it to rewrite the left- and right-hand sides of Equation (5). The left hand side becomes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Similarly the right-hand side becomes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

Both Equations (6) and (7) expand out to the same expression.

Theorem 5.4 If K [subset or equal to] [DELTA] is a connected root subsystem of type [A.sub.n], [B.sub.n], [C.sub.n], [F.sub.4], or [G.sub.2] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If K is a connected root subsystem of type [D.sub.n] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If K is a connected root subsystem of type [E.sub.n] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our proof of this theorem is combinatorial and treats each Lie type as its own case. The uniformity across Lie types suggests that a uniform proof exists. Such a proof might shed light on the topology of these varieties. In fact, the theorem can be stated in an even more uniform manner.

Theorem 5.5 If K [subset or equal to] [DELTA] is a connected root subsystem of any Lie type and [absolute value of R([v.sub.K])] is the number of reduced words for [v.sub.K] then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: Given Theorem 5.4 it is sufficient to show that [absolute value of R([v.sub.K])] = 1 if K is type A, B, C, F, or G, that [absolute value of R([v.sub.K])] = 2 for type D and that [absolute value of R([v.sub.K])] = 3 for type E. Given one reduced word any other reduced word can be obtained by a series of braid moves and commutations [4]. If K is type A, B, C, F, or G then [s.sub.i] and [s.sub.i+1] do not commute for any i. Therefore [s.sub.1][s.sub.2] ... [s.sub.n-1][s.sub.n] is the only reduced word for [v.sub.K].

If K is of type D then [s.sub.i] and [s.sub.i+1] commute if and only if i = n - 1. Also [s.sub.n-2] and [s.sub.n] do not commute. The only two reduced words for [v.sub.K] are [s.sub.1][s.sub.2] ... [s.sub.n-2][s.sub.n-1][s.sub.n] and [s.sub.1][s.sub.2] ... [s.sub.n-2][s.sub.n][s.sub.n-1] so [absolute value of R([v.sub.K])] = 2.

If K is type [E.sub.n] then we start with the word [v.sub.K] = [s.sub.1][s.sub.2][s.sub.3][s.sub.4] ... [s.sub.n] with the labels given as in Figure 2. The reflection [s.sub.2] commutes with [s.sub.1] and [s.sub.3] but not [s.sub.4]. The reflection [s.sub.3] does not commute with [s.sub.1]. When i > 2, [s.sub.i] and [s.sub.i+1] do not commute. Thus [v.sub.K] has exactly 3 reduced words: [s.sub.1][s.sub.2][s.sub.3][s.sub.4] ... [s.sub.n] and [s.sub.1][s.sub.3][s.sub.2][s.sub.4] ... [s.sub.n] and [s.sub.2][s.sub.1][s.sub.3][s.sub.4] ... [s.sub.n].

We can now give Giambelli's formula explicitly for all Peterson Schubert classes.

Corollary 5.6 If K [subset or equal to] [DELTA] and K = [K.sub.1] x ... x [K.sub.m] where each [K.sub.l] is a maximally connected subset then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Acknowledgements

Thank you to Julianna Tymoczko for her help and guidance at all stages of this project. Thank you to David Anderson, Tom Braden, David Cox, Allen Knutson, Aba Mbirika, Jessica Sidman, and Alex Yong for helpful comments and conversations. I am grateful to Erik Insko for getting me started with Sage and to Hans Johnston for the computer access and programming help. Thank you to the anonymous reviewers for their insightful comments and to Christophe Gole for help with the French version of the abstract.

References

[1] Bayegan, Darius; Harada, Megumi. A Giambelli formula for the [S.sup.1]-equivariant cohomology of type A Peterson varieties. eprint arXiv:1012.4053, 12/2010.

[2] Bergeron, Nantel; Sanchez-Ortega, Juana; Zabrocki, Mike. The Pieri Rule for Dual Immaculate Quasi-Symmetric Functions. eprint arXiv:1307.4273v1, 16 Jul2013.

[3] Billey, Sara. Kostant polynomials and the cohomology ofG/B. Duke Math J. 96 no. 1, (1999).

[4] Bjorner, Anders; Brenti, Francesco. "Combinatorics of Coxeter Groups." Graduate Texts in Mathematics. Springer (2000).

[5] Buch, Anders Skovsted. Direct proof of the quantum Monk's formula Proceedings of the AMS. Volume 131, No 7, 2037-2042. (2003).

[6] Fulton, William. "Young Tableaux." London Mathematical Society Student Texts 35. Cambridge University Press (1997).

[7] Goresky, Mark; Kottwitz, Robert; MacPherson, Robert. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131, 25-83 (1998).

[8] Graham, William. Positivity in equivariant Schubert calculus. Duke Math. J. 109, 599-614. (2001).

[9] Harada, Megumi; Tymoczko, Julianna. A positive Monk formula in the [S.sup.1]-equivariant cohomology of type A Peterson varieties. eprint arXiv:0908.3517, 08/2009.

[10] Harada, Megumi; Tymoczko, Julianna. Poset pinball, GKM-compatible subspaces, and Hessenberg varieties. eprint arXiv:1007.2750, 07/2010.

[11] Ikeda, Takeshi; Naruse, Hiroshi. Excited Young diagrams and equivariant Schubert calculus. Trans. Amer. Math. Soc. 361, 5193-5221, (2009).

[12] Insko, Erik; Yong, Alexander. Patch ideals and Peterson varieties Transform. Groups 17, No. 4, 1011-1036(2012).

[13] Kostant, Bertram. Flag Manifold Quantum Cohomology, the Toda Lattice, and the Representation with Highest Weight p. Selecta Mathematica. Volume. 2, No.1 (1996).

[14] Lam, Thomas; Shimozono, Mark. Equivariant Pieri Rule for theHomology of the Affine Grassmannian. Journal of Algebraic Combinatorics. Volume 36, Issue 4. (Dec. 2012).

[15] Precup, Martha. Affine pavings of Hessenberg varieties for semisimple groups. Selecta Mathematica. (Nov. 2012).

[16] Rietsch, Konstanze. A mirror construction for the totally nonnegative part of the Peterson variety. Nagoya Math J. Volume 183 (2006).

Elizabeth Drellich *

University of Massachusetts, Amherst, USA

* Email: drellich@math.umass.edu. Partially supported by NSF grant DMS-1248171
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