ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 15 (2018) 523-542 https://doi.org/10.26493/1855-3974.1062.ba8
(Also available at http://amc-journal.eu)
Wonderful symmetric varieties and Schubert
polynomials*
Mahir Bilen Can, Michael Joyce
Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
Benjamin Wyser
Department ofMathematics, Oklahoma State University, Stillwater, OK 74078, USA
Received 9 June 2016, accepted 25 October 2017, published online 11 September 2018
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of the general linear group on the flag variety. Combining this with a slight extension of results of Can, Joyce and Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.
Keywords: Symmetric varieties, Schubert polynomials, wonderful compactification, equivariant co-homology, weak order, parabolic induction. Math. Subj. Class.: 14M27, 05E05, 14M15
1 Introduction
Suppose that G is a connected reductive algebraic group over C. Suppose that B D T are a Borel subgroup and a maximal torus of G, respectively, W is the Weyl group, and let t denote the Lie algebra of T. By a classical theorem of Borel [1], the cohomology ring of G/B with rational coefficients is isomorphic to the coinvariant algebra Q[t*]/ZW, where
* We thank Michel Brion for many helpful conversations and suggestions. We thank the referee for their careful reading and many helpful suggestions. The first and second author were supported by NSA-AMS Mathematical Sciences Program grant H98230-14-1-142. The second is partially supported by the Louisiana Board of Regents Research and Development Grant 549941C1. The third author was supported by NSF International Research Fellowship 1159045 and hosted by Institut Fourier in Grenoble, France.
E-mail address: mcan@tulane.edu (Mahir Bilen Can), mjoyce3@tulane.edu (Michael Joyce), bwyser@okstate.edu (Benjamin Wyser)
Abstract
©® This work is licensed under https://creativecommons.org/licenses/by/3.0/
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IW denotes the ideal generated by homogeneous W-invariant polynomials of positive degree. Any subvariety Y of G/B defines a cohomology class [Y] in H*(G/B). It is then natural to ask for a polynomial in Q[t*] which represents [Y]. In this paper, for certain families of subvarieties of certain G/B, we approach and answer this question in two different ways. Relating the two answers leads in the end to our main result, Theorem 4.1, which, roughly stated, says that certain non-negative linear combinations of Schubert polynomials factor completely into linear forms.
Our group of primary interest is G = GLn, with B its Borel subgroup of lower-triangular matrices, and T its maximal torus of diagonal matrices. In this case, there is a canonical basis xi,... ,xn of t* that correspond to the Chern classes of the tautological quotient line bundles on the variety of complete flags G/B. Let Zn denote the center of GLn, consisting of diagonal scalar matrices. Let On denote the orthogonal subgroup of GLn, and let Sp2n denote the symplectic subgroup of GL2n. Denote by GOn (resp. GSp2n) the central extension ZnOn (resp. Z2nSp2n). For any ordered sequence of positive integers m = (pi,... ,Ms) that sum to n, GLn has a Levi subgroup Lm := GLM1 x • • • x GLMs, as well as a parabolic subgroup PM = LM x UM containing B, where UM denotes the unipotent radical of PM.
The subgroup
Hm := (GOMi x---x GOMs) x UM
of GLn is spherical, meaning that it acts on GLn/B with finitely many orbits. Moreover, there is a unique closed HM-orbit YM on GLn/B, which is our object of primary interest.
The reason for our interest in this family of orbits is that they correspond to the closed B-orbits on the various G-orbits of the wonderful compactification of the homogeneous space GLn/GOn. This homogeneous space is affine and symmetric, and it is classically known as the space of smooth quadrics in Pn-i. Its wonderful compactification, classically known as the variety of complete quadrics [9, 13], is a G-equivariant projective embedding X which contains it as an open, dense G-orbit, and whose boundary has particularly nice properties. (We recall the definition of the wonderful compactification in Section 2.1.)
It turns out that, with minor modifications, our techniques apply also to the wonderful compactification X' of the space GL2n/GSp2n, which parameterizes non-degenerate skew-symmetric bilinear forms on C2n, up to scalar. Letting G = GL2n in this case, the G-orbits on X' are again parametrized by compositions
M = (Mi,..., Ms)
of n; note that this is of course equivalent to parametrizing them by compositions of 2n with each part being even. Each G-orbit has the form G/H^, with
H := (GSp2Mi x---x GSp2Ms) x UM,
a spherical subgroup which again acts on GL2n/B with a unique closed orbit Y^.
Let us consider two ways in which one might try to compute a polynomial representative of [Ym] (or [Y^]). For the first, note that YM, being an orbit of HM, also admits an action of a maximal torus SM of HM. Thus YM admits a class [YM]S^ in the SM-equivariant cohomology of GLn/B, denoted by HS (GLn/B). In brief, this is a cohomology theory which is sensitive to the geometry of the SM-action on GLn/B. It admits a similar Boreltype presentation, this time as a polynomial ring in two sets of variables (the usual set of x-variables referred to in the second paragraph, along with a second set which consists of
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y and z-variables) modulo an ideal. Moreover, the map HS (GLn/B) ^ H*(GLn/B) which sets all of the y and z-variables to 0 sends the equivariant class of any SM-invariant subvariety of GLn/B to its ordinary (non-equivariant) class. Thus if a polynomial representative of [Ym]sm can be computed, one obtains a polynomial representative of [YM] by specializing y, z ^ 0.
In [15], this problem is solved for the case in which ^ has only one part, in which case Hm = GOn. Here, we extend the results of [15] to give a formula for the equivariant class [Y„l sm (and [Y' ]s' ) for an arbitrary composition The main general result is Proposition 3.4; it, together with Proposition 3.5, imply the case-specific equivariant formulas given in Corollaries 3.6 and 3.8.
The formulas for [YM] and [Y^] obtained from these corollaries (by specializing y and z-variables to 0) are as follows:
Corollary 1.1. The ordinary cohomology class of [YM] is represented in H* (G/B) by the formula
... _ _ .
Vi=1	/ i= 1 Vi+1<j<k<Vi+i-j
2d(M) mxR^^) ^	Y[	(Xj + Xk).
Corollary 1.2. The ordinary cohomology class of [Y^] is represented in H* (G/B) by the formula
(2n	\ s
n xR(M,iM n n +xk).
i=1	) i= 1 Vi + 1<j<k<Vi+1-j
The notations vi, d(p), R(p, i), S(p, i), etc. will be defined in Sections 2 and 3. For now, note that the representatives we obtain are factored completely into linear forms. In fact, the formulas reflect the semi-direct decomposition of HM (resp., Hp as we will detail in Section 3.
A second possible way to approach the problem of computing [YM] is to write it as a non-negative integral linear combination of Schubert classes. For each Weyl group element w in the symmetric group W = Sn, there is a Schubert class [Xw ], the class of the Schubert variety Xw = B+wB/B in GLn/B, where B+ denotes the Borel subgroup of upper-triangular elements of GLn. The Schubert classes form a Z-basis for H* (GLn/B).
Assuming that one is able to compute the coefficients in
[Ym] = £ Cw[Xw],	(1.1)
then one may replace the Schubert classes in the above sum with the corresponding Schubert polynomials to obtain a polynomial in the x-variables representing [YM]. The Schubert polynomials Sw are defined recursively by first explicitly setting
__ n—1 n—2	2
wo • X1 x2 * * * xn—2Xn—1,
and then declaring that Sw = di&wsi if wsi < w in Bruhat order. Here, si = (i,i + 1) represents the ith simple reflection, and di represents the divided difference operator defined by
d (f )(x	x ) _ f (x1,..., xn) f (x 1, . . . , xi — 1, Xi+1, xi, Xi+2 , . . . , xn)
xi Xi+1
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It is well-known that Sw represents [Xw] [10], and so if (1.1) can be computed, [Ym] is represented by the polynomial
^ ^ cw Sw • wEW
In fact, a theorem due to M. Brion [2] tells us in principle how to compute the sum (1.1) in terms of certain combinatorial objects. More precisely, to the variety Ym there is an associated subset of W, which we call the W-set of Ym, and denote by W (Ym). For each w e W (Ym), there is also an associated weight. In fact, this weight is always a power of 2, and the aforementioned theorem of Brion says that the sum (1.1) can be computed as
[Ym] = £ 2d(Y-w)[Xw ]	(1.2)
wew (ym)
for non-negative integers d(YM, w). This can be turned into an explicit polynomial representative via the aforementioned Schubert polynomial recipe, assuming that one can compute the sets W(Ym), and the corresponding exponents d(YM,w) explicitly. In fact, in Section 2.4, we recall explicit descriptions of the W-sets W (Ym) which have already been given in [5, 6], slightly extending those results to also give an explicit description of W(Y') for arbitrary p. (Previous results of [6] only described W(Y') when p consisted of a single part.) And as we will note, the exponents d(YM, w) are straightforward to compute.
This gives a second answer to our question, but note that it comes in a different form. Indeed, the formulas of Corollaries 1.1 and 1.2 are products of linear forms in the x-variables which are not obviously equal to the corresponding weighted sums of Schubert polynomials. Of course, it is a priori possible that the two polynomial representatives are actually not equal, but simply differ by an element of IW, the ideal defining the Borel model of H*(G/B). However, our main result, Theorem 4.1, states that in fact the apparent identity in H *(G/B) is an equality of polynomials.
The paper is organized as follows. Section 2 is devoted mostly to recalling various background and preliminaries: We start by recalling necessary background on the wonderful compactification in Section 2.1. We then give the explicit details of the examples which we are concerned with in Section 2.2; this includes our conventions and notations regarding compositions, as well as our particular realizations of all groups, including the groups Hm and H'. In Section 2.4, we review the notion of weak order and W-sets. We recall results of [5, 6] which are relevant to the current work, giving a slight extension of those results to the case of W (Y') for arbitrary p.
In Section 3, we briefly review the necessary details of equivariant cohomology and the localization theorem. We then use those facts to extend the formulas of [15] to the more general cases of this paper, obtaining Proposition 3.4 in a general setting, and its case-specific Corollaries 3.6 and 3.8. Corollaries 1.1 and 1.2 are immediate consequences of these.
Finally, in Section 4, we compare the representatives of [Ym] and [Y'] obtained via our two different approaches, obtaining Theorem 4.1.
2 Background, notation, and conventions
Throughout the text, we use italicized notation when we give arguments that apply to general reductive groups. In that case, G is an arbitrary connected, reductive algebraic group
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defined over C. We fix a Borel subgroup B of G and a maximal torus T of G contained in B. We let W denote the Weyl group of G and for every simple root a of T, we let sa e W denote the associated simple reflection and Pa = B U BsaB denote the minimal parabolic subgroup containing B associated to a.
By contrast, we use bolded notation to denote our two main examples
G/H = GL„/O„ and G/H = GL2„/Sp2„.
In these examples, we use the Borel subgroup B consisting of lower-triangular matrices and the maximal torus T consisting of diagonal matrices. Furthermore, the Weyl group W is isomorphic to the symmetric group Sn (respectively, S2n). We hope this helps the reader distinguish our case-specific results from the general results that we need along the way.
2.1 The wonderful compactification
We review the notion of the wonderful compactification of a general spherical homogeneous space. Using our convention outlined above, let G be a connected, reductive algebraic group defined over C. An algebraic subgroup H of G, as well as the homogeneous space G/H, is called spherical if a Borel subgroup B has finitely many orbits on G/H (or equivalently, if H has finitely many orbits on G/B). Some such homogeneous spaces, namely partial flag varieties, are complete, while others (for example, symmetric homogeneous spaces) are not. In the event that G/H is not complete, a completion of it is a complete G-variety X which contains an open dense subset X0 G-equivariantly isomorphic to G/H. X is a wonderful compactification of G/H if it is a completion of G/H which is a "wonderful" spherical G-variety; this means that X is a smooth spherical G-variety whose boundary (the complement of X0) is a union of smooth, irreducible G-stable divisors Di,..., Dr (the boundary divisors) with normal crossings and non-empty transverse intersections, such that the G-orbit closures on X are precisely the partial intersections of the Dj's. The number r is called the rank of the homogeneous space G/H.
The number of G-orbits on X is then 2r, and they are parametrized by subsets of {1,..., r}, with a given subset determining the orbit by specifying the set of boundary divisors containing its closure. It is well-known that the subsets of {1, 2,..., r} are in bijection with the compositions of r +1. A composition of n is simply a tuple p = ..., with£i = n. For a given composition p = ..., we define the integers v2,..., vs by the formula vi = J2j—i Mj for i = 2,..., s. By convention, we set vi = 0. In words, vi is the sum of the first i -1 parts of the composition We parametrize the G-orbits (G = GLn or G = GL2n) in our examples by compositions of n, with n - 1 being the rank of both of the symmetric spaces GLn/GOn and GL2n/GSp2n.
2.2 Our examples
We now describe the two primary examples to which we will directly apply the general results of this paper. The first is the wonderful compactification X of the space of all smooth quadric hypersurfaces in Pn -1, i.e. G/H, where (G, H) = (GLn, GOn), classically known as the variety of complete quadrics.
Choose B to be the lower-triangular subgroup of GLn, and T to be the maximal torus consisting of diagonal matrices. We realize H0 = On as the fixed points of the involution given by 0(g) = J(g4)-1 J, where J is the n x n matrix with 1's on the antidiagonal, and 0's elsewhere. When H0 is realized in this way, H0 n B, the lower-triangular subgroup of
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Ho, is a Borel subgroup, and S0 := H0 n T is a maximal torus of H0, consisting of all elements of the form
diag(ai, ..., am, amx,. .., a-1),	(2.1)
where a» G C* for i = 1,..., m when n = 2m is even, and of the form
diag(ai,.. ., am, 1, a^1, .. ., a-1),
where a» G C* for i = 1,..., m when n = 2m +1 is odd. The Lie algebra s0 of S0 then takes the form
diag(ai, .. ., am, -am, .. ., -ai),	(2.2)
where G C for i = 1,..., m in the even case, and
diag(ai,. .., am, 0, -am,. .., -ai),
where a4 G C for i = 1,..., m in the odd case.
Note that the diagonal elements of H form a maximal torus S of dimension one greater than dim S0. The general element of S is of the form
diag(Aai,..., Aam, Aa^,..., Aa-i)	(2.3)
in the even case, and of the form
diag(Aai,..., Aam, A, Aami,..., Aa-i)	(2.4)
in the odd case. Here, A is an element of C* and the aj's are as before. The Lie algebra s of S then consists of diagonal matrices of the form
diag(A + ai,..., A + am, A — am,..., A — ai)	(2.5)
in the even case, and of the form
diag(A + ai,..., A + am, A, A — am,..., A — ai).	(2.6)
in the odd case. Here, A is an element of C and the aj's are as before.
Thus we have described the homogeneous space G/H, where G = GLn and H = GOn, which is the dense G-orbit on X. We now describe the other G-orbits. As mentioned in Section 2.1, they are in bijection with compositions ^ of n.
Corresponding to we have a standard parabolic subgroup PM = LM x UM containing B whose Levi factor LM is GLM1 x • • • x GLMs, embedded in GLn in the usual way, as block diagonal matrices. The G-orbit corresponding to ^ is then isomorphic to G/HM, where HM is the group
(GOMi x ••• x GOMs) x UM,
where GOMi = ZMi OMi is realized in GLMi as described above. Then B n HM is a Borel subgroup of Hm, and SM := T n HM is a maximal torus of HM.
Note that SM is diagonal, and consists of s "blocks", the ith block consisting of those diagonal entries in the range v + 1,..., v + If = 2m is even, then the ith block is of the form
diag(Aiaiji,..., Ajaijm, Aja-m,..., Aia-ii).	(2.7)
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The corresponding ith block of an element of sM is then of the form
diag(Aj + aj,!,..., Aj + aim, Aj - a'j,m,..., Aj - aj,i).	(2.8)
If pi = 2m + 1 is odd, then the ith block is of the form
diag(Ajaj,i,..., Aj aj,m, Aj, A ja-^,..., Aj a-).	(2.9) The ith block of an element of sM is correspondingly of the form
diag(Aj + aj,!,..., Aj + aj,m, Aj, Aj - aj,m,..., Aj - a^).	(2.10)
Our second primary example is the wonderful compactification of (G, H'), (G, H') = (GL2n, GSp2n). H' is a central extension of H0 = Sp2n, the latter group being realized as the fixed points of the involutory automorphism of GL2n given by g ^ J(gt)-! J, where J is the 2n x 2n antidiagonal matrix whose antidiagonal consists of n 1's followed by n - 1's, reading from the northeast corner to the southwest. Let S0 := Sp2n n T.
Once again taking B to be the lower-triangular Borel of GL2n, and T to be the diagonal maximal torus of GL2n, one checks that H0 n B is a Borel subgroup of H0, and that S0 := H0 n T is a maximal torus of H0. The corresponding torus S' of H' is then of exactly the same format as indicated in (2.3), while its Lie algebra s' is as indicated by (2.5).
The additional G-orbits on the wonderful compactification X' of GL2n/GSp2n again correspond to compositions p = (p!,..., ps) of n. For such a composition, we let PM = Lm x UM be the standard parabolic subgroup whose Levi factor is GL2mi x • • • x GL2Ms, embedded in GL2n as block diagonal matrices. Then the G-orbit corresponding to p is isomorphic to G/H^, where
H = (GSp2Ml x ••• x GSp2Ms) x UM,
with each GSp2M. = Z2m. Sp2^ embedded in the corresponding GL2m. just as described above.
The torus S^ then consists of s "blocks", just as in the orthogonal case. This time, each block is of even dimension, so each is of the form described by (2.7). The corresponding block of the Lie algebra s^ is then of the form indicated in (2.8).
2.3 Some general results
We now introduce several general observations that are applicable to our chosen examples. In this subsection, returning to the conventions set forth in the beginning of Section 2, let G be an arbitrary connected, reductive algebraic group over C and let P be a parabolic subgroup of G containing the Borel subgroup B with Levi decomposition P = L x U, where L is a Levi subgroup of G containing T and U is the unipotent radical of P. Let Hl be a subgroup of L and consider the G-variety V = G xP L/Hl, the quotient of G x L/Hl by the action of P, where P acts on G by right multiplication and on L/Hl via its projection to L. Note that V is a G-variety via left multiplication of G on the first factor. V is a homogeneous G-variety, and the stabilizer subgroup of the point [1,1HL/HL] is H := HlU C P. The construction of V = G/H from L/Hl (or equivalently of H C G from Hl C L) is called parabolic induction.
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Note that in our examples, HM is obtained via parabolic induction from GOMl x • • • x GOMfc (playing the role of HL) and H^ is obtained via parabolic induction from GSpMi x • • • x GSpMfc (likewise playing the role of HL). We now summarize some results from the literature that describe the role of parabolic induction for wonderful varieties.
Definition 2.1. Let G be a reductive algebraic group. A subgroup H of G is said to be symmetric if there exists an algebraic involution 0: G ^ G such that, letting K = Ge =
{g € G : 0(g) = g} and letting Z denote the center of G, we have ZK0 C H C ZK.
Proposition 2.2.
1.	Let H be a symmetric subgroup of G such that G/H has a wonderful compactifi-cation X. Then every G-orbit of X is obtained via parabolic induction from some symmetric homogeneous space L/Hl associated to some Levi subgroup L of G.
2.	If Hl is a spherical subgroup of L such that L/Hl contains a single closed BL-orbit (with Bl a Borel subgroup of L) and H is the subgroup of G obtained by parabolic induction, i.e. G/H = G xP L/Hl, then H is a spherical subgroup of G and G/H contains a single closed B-orbit.
Proof. The first result is a reformulation of a result of de Concini and Procesi [7, Theorem 5.2]. They show that a G-orbit V has a G-equivariant map V ^ G/P with fiber L/Hl. (In fact, they show that the closure of V maps G-equivariantly to G/P with fiber the wonderful compactifcation of L/Hl, from which our statement follows by restricting the map to V.) It follows that there is a bijective morphism $: G xP L/Hl ^ V, which is an isomorphism if $ is separable [14, discussion after Theorem 2.2].
The second result follows from [4, Lemma 6], but we give a direct proof for completeness. We again consider the G-equivariant fibre bundle n: G/H ^ G/P, with fiber L/Hl. Since HL is a spherical subgroup of L and P is a spherical subgroup of G (by the classical Bruhat decomposition), it follows that H is a spherical subgroup of G. Moreover, since there is a unique closed BL-orbit in L/Hl and a unique closed B-orbit in G/P, there is a unique closed B-orbit in G/H, namely the B-orbit which maps via n to the closed B-orbit of G/P and whose fiber over the base point 1P/P is identified with the closed BL-orbit in L/Hl.	□
2.4 Weak order and W-sets
Let H be a spherical subgroup of the connected, reductive algebraic group G and assume that there exists a wonderful compactification X of G/H. We continue with the notation of the previous subsection. In this subsection we review the notion of the weak order on the set of B-orbit closures of X. Note that X is a spherical variety, meaning that B has finitely many orbits on X, so the set of B-orbits equipped with the weak order is a finite poset.
The weak order on the set of B-orbit closures of X is the one whose covering relations are given by Y -< Y' if and only if Y' = PaY = Y for some simple root a of T relative to B. In general, Y < Y' if and only if Y' = Pas • • • Pai Y for some sequence of simple roots ai,... ,as.
When considering the weak order on X, it suffices to consider it on the individual Gorbits separately. Indeed, if Y and Y' are the closures of B-orbits Q and Q', respectively, and if Y < Y' in weak order, then Q and Q' lie in the same G-orbit. Therefore, we focus on the weak order on B-orbit closures on a homogeneous space G/H. The Hasse diagram
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of the weak order poset can be drawn as a graph with labeled edges, each edge with a weight of either 1 or 2. This is done as follows: For each cover Y -< Y' with Y' = PaY, we draw an edge from Y to Y', and label it by the simple reflection sa. If the natural map Pa xB Y ^ Y' is birational, then the edge has weight 1; if the map is generically 2-to-1, then the edge has weight 2. (These are the only two possibilities.) The edges of weight 2 are frequently depicted as double edges [4].
In the graph described above, there is a unique maximal element, since G/H is the closure of its dense B-orbit. Given a B-orbit closure Y, its W-set, denoted W(Y), is defined as the set of all elements of W obtained by taking the product of edge labels of paths which start at Y and end at G/H. The weight d(Y, w) alluded to before (1.2) is defined as the number of double edges in any such path whose edge labels multiply to w. (Note that there is one such path for each reduced expression of w, but all such paths have the same number of double edges, so that d(Y, w) is well-defined [4].) We have now recalled all explanation necessary to understand (1.2).
Next, we briefly recall results of [5, 6] which give explicit descriptions of these W-sets in the cases described in Section 2.2.
We begin by addressing the case of the extended orthogonal group H = GOn and the variants HM. For a set A C [n] := {1,2,..., n}, say that a < b are adjacent in A if there does not exist c G A such that a < c < b. Let Wn denote the set of permutations w G Sn that have the following recursive property. Initialize Ai = [n]. For 1 < i < |_n/2j, assume that w(1),..., w(i - 1) and w(n + 2 - i),..., w(n) have already been defined. (This condition is vacuous in the case i = 1.) Then w(i) and w(n +1 - i) must be adjacent in Aj and w(i) must be greater than w(n +1 - i). Define Aj+1 := Aj \{w(i),w(n +1 - i)}. This completely defines w when n is even, and if n = 2k + 1 is odd, then Ak+1 will consist of a single element m, so define w(k + 1) = m. For example, W5 consists of the eight elements of S5 given in one-line notation by 24531, 25341, 34512, 35142, 42513, 45123, 52314, 53124.
Proposition 2.3 ([5]). Let Y denote the closed B-orbit in G/H where G = GLn and H = GOn. Then W(Y) = Wn.
Remark 2.4. The "W-set" of [5], which is denoted by Dn there, differs slightly from ours. More precisely, the relationship between Dn and our W-set is
Wn = {wow-1wo : w G Dn}.
Let us explain the reason for the discrepancy. First, the partial order considered in [5] is the opposite of the weak order on Borel orbits considered here, which necessitates inverting the elements of Dn. Second, we consider here B to be the Borel subgroup of lower triangular matrices in GLn, while [5] uses the Borel subgroup of upper triangular matrices. This necessitates conjugating the elements by w0.
Similarly, we define a set WM C Sn associated with a composition m = (^1,..., of n. We begin by recalling the notion of a ^-string [5]. Recall that we have defined vk = Xj=o Mj for 2 < k < s; by convention v1 = 0. The ith ^-string of a permutation w G Sn, denoted by strj(w) is the word w(v + 1) w(vj +2) ... w(vj+1). For example, if w = 3715462 is a permutation from S7 (written in one-line notation) and m = (2,4,1), then the second ^-string of w is the word 1546. Let A C [n] have cardinality k and assume a word w of length k is given that uses each letter of A exactly once. Define a
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bijection between [k] and A by associating to i e [k] the ith largest element of A. Under this bijection, the word w corresponds to the one-line notation of a permutation w in Sk. Call w the permutation associated to the word w. Continuing the example, the permutation associated to the word w = 1546 is 1324 e S4 (in one-line notation).
The set Wp consists of all w e Sn such that the letters of strj(w) are precisely those j such that n - vj+1 < j < n - v and the permutation associated to str j (w) is an element of Wpi. For example, W(4,2) consists of the three elements of S6 given in one-line notation by 465321, 563421, 643521.
Proposition 2.5 ([5]). Let Yp denote the closed B-orbit in G/Hp where G = GLn and Hp = (GOPl x • • • x GOPs) x Up, i.e. Hp is obtained by parabolic induction from HL = GOPl x • • • x GOp C L = GLPl x • • • x GLp to G = GLn. Then
w (ym) = wp.	s	s
Just as in Remark 2.4, the relation between Wp and the set Dp defined in [5] is Wp = {wow-1wo : w e Dp}.
We now turn to the extended symplectic case H' = GSp2n and its variants Hp. Consider the inclusion of Sn into S2n via the map u ^ ^(u) = v = v1v2 • • • v2n, where
[vi,v2, . . . ,v„,v„+i, . . . ,V2n-1, V2n] =
[2u(1) - 1, 2u(2) - 1,..., 2u(n) - 1, 2u(n),..., 2u(2), 2u(1)].
Let W2n = {^(u) e S2n : u e Sn}. For example, W6 consists of the six elements of S6 given in one-line notation by 135642, 153462, 315624, 351264, 513426, 531246.
Proposition 2.6 ([6, 12]). Let Y' denote the closed B-orbit in G/H' where G = GL2n and H' = GSp2n. Then W(Y') = Wn.
We now proceed to define a set Wp C S2n for any composition p = (p1,..., ps) of n. The set Wp consists of all w e Sn such that the letters of strj(w) are precisely those j such that n - vi+1 < j < n - v and the permutation associated to str^w) is an element of W2 p.. For example, W'2 4) consists of the two elements of S6 given in one-line notation
by 1235564, 125346. '
Proposition 2.7. Let Y'p denote the closed B-orbit in G/Hp where G = GL2n and Hp = (GSp2pi x • • • x GSp2p) x Up, i.e. H'p is obtained by parabolic induction from Hl = GSp2p 1 x • • • x GSp2ps C L = GL2p 1 x • • • x gl2ps to G = GL2n. Then
w (y; ) = wp .
Proposition 2.7 is proved in exactly the same manner as Proposition 2.5 is proven in [5, Theorem 4.11], so we omit its proof. Alternatively, it can be obtained as a corollary of Proposition 2.6 by applying a general result of Brion on W-sets for homogeneous spaces obtained by parabolic induction [2, Lemma 1.2].
3 Equivariant cohomology computations 3.1 Background
We start by reviewing the basic facts of equivariant cohomology that we will need to support our method of computation. All cohomology rings use Q-coefficients. Results of this
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section are generally stated without proof, as they are fairly standard. To the reader seeking a reference we recommend [15] for an expository treatment, as well as references therein.
We will apply our results to equivariant cohomology with respect to the action of SM (respectively, Sp on G/B, these tori having been defined in Section 2.2. Given a variety X with an action of an algebraic torus S with Lie algebra s, the equivariant cohomology is, by definition,
H*s(X) := HS((ES x X)/S),
where ES denotes a contractible space with a free S-action. HS (X) is an algebra for the ring A S := HS({pt.}), the A S-action being given by pullback through the obvious map X ^ {pt.}. The ring A S is naturally isomorphic to the symmetric algebra Sym(sS) on sS. Thus, if yi,... ,yn are a basis for sS, then AS ~ Sym(sS) is isomorphic to the polynomial ring Q[y] = Q[yi,..., yn]. When X = G/B with G a reductive algebraic group and B a Borel subgroup, and if S C T C B with T a maximal torus in G, then we have the following concrete description of HS (X):
Proposition 3.1. Let R = Sym(ts), R' = Sym(sS). Then HS(X) = R' <rw R. If Xi,..., Xn are a basis for ts, and Yi,..., Ym are a basis of sS, elements of HS (X) are thus represented by polynomials in variables xj := 1 < Xj and yj := Y < 1.
To make this clear in the setting of our examples (cf. Section 2.2), if S is taken to be the full maximal torus T of GLn, we let Xj (i = 1,..., n) be the function on t which evaluates to aj on the element
t = diag(ai,..., an).
We denote by Yj (i = 1,..., n) a second copy of the same set of functions. We then have two sets of variables as in Proposition 3.1, typically denoted x = xi,... ,xn and y = yi,...,yn, and T-equivariant classes are represented by polynomials in these variables.
If p = (pi,..., ps) is a composition of n, let T be the full torus of GLn, and let SM be the torus of HM, as in Section 2.2. We denote by Xj the same function on t as described above. We denote by Yj j the function on sM which evaluates to aj j on an element of the form in (2.8) (if is even) or (2.10) (if is odd). We denote by Zj the function which evaluates to Aj on an element of the form (2.8) or (2.10). Then letting lower-case x, y, and z-variables correspond to these coordinates (with matching indices), H| (G/B) is generated by these variables, and when we seek formulas for certain SM-equivariant classes, we are looking for polynomials in these particular variables.
We next recall the standard localization theorem for torus actions. For more on this fundamental result, the reader may consult, for example, [3].
Theorem 3.2. Let X be an S-variety, and let i: XS ^ X be the inclusion of the S-fixed locus of X. Then the pullback map of AS-modules
is: HS(X) ^ HS(XS)
is an isomorphism after a localization which inverts finitely many characters of S. In particular, if HS (X) is free over AS, then is is injective.
When X = G/B, HS(X) = R' <rw R is free over R (as R is free over RW), so any equivariant class is entirely determined by its image under is. We will only apply this
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result in the event that S = T is the full maximal torus of G, so the S-fixed locus is finite, being parametrized by W. Then for us,
Hg (XS) = 0 As,
wew
so that in fact a class in HS(X) is determined by its image under i*w for each w e W, where here iw denotes the inclusion of the S-fixed point wB/B in G/B. Given a class ft e Hg(X) and an S-fixed point wB/B, we may denote the restriction i*w (ft) at wB/B by ft|w.
We end the section by recalling how the restriction maps are computed.
Proposition 3.3 ([15]). With the notation of the preceding paragraph, suppose that ft e Hg (X) is represented by the polynomial f = f (x, y) in variables x, y. Then ft|w e AS is the polynomial f (wY|s, Y).
3.2 A general result
We continue with the general setup given at the beginning of Section 2. Let P = LU be a parabolic subgroup of G containing B, with L a Levi factor containing T and U the unipotent radical of P contained in B. Let HL denote a spherical subgroup of L and let S := T n Hl be a maximal torus of HL. Let X denote the generalized flag variety G/B.
Let H be the subgroup obtained by parabolic induction from HL C L to G, i.e. H = HlU. As proven in Proposition 2.2, H is a spherical subgroup of G with a unique closed B-orbit Q. The torus S is a maximal torus of H, and we seek to describe [Q] e Hg(X).
Denote by BL the Borel subgroup B n L of L, and let Y denote the generalized flag variety L/Bl. Note that there is an S-equivariant embedding j: Y ^ X, which induces a pushforward map in cohomology jg: Hg (Y) ^ Hg (X). Given our setup, the orbit Q' = H • 1BL/Bl is closed in Y. In fact, j (Q') = Q. To see this we just observe that 1B/B = 1BL/BL under the embedding j. Nonetheless, to avoid possible confusion, we refer to the closed orbit as Q' when thinking of it as a subvariety of Y, and as Q when thinking of it as a subvariety of X.
Let WL C W denote the Weyl group of L. In the root system $ for (G, T), choose to be the positive system such that the roots of B are negative. Similarly, let $L be the root system for (L, T) and let = $L n be the positive roots of L such that the roots of Bl are negative. Given a root r e $ (resp., r e $L), let gr denote the associated one-dimensional root space in g (resp., l).
The next result relates the class of Q' in Hg(Y) to the class of Q in Hg(X).
Proposition 3.4. With notation as above, the classes j*[Q] and [Q'] are related via multiplication by the top S-equivariant Chern class of the normal bundle NY X, i.e. j*[Q] = cS (NY X) n [Q']. This Chern class is the restriction of a T-equivariant Chern class for the same normal bundle, and the latter class, which we denote by a, is uniquely determined by the following properties:
= i rire$+\$+ wr if w e WL, 10	otherwise.
Proof. The first statement follows easily from the equivariant self-intersection formula [8, p. 621, (4)], since [Q] = jg [Q'\. Since both X and Y have T-actions (not simply S-actions,
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as is the case for Q and Q'), there does exist a top T-equivariant Chern class of NX Y, and clearly the S-equivariant version is simply the restriction of the T-equivariant one.
The properties which define a follow from analysis of tangent spaces at various fixed points. Indeed, it is clear that the T-fixed points of Q' lying in Y are those which correspond to elements of WL Ç W. Thus for w e WL, we have a|w = 0. For w e WL, the restriction is cTd (NxY)|w = cj (NxY|w) = cTd (TwX/TwY), where d = codimx(Y). Since both X and Y are flag varieties, it is straightforward to compute these two tangent spaces, and their decompositions as representations of T. Indeed, Tw X is simply 0re$+ A«"-, while TwY is 0re$+ 0wr. The quotient of the two spaces is then
© rg$+\$+ flwr, which implies our claim on a|w.
Finally, that these restrictions determine a follows from the localization theorem, Theorem 3.2.	□
We now determine an explicit formula for the T-equivariant Chern class a that is defined in Proposition 3.4 when G is of type A. Let ^ = (p1,..., be a composition of n, let L = GLM1 x • • • x GLMs, and let T be the full diagonal torus of GLn.
For each 1 < k < n, let ek G t* be given by ek(diag(t1,... ,tn)) = tk. For any 1 < k < I < n, let akl = ek - ee G $+. Finally, let
Mx, y) = n (xk - ve).
ak,eG<£+\<£+
An equivalent definition of showing the explicit dependence on the composition ^ is as follows. For each pair i, j with 1 < i < j < s, we define a polynomial
hi,j(x, y) :=	n (xe - Vk).
k=Vj + 1 e=vi + 1
Then it follows immediately that
Mx y)= n hi'j(x'y).
1 <i<j<s
Proposition 3.5. The T-equivariant class a is represented by the polynomial hM(x, y).
Proof. It is straightforward to verify that the polynomial representative we give satisfies the restriction requirements of Proposition 3.4. Indeed, the Weyl group WL of L is a parabolic subgroup of the symmetric group on n-letters, embedded as those permutations preserving separately the sets {vi + 1,..., vi + for i = 0,..., s - 1. Applying such a permutation to the representative above (with the action by permutation of the indices on the x-variables, as in Proposition 3.3) gives the appropriate product of weights. On the other hand, applying any w e WL will clearly give 0, since such a permutation necessarily sends some l e {vi + 1,..., vi + (for some i) to some k e {vj + 1,..., Vj + ^} with i < j. This permutation forces the factor - yk to vanish, which, in turn, forces hM(x, y) to vanish. By Proposition 3.4, hM represents a.	□
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3.3 The orthogonal case
We now apply these computations specifically to the two type A cases described in Section 2.2, using all of the notational conventions defined there. We start with the case of (G, H) = (GLn, GOn). Let ^ = (^i,..., be a composition of n, and let B, T, and SM be as defined in Section 2.2. Let PM = LMUM be the standard parabolic subgroup containing B whose Levi factor LM corresponds to
With these choices made, let the x, y, and z variables be the generators for (G/B) explicitly described after the statement of Proposition 3.1. We seek a polynomial in the x, y, and z-variables which represents the class of HM • 1B/B g (GLn/B).
To find such a formula, we use a known formula for the closed H0 = On-orbit on GLn/B to deduce a formula for the SM-equivariant class of HM • 1B/B in (Lm/Bl^ ), and then apply Proposition 3.4. Indeed, we know from [15, 16] that when H0 = On, the class of H0 • 1B/B in Hgo (GLn/B) (here, S0 is the maximal torus of H0 described in Section 2.2) is given by
[Ho • 1B/B]so = n (xi + xj) = 2L"/2J fl xi fl (x< + Xj). (3.1)
1<i<j<n-i	i<n/2 1<i<j<n-i
The formula of (3.1) is given in [15] in the case that n is even, while an alternative formula is given in the case that n is odd. It is observed in [16] that the above formula applies equally well when n is odd.
Recall that when H = GOn, a maximal torus S of H has dimension one greater than the corresponding maximal torus S0 of H0. Thus the S-equivariant cohomology of GLn/B has one additional "equivariant variable", which we call z. In this case, it is no harder to show that the class of H • 1B/B in (GLn/B) is given by
[H • 1B/B]s= Pn(x, y,z):= fl (xi + Xj - 2z).	(3.2)
1<i<j<n-i
Note that by restricting from H|(G/B) to (G/B) (which amounts to setting the additional equivariant variable z to 0), we recover the original formula (3.1).
Combining these formulae with Proposition 3.4, for each composition ^ of n we are now ready to give case-specific formulae for the unique closed HM-orbit HM • 1B/B on GLn/B. (Recall that each of these orbits corresponds to the unique closed B-orbit on the corresponding G-orbit on the wonderful compactification of G/H.)
We consider the variables x = (x1,..., xn), y = (y1,..., yn), and z = (z1,..., zs), where s is the number of parts in the composition We divide the variables into smaller clusters dictated by the composition Let x(i) = (xVi+1,..., xVi+1) and y(i) = (yVi+1, ..., yVi+1). Then define
s
PM(x, y, z) := fl PMi(x(i), y(i),zi), i=1
where PMi is given by (3.2).
We now introduce an equivalent, but more explicit, description of the class [Hm • 1B/B]S^ which reflects the block decomposition associated to To this end, we introduce new notation for a fixed composition ^ = (^1,..., ^s) of n. First, for each of
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the s blocks of SM, define a polynomial /¿(x, z) as follows:
z) = JJ (xj - zi).
j = Vi+l
In words, the xj occurring in the terms of this product are those occurring in the first half of their block, and from each, we subtract the z-variable corresponding to that block. So for example, if n = 11, m = (6,5), then
/l(x, z) = (xi - Zl)(x2 - Zl)(x3 - Zl),
while
/2(x, z) = (xr - Z2)(xg - Z2). Next, for each block, define g(x, z) as follows:
gi(x, z) =	(xj + xfc - 2zj).
(Note that gj(x, z) = 1 unless m > 3.) So for m = (6,5) as above, we have that
gl(x, z) = (xi + x2 - 2zi)(xi + x3 - 2zi)(xi + x4 - 2zi)(xi + x5 - 2zi)
(x2 + x3 - 2zi)(x2 + x4 - 2zi),
and
g2 (x, z) = (xr + xg - 2z2)(xr + xg - 2z2)(xr + xio - 2z2)(xg + xg - 2Z2).
Finally, we define a third polynomial hM(x, y, z) in the x, y, and z-variables to simply be hM(x, p(y)), where p denotes restriction from the variables yl,..., yn corresponding to coordinates on the full torus T to the variables yitj, zi on the smaller torus SM. To be more explicit, for each i, j with 1 < i < j < s define hi,j (x, y, z) to be
rifc=l nr=jl2(xVi+fc - y^- zj x^+k+y^- zj)	if mis even,
nr= l (xVi+k - zj) n[=l/2J (xVi+k - yj,i - zj)(xVi+k + yj,i - zj) if Mjis odd. So for the case n = 4, m = (2,2), we have
hl,2(x, y, z) = (xi - y2,l - Z2)(xi + y2,l - Z2)(x2 - y2,l - Z2)(x2 + y2,l - Z2), while for the case n = 5, m = (2, 3), we have
hl,2(x, y, z) = (xl - Z2)(x2 - Z2)(xl - y2,l - Z2)(xi + y2,l - Z2)
(x2 - y2,l - Z2)(x2 + y2,l - Z2).
Then we define
hM(x, y, z) = JJ hj,j(x, y, z).
l<i<j<s
Propositions 3.4 and 3.5 then imply the following formula for the SM-equivariant class of Hm • 1B/B in this case.
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Corollary 3.6. The S'-equivariant class of the unique closed H'-orbit HM • 1B/B on G/B is represented by the polynomial
s
2d(M)hM(x, y, z) H fi(x, z)gi(x, z),
i=i
where d(^) = ES=iLMi/2J.
Proof. The fact that the product	si=1 /¿(x, z)gj(x, z) is equal to the formula for
j* [Q] follows from the formula of (3.2). It follows from Proposition 3.4 and Proposition 3.5 that the representative of [Q] is obtained from j * [Q] by multiplying with the top SM-equivariant Chern class of the normal bundle, which is represented by the polynomial
hM(x, y, z).	□
Corollary 3.7. The S'-equivariant class [HM • 1B/B] is represented by the polynomial P'(x, y, z)hM(x, y, z).
Proof. This follows immediately from the observation that
(x(i), y(i), zi) = /2f(x, z)ffi(x, z).	□
3.4 The symplectic case
We give similar (but simpler) formulas for the case when (G, H') = (GL2n, GSp2n). Recall from [15] that for the case (G, H0) = (GL2n, Sp2n), the S0-equivariant class (S0 the maximal torus of H0 described in Section 2.2) of the unique closed orbit H0 • 1B/B is given by
[H0 • 1B/B]s0 = H (xi + xj).	(3.3)
1<i<j<2n-i
As before, it is no harder to see that if S' is the maximal torus of diagonal elements of H', then the S'-equivariant class of the closed orbit H' • 1B/B is represented by
[H' • 1B/B]s' = Pn(x, y,z):= H (xi + Xj - 2z).	(3.4)
1<i<j<2n-i
Now let ^ = (^1,..., ^s) be a composition of 2n with all even parts. Let H' be the spherical group
(GSpMi x---x GSpMs ) k UM,
as defined in Section 2.2. Let S'M be the maximal torus of H', with the y and z-variables as defined above.
Corollary 3.8. In HS*, (G/B), the class of the closed H'-orbit H' • 1B/B is represented by
s
hM(x, y, z^ gi(x, z).
i=1
Proof. The proof is identical to that of Corollary 3.6, using Proposition 3.4 combined with (3.4) in the same way.	□
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Remark 3.9. Note that gj(x, z) = P^. (x(i), y(i), zj), where P^. is given by (3.4), so that again [H^ • 1B/B] is equal to P^(x, y, z)hM(x, y, z), where
s
P> (x, y, z) := n P* (x(j), y(j),Zj). i=i
We now specialize these formulas to ordinary cohomology (by setting all y and z-variables to 0), in order to prove Corollaries 1.1 and 1.2. First, we define the notations used in those formulas which have not yet been defined. For each i = 1,..., n, let B(p,i) denote the block that the variable xi occurs in, i.e. b(v, i) is the smallest integer j such that
j
-i.
i=\
Then for each i = 1,..., n, define r(v, i) to be
R(v,i):=	Vj.
B(n ,i)<j<s
This is the combined size of all blocks occurring strictly to the right of the block in which xi occurs.
Finally, again for each such i, define S(p, i) to be 1 if and only if xi occurs in the first half of its block, and 0 otherwise. Note that by the "first half" we mean those positions less than or equal to 1/2 where I is the size of the block; in particular, for a block of odd size, the middle position is not considered to be in the first half of the block.
Proof of Corollaries 1.1 and 1.2. The formula of Corollary 1.1 comes from that of Corollary 3.6; we simply set y = z = 0. The binomial terms Xj + xk come from the polynomials gi(x, 0). The monomial terms come from the polynomials fi(x, 0) and hi,j (x, 0,0). The x^O,i) term comes from fi(x, 0), the latter being xi if this variable occurs in the first half of its block, and 1 otherwise. The remaining xR(M,i) comes from the hi,j (x, 0,0). Indeed, it is evident that for an x-variable in block i, for each j > i the given x-variable appears in precisely ^j linear forms involving y, z terms associated with block j.
The proof of Corollary 1.2 is almost identical, except simpler.	□
4 Factoring sums of Schubert polynomials
We end by establishing explicit polynomial identities involving sums of Schubert polynomials, using the cohomological formulae of the preceding section together with the results of [5, 6] which were recalled in Section 2.4.
Note that by Brion's formula (1.2) combined with the fact that the Schubert polynomial Sw is a representative of the class of the Schubert variety Xw in H*(G/B), we have the following two families of identities in H*(G/B):
^ 2d(Y-w)6w = 2d(M) J]xf(^i)+5(^i) n ( n (xj + xk)Y (4.1)
wew(Ym)	i=l	i=l \ Vi + l<j<k<vi+i-j	)
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2n	s /	\
2 6w=n xf(M,î) n n (**+. (4.2)
w£W(Y^)	i=1	i=1 V Vi + 1<j<fc<Vi+i-j	/
Equation (4.1) above simply combines (1.2) with Corollary 1.1. Likewise, (4.2) combines (1.2) with Corollary 1.2, together with the fact that all B-orbit closures in the sym-plectic case are known to be multiplicity-free, meaning d( Y^, w) = 0 for all w e W(Y^).
In fact, also in (4.1) above, the powers of 2 can be completely eliminated from both sides of the equation. This follows from a result of Brion [4, Proposition 5], which states that whenever G is a simply laced group (recall that for us, G = GLn is simply laced), all of the coefficients appearing in (1.2) are the same power of 2. It is explained in [5, Section 5] that the coefficients appearing on the left-hand side of (4.1) are in fact all equal to 2d(M). Since H*(G/B) has no torsion, we have the simplified equality
2 S n
w£W(Ym)	i=i	i=1 \ Vi + 1<j<fc<vi+i
Sw=n xf^)^^ n n ^+xfc). (4.3)
Now, note that a priori, the identities (4.2) and (4.3) hold only in H* (G/B). That is, we know only that the left and right-hand sides of the identities are congruent modulo the ideal IW. We end with a stronger result.
Theorem 4.1. Both (4.2) and (4.3) are valid as polynomial identities.
Proof. We use the fact that the Schubert polynomials |6w | w g Sn} are a Z-basis for the Z-submodule r of Z[x] spanned by monomials f] xf with cj < n — i for each i [11, Proposition 2.5.4]. We claim that on the right-hand side of (4.3) (resp. (4.2)), each xj does in fact occur with exponent at most n — i (resp. 2n — i). To see this, note that since the right-hand side of each identity is a product of linear forms, it suffices to count, for each i, the number of these linear factors in which xj appears.
In (4.3), we claim that xj appears in precisely	j) Mi) — i of the linear factors.
Since J2s=Bo j) Mj < J2s=i Mj = n, this establishes our claim that the right-hand side lies in r. Indeed, clearly xj appears r(m, i) + ¿(m, i) times as a monomial factor, so we need only count the number of binomial factors of the form Xj + xk that it appears in. One checks easily that it appears in Mj — i — 1 such factors if xj occurs in the first half of its block, and in Mi — i such factors otherwise. In other words, if N is the number of binomial factors involving xj, then we have ¿(m, i) + N = Mi — i. Thus
r(m, i) + ¿(m, i) + Nj = r(m, i) + Mi — i
= (	Mi I + Mi — i
Vj=B(M,j) + i )
2
- »,
as claimed.
Clearly, the right-hand side of (4.2) also lies in r, applying the same argument with n replaced by 2n. Indeed, the only difference is in the lack of the additional monomial factor
j
M. B. Can et al.: Wonderful symmetric varieties and Schubert polynomials	541
xf(m'") ; thus xi occurs in either (Xj=B(Mli) Pj) - i or (Zj=B(M,i) Pi) - i -1 of the linear factors on the right-hand side of (4.2). In either event, this is at most 2n - i, as required.
Now, since the right-hand side of each of (4.2) and (4.3) are in r, they are expressible as a sum of Schubert polynomials whose indexing permutations lie in S2n (for (4.2)) or Sn (for (4.3)) in exactly one way. Furthermore, since the Schubert classes {[Xw]} are a Z-basis for H* (G/B), the cohomology class represented by the right-hand side of (4.2) and (4.3) is a Z-linear combination of Schubert classes in precisely one way. Clearly, the same indexing permutations must arise with the same multiplicities in both the polynomial expansion and the cohomology expansion. Then since (4.2) and (4.3) are correct cohomologically, they must also be polynomial identities.	□
Example 4.2. In the orthogonal case when p = (3,4), the identity (4.3) becomes
S6752431 + S6753412 + S6754213 + S7562431 + S7563412 + S7564213 =
x1x|x|x4x5(xi + X2)(X4 + X5 )(x4 + X6).
References
[1]	A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953), 115-207, doi:10.2307/1969728.
[2]	M. Brion, The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv. 73 (1998), 137-174, doi:10.1007/s000140050049.
[3]	M. Brion, Equivariant cohomology and equivariant intersection theory, in: A. Broer and G. Sabidussi (eds.), Representation Theories and Algebraic Geometry, Kluwer Academic Publishers, Dordrecht, volume 514 of NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pp. 1-37, 1998, doi:10.1007/978-94-015-9131-7_1, proceedings of the NATO Advanced Study Institute and Seminaire de Mathematiques Superieures held at the Universite de Montreal, Montreal, PQ, July 28 - August 8, 1997.
[4]	M. Brion, On orbit closures of spherical subgroups in flag varieties, Comment. Math. Helv. 76 (2001), 263-299, doi:10.1007/pl00000379.
[5]	M. B. Can and M. Joyce, Weak order on complete quadrics, Trans. Amer. Math. Soc. 365 (2013), 6269-6282, doi:10.1090/s0002-9947-2013-05813-8.
[6]	M. B. Can, M. Joyce and B. Wyser, Chains in weak order posets associated to involutions, J. Comb. Theory Ser. A 137 (2016), 207-225, doi:10.1016/j.jcta.2015.09.001.
[7]	C. De Concini and C. Procesi, Complete symmetric varieties, in: F. Gherardelli (ed.), Invariant Theory, Springer, Berlin, volume 996 of Lecture Notes in Mathematics, 1983 pp. 1-44, doi:10.1007/bfb0063234, proceedings of the 1st 1982 Session of the Centro Internazionale Matematico Estivo (CIME) held at Montecatini, June 10- 18, 1982.
[8]	D. Edidin and W. Graham, Localization in equivariant intersection theory and the Bott residue formula, Amer. J. Math. 120 (1998), 619-636, doi:10.1353/ajm.1998.0020.
[9]	D. Laksov, Completed quadrics and linear maps, in: S. J. Bloch (ed.), Algebraic Geometry, Bowdoin, 1985, Part 2, American Mathematical Society, Providence, RI, volume 46 of Proceedings of Symposia in Pure Mathematics, 1987 pp. 371-387, proceedings of the summer research institute held at Bowdoin College, Brunswick, Maine, July 8 - 26, 1985.
[10] A. Lascoux and M.-P. Schutzenberger, Polynfimes de Schubert, C. R. Acad. Sci. Paris Ser. I Math. 294 (1982), 447-450, doi:10.1090/conm/088/1000001.
542
Ars Math. Contemp. 15 (2018) 441-466
[11]	L. Manivel, Symmetric Functions, Schubert Polynomials and Degeneracy Loci, volume 6 of SMF/AMS Texts and Monographs, American Mathematical Society, Providence, RI, 2001, translated from the 1998 French original by J. R. Swallow.
[12]	N. Ressayre, Spherical homogeneous spaces of minimal rank, Adv. Math. 224 (2010), 1784— 1800, doi:10.1016/j.aim.2010.01.014.
[13]	J. G. Semple, On complete quadrics, J. London Math. Soc. 23 (1948), 258-267, doi:10.1112/ jlms/s1-23.4.258.
[14]	D. A. Timashev, Homogeneous Spaces and Equivariant Embeddings, volume 138 of Encyclopaedia of Mathematical Sciences, Springer, Heidelberg, 2011, doi:10.1007/ 978-3-642-18399-7.
[15]	B. J. Wyser, K-orbit closures on G/B as universal degeneracy loci for flagged vector bundles with symmetric or skew-symmetric bilinear form, Transform. Groups 18 (2013), 557-594, doi: 10.1007/s00031-013-9221-1.
[16]	B. J. Wyser and A. Yong, Polynomials for symmetric orbit closures in the flag variety, Transform. Groups 22 (2017), 267-290, doi:10.1007/s00031-016-9381-x.