Multiplicity on a Richardson variety in a cominuscule G/P

We show that in a cominuscule partial flag variety G/P, the multiplicity of an arbitrary point on a Richardson variety X_w^v = X_w \cap X^v is the product of its multiplicities on the Schubert varieties X_w and X^v.


Introduction
Richardson varieties, named after [33], are intersections of a Schubert variety and an opposite Schubert variety inside a partial flag variety G/P (G a connected complex semi-simple group, P a parabolic subgroup). They previously appeared in [9, Ch. XIV, §4] and [36], as well as the corresponding open cells in [6]. They have since played a role in different contexts, such as equivariant K-theory [24], positivity in Grothendieck groups [3], standard monomial theory [4], Poisson geometry [8], positroid varieties [13], and their generalizations [14,1].
On the other hand, singularities of Schubert varieties have been extensively studied in the last decades. The singular locus of Schubert varieties in Grassmannians has been determined independently in [37] and [27], and more generally in a minuscule G/P in [26]. In the full flag variety of type A n , it has been determined independently in [2], [5], [12], and [29].
Moreover, the multiplicity of a singular point on a Schubert variety is known in several cases: when G/P is minuscule of arbitrary type, or cominuscule of type C n , a recursive formula was given in [26]. A direct determinantal formula was given in [34] for G/P a Grassmannian; it has been subsequently interpreted in terms of nonintersecting lattice paths [17]. The multiplicity problem has also been studied in relationship with Hilbert functions and Gröbner degenerations [7,16,18,23,31,32], as well as with T -equivariant cohomology [10,11,15,20,21,25]. The problem of determining the multiplicity of a point in a Schubert variety in the full flag variety is more complicated; see [39,28,40,41].
For Richardson varieties in a minuscule G/P , the multiplicity of a T -fixed point (T ⊂ P a maximal torus in G) has been determined by Kreiman and Lakshmibai [22] (for the Gröbner point of view, see also [19] in type A n and [38] in orthogonal types).
In this paper, we determine the multiplicity of an arbitrary point 1 on a Richardson variety in a cominuscule G/P .
Before stating the main result, let us fix some notation. Let G, P, T be as above, with G adjoint. Let X(T ) be the character group of T , R ⊂ X(T ) the root system, Date: April 21, 2011. 2010 Mathematics Subject Classification. Primary 14M15; Secondary 14B05 14L30. 1 Note that unlike in the case of a Schubert variety, this no longer follows from information about T -fixed points, as pointed out in the introductions of [22] and [19]. and W = N G (T )/T its Weyl group. Let B ⊂ G be a Borel subgroup such that T ⊂ B ⊂ P : it determines a system of positive roots R + and a system of simple roots S. Denote by B − the opposite Borel subgroup (i.e. such that B ∩ B − = T ).
Let W P ⊂ W be the subgroup associated to P (so that W G = W and W B is the trivial subgroup). In the quotient W P = W/W P , every coset wP contains a unique minimal element for the Bruhat order ≤ on W , so we shall identify W P with the set of minimal representatives. The B-orbit (resp. the B − -orbit) of a T -fixed point e τ = τ P is called a Schubert cell (resp. an opposite Schubert cell) in G/P , and denoted by C τ (resp. C τ ). Its closure is the Schubert variety X τ (resp. the opposite Schubert variety X τ ).
If v, w ∈ W P , then the intersection X v w = X w ∩X v is called a Richardson variety; it is non-empty if and only if v ≤ w (note that Schubert varieties are the particular cases X w = X e w and X v = X v w0 , where e, w 0 ∈ W are the identity and the longest element, respectively). Now assume P to be maximal, and let α be the associated simple root (so that W P is generated by the reflections s δ with δ ∈ S \ {α}). Then P (or α) is said to be • cominuscule if α occurs with a coefficient 1 in the decomposition of the highest root of R + ; • minuscule if α ∨ is cominuscule in the dual root system R ∨ .
The main result of this paper is the following Theorem 0.1. Assume P is cominuscule. Let m ∈ X v w be arbitrary, and denote by µ w (resp. µ v , µ v w ) the multiplicity of m on X w (resp. X v , X v w ). Then This result indeed determines the multiplicities on X v w , since those on X w and X v are known: types A n , D n , E 6 , E 7 are covered by [26], Section 3 (since cominuscule is equivalent to minuscule in those types), and type C n is covered by [26], Section 4. The only remaining case, in type B n (cf. the table below), is elementary, and covered in the Appendix of the present paper for the sake of completeness.
Note that (1) is exactly the result obtained in [22] for a T -fixed point in a minuscule G/P .
To prove the theorem, we shall use a description of the multiplicity using a central projection: namely, given a projective variety X ⊂ P N and a point m ∈ X, we consider the projection p m , of centre m, onto a hyperplane not containing m. Then the multiplicity of m on X is the difference between the degree of X and the projective degree of p m . Note that the projective degree of p m is zero when X is a cone. We apply this description for X the projective closure of the affine trace is an affine open subset of G/P identified with A N . One then needs to know whether the affine traces of X w , X v , X v w are cones or not. In this setting, we can explain why we assume that P is cominuscule: • it implies that X w ∩ O τ is a cone over any point of the cell C τ (though this may not be the case for X v ∩ O τ ); • we relate the central projection p m to a map which turns out to be a Caction if P is cominuscule. It is this C-action which allows to prove all the necessary properties for p m .
In Section 1, we give a system of local coordinates in which X w ∩O τ is a cone over both e τ and m, and X v ∩ O τ over e τ . In Section 2, we prove Theorem 0.1 assuming certain formulas for the degrees involved and that X v ∩ O τ is not a cone over m. These assumptions are summarized in Proposition 2.1, and proved in Sections 4 and 5. The proofs are based on a C-action linking the central projections of centres m and e τ ; this action is defined and studied in Section 3.
For the convenience of the reader, we give the minuscule and cominuscule weights in the following table: There are no minuscule nor cominuscule fundamental weight in type E 8 , F 4 , G 2 .
Assumption. For the rest of the paper, the parabolic subgroup P is assumed to be cominuscule.
Acknowledgements. I would like to thank Christian Ohn for helpful discussions, and Takeshi Ikeda for pointing out several references in the literature. I am also grateful to the referee for his valuable remarks, and especially for pointing out a gap in the proof of Proposition 4.5 and for providing a way to fill it.

Local coordinates
The notations are as in the Introduction. Moreover, R P denotes the root system associated with P : where R u (P ) is the unipotent radical of P , and U β is the root subgroup associated with β.
Let m ∈ X v w . Then m lies in a Schubert cell C τ for some τ ∈ W P . Let for all t ∈ T and all x ∈ C. Let N be the cardinality of R + \ R + P . We identify O τ with the affine space A N via the isomorphism (In particular, N is the dimension of G/P .) Proof. Since B is the semi-direct product of T and the unipotent subgroup U , we have C τ = U.e τ . Moreover, for any ordering of positive roots {β 1 , . . . , β p }, We choose an ordering such that the positive roots β with −β / ∈ τ (R + \ R + P ) appear at the end. Then, by the preceding lemma, we have: The following lemma will be useful for the next section. Lemma 1.3. For all β, γ ∈ τ (R + \ R + P ) and for all x, y ∈ C, the elements θ β (x) and θ γ (y) commute.
Proof. We use the following expansion for the commutator (cf. [35], proposition 8.2.3): where c β,γ,i,j are some constants in C. Since the commutator must lie in U − τ , it suffices to prove that the roots of the form iβ + jγ do not lie in τ (R + \ R + P ). Now, P is the parabolic subgroup associated with the simple root α. Since α is cominuscule, a positive root δ lies in R + \ R + P if and only if α occurs with coefficient 1 in the expression of δ. Clearly, α occurs with a coefficient i + j in τ −1 (iβ + jγ).
It is a group of type A n−1 . The torus T is the group of diagonal matrices of determinant 1, and the Borel subgroup B is the group of upper triangular matrices of determinant 1. The roots are denoted α i,j , where The positive roots are the α i,j with i < j, and the simple roots are the α i = α i,i+1 (i = 1, . . . , n − 1). Let ω = ω d be the fundamental weight associated with the simple root α d . The corresponding parabolic subgroup P is The group G acts transitively on the Grassmannian G d,n of d-spaces in C n , and P is the isotropy subgroup of the vector space generated by e 1 , . . . , e d , where (e 1 , . . . , e n ) is the canonical basis of C n . The Weyl group W of this root system is S n , and W P is isomorphic to S d × S n−d , so The Lie algebra g of G is the space of traceless matrices. Let t be the Lie algebra of the torus T . We have the weight decomposition of g: where E i,j is the elementary matrix with a 1 on the row i and column j, and zero elsewhere. Thus, the root subgroups are given by and the isomorphism θ αi,j is just x → exp(xE ij ). Moreover, Returning to the general case, we denote by (m −β |β ∈ τ (R + \ R + P )) the coordinates of m, that is, Notations 1.6. We set: We now investigate if these affine varieties are cones over m.
w are cones over e τ . Proposition 1.8. The variety Y w is a cone over m.
Proof. Consider the translation that maps e τ to m. It is given in coordinates by ( Since m −β = 0 for all β > 0, we have b ∈ B according to Lemma 1.2. Now b leaves Y w invariant and maps e τ to m.
However, the opposite Schubert variety Y v need not be a cone over m. Example 1.9. We take the same notations as in Example 1.5. In particular, using the identification W P = I d,n , we denote a Schubert variety in G d,n by X i1...i d , and similarly for opposite Schubert and Richardson varieties. In the Grassmannian G 3,7 , consider the Richardson Here, a matrix between brackets actually stands for the 3-space in C 7 generated by its columns. The equations of X 356 are: The equations of X 125 are: We set: If Y v is indeed a cone over m, then we have the following result. The proof is taken from [22], Remark 7.6.6.
as well, so we may consider the projective varieties P(Y w ), P(Y v ) and P(Y v w ), consisting of lines through m. Then µ w (resp. µ v , µ v w ) is just the degree of P(Y w ) (resp. P(Y v ), P(Y v w )). We conclude with Bézout's theorem since P(Y w ) and P(Y v ) intersect transversely (cf. [33], Corollary 1.5).
Assumption 1.11. For the rest of the paper, we assume that Y v is not a cone over m.
It is not clear however whether Y v w is a cone or not. This problem will be solved in Section 4.

2.
Central projection and proof of Theorem 0.1 We shall compute the multiplicity of a point m ∈ Y v w by relating it to degrees of projections, which requires us to work in a projective setting. More precisely, embed A N into P N via ι : and consider the projective closures We also identify P N −1 with the hyperplane at infinity ξ = 0 and consider the central projection p m : P N → P N −1 , sending any point x = m to the intersection of the line (mx) with P N −1 . If X ⊂ P N is any projective variety and m ∈ X, then we have the following formula (cf. [30], Theorem 5.11): where deg X is the degree of X, deg(p m ) |X is the degree of the rational map p m restricted to X, and p m X denotes the Zariski closure of p m (X \ {m}).
We defer the proof to Section 4.
Proof of Theorem 0.1. Using (5) and Proposition 2.1, we obtain Remark 2.2. In particular, this result enables us to find the singular locus of X v w in terms of those of X w and X v : the point m is smooth on X v w if and only if µ v w = 1 if and only if µ w = µ v = 1, that is, if and only if m is smooth on both X w and X v . Note that this may also be seen more directly, using the fact that X w and X v intersect properly and transversely at any point at which µ w = µ v = 1 (cf. [33] Corollary 1.5, or [1] Corollary 2.9).

C-action on G/P
In this section, we introduce the main tool that will permit us to prove Proposition 2.1 in the next section. Let e τ , m ∈ O τ be as before: we shall construct an action of (the additive group) C on G/P for which e τ and m are in the same orbit.
Consider first the map where b ∈ B ∩ U − τ is the element defined in the proof of Proposition 1.8. The computation (3) shows that this map extends to a group homomorphism ϕ : C → B. The natural B-action on G/P thus induces a C-action: Moreover, O τ is invariant under this action (again by (3)). Actually, C acts on O τ = A N by translations: indeed, we get the following commutative diagram Thus, the commutative diagram (6) restricts to Remark 3.1. Since (6) is a fibre product diagram, any fibre Φ −1 (λy) (for λ = 0 and [y] ∈ P N −1 ) is mapped isomorphically via f to the fibre p −1 m ([y]). Since we have the equalities In the next section, this remark will allow us to relate the degree of p m in diagram (7) to that of Φ.

Proof of Proposition 2.1
Proof of (a). Since Y w , Y v , and Y v w are (affine) cones over e τ , it is clear that  • Z v w is a cone over m, Proof. By Remark 3.1, we see that the dimension of a generic fibre of Φ equals the dimension of a generic fibre of p m . Now Z v w is a cone over m if and only if every fibre of p m has dimension 1, if and only if dim Proof of (b) and (c). By Proposition 4.2, it suffices to compare the degree In particular, a point lies in Im There exists an open set Ω v of F v in which the fibre of every point y consists of d v points. Then d v is just the number of points in the C-orbit of y that belong to Y v . Now set y = (y −β ) β∈τ (R + \R + P ) and let c = so we have c.e τ = c − .y =: x. Since c ∈ B, x ∈ C τ ⊂ Y w . Now c − commutes with ϕ ξ for all ξ ∈ C, hence every point in c − (Ω v ) has a C-orbit which meets Y v in exactly d v points. In particular, F v w = Y v w , since otherwise every fibre of Φ v w would have dimension 1 (by Lemma 4.3), which is not the case for the fibre of x. This already shows (b), so it makes sense to talk about the degree d v w of Φ v w . Thus, let Ω v w be an open set of F v w such that for every point z in Ω v w , the fibre of z consists and Ω v w are non-empty open sets of the irreducible variety F v w , so they must meet. Taking z in this intersection, we see that d v w = d v , which shows (c).
Indeed, take a smooth point x of F v belonging to Φ(C × Y v ). We have seen in the previous proof that from x we can construct an isomorphism c − of F v mapping x to a point of F v w , which thus remains smooth on F v . The two non-empty open subsets Ω w and Ω v of the irreducible variety F v w have a non-empty intersection Ω Let us summarize the properties of Ω: it is a non-empty open subset of F v w , whose every point y is smooth in both F w and F v , and y = Φ(p) with p smooth in ∈ Ω be such a point. We view the map Φ : x as a map Φ : C N +1 → C N . It is linear and surjective. Thus, This transversality result proves that the intersection is proper: indeed, on one Proposition 4.5. We have the equality F v w = F w ∩ F v . In particular, the intersection F w ∩ F v is generically transverse.
This result will be proved in the next section.
Proof of (d). Since y = Φ(ξ, x) implies zy = Φ(zξ, zx) for all z ∈ C, Φ(C× Y v w ) is a cone over e τ , and so is its closure F v w . But by the commutative diagram (7), , and similarly deg(p m Z w ) = deg(F w ) and deg(p m Z v ) = deg(F v ). Equality (d) now follows from Proposition 4.4 and Bézout's theorem, noting that deg(p m Z w ) = deg(Z w ).

Proof of Proposition 4.5
Since x))|ξ ∈ C, x ∈ G/P } and Γ be its closure in P 1 ×G/P ×G/P (so Γ is the graph of Φ viewed as a rational map). We have a commutative diagram: The morphism π 1 × π 2 : Γ → P 1 × G/P is surjective, and restricts to an isomorphism between U and C × G/P . In particular, Γ is an irreducible projective variety of dimension N + 1.
Likewise, let U w = {(ξ, x, Φ(ξ, x))|ξ ∈ C, x ∈ X w } and Γ w be its closure, and similarly for We now need to study the π 3 -fibre of a point in F w . Actually, if y is in Y w , then its fibre lies entirely in Γ w . Indeed, U − τ naturally acts on G/P and on Γ via g.(ξ, x, y) = (ξ, g.x, g.y) (since U − τ is Abelian), and the morphism π 3 is U − τequivariant. It follows that whenever two points in G/P belong to the same U − τorbit, their fibres are isomorphic. Now since π 3 : Γ → G/P is dominant, there is an open set in G/P in which every point has a fibre of pure dimension 1. Since O τ is open in G/P , it meets this open set, and since O τ is a U − τ -orbit in G/P , y itself has a fibre of pure dimension 1. Now fix an irreducible component C of π −1 3 (y). Then π 1 × π 2 (C) ∩ (C × G/P ) ⊂ Φ −1 (y).
If C ∩ U = ∅, then the left hand side of this inclusion is non-empty and of dimension 1. Since Φ −1 (y) is isomorphic to the C-orbit of y, it is itself irreducible of dimension (at most) 1, hence the inclusion becomes an equality. Taking closures, we then obtain C = {(ξ, x, y)|(ξ, x) ∈ Φ −1 (y)}; in particular, C is the unique irreducible component of π −1 3 (y) that intersects U. Note also that C ⊂ Γ w . Now let C ′ be an irreducible component of π −1 3 (y) different from C, so that C ′ ⊂ {∞} × G/P × {y}. Let Γ ∞ ⊂ Γ be the subvariety π −1 1 (∞). We have a U − τ -equivariant morphism π : Γ ∞ → G/P : (∞, x, y) → y, so C ′ is an irreducible subvariety of the fibre π −1 (y). Since Γ ∞ Γ, its dimension is at most N . Because of the equivariance of π, we see that O τ is in the image of π, so π is surjective. Decomposing Γ ∞ into irreducible components Γ ∞ = C 1 ∪ · · · ∪ C r , we obtain G/P = π(C 1 ) ∪ · · · ∪ π(C r ), so that for some i, π : C i → G/P is onto. Renumbering the C i , we may assume that for some t ≥ 1, C 1 , . . . , C t are mapped surjectively to G/P , and C t+1 , . . . , C r are not. For i ≤ t, there is an open set U i of G/P such that each element on U i has a finite fibre in C i . For i > t, let U i be the open set G/P \ π(C i ). Taking the intersection U = n i=1 U i , we obtain a non-empty open set of G/P satisfying the following property: for each z ∈ U , the fibre of z in Γ ∞ consists of a finite number of points. Again, U meets the open orbit O τ , so this property is true for every point in O τ , in particular for y. So C ′ is included in the finite fibre π −1 (y): a contradiction. Therefore, C ′ cannot exist, i.e. π −1 3 (y) = C ⊂ Γ w is irreducible, and not contained in {∞} × G/P × G/P .
Appendix. Singularities of Schubert varieties in SO(2n + 1)/P 1 In this Appendix, we shall determine the singular locus of Schubert varieties in G/P , where G is of type B n and P is cominuscule. So let V = C 2n+1 together with a non-degenerate symmetric bilinear form (., .) given in the canonical basis (e 1 , . . . , e 2n+1 ) by the anti-diagonal matrix E with 1's all along the anti-diagonal. The expression of the quadratic form Q associated with (., .) is x i x 2n+2−i .
The Jacobian criterion easily shows that Sing X i is equal to X 2n+1−i if i > n + 1, and empty if i < n+1. Moreover, since X i is defined by a single quadratic equation, the multiplicity of a singular point must be equal to 2. Hence there are two cases for the multiplicity µ i (x) of a point x = [x 1 : · · · : x i : 0 : · · · : 0] on X i : • Case 1: i < n + 1. Then µ i (x) = 1.
Note that a Richardson variety X j i (j ≤ i) also is a quadric in a projective space, so the multiplicity of a point m ∈ X j i must be at most 2. But by Theorem 0.1, if m were singular in both X i and in X j , then its multiplicity would be 4. This means that Sing X i ∩Sing X j = ∅, a fact that can also be verified directly: indeed, if this intersection is non-empty, then 2n+3−j ≤ 2n+1−i, so j ≤ i ≤ j −2, a contradiction.