EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS

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1 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS TAKESHI IKEDA AND HIROSHI NARUSE Abstract. We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call excited Young diagrams and the second one is written in terms of factorial Schur Q- or P -functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety. 1. Introduction In this paper, we give explicit descriptions of the Schubert classes in the (torus) equivariant cohomology ring of the Grassmannians as well as the maximal isotropic Grassmannians of both symplectic and orthogonal types. Our main results express the image of an equivariant Schubert class under the localization map to the torus fixed points. Now let us fix some notation. Let G be a complex semisimple connected algebraic group. Choose a maximal torus T of G and a Borel subgroup B containing T. Let P be a maximal parabolic subgroup of G containing B. We are interested in the (integral) T -equivariant cohomology ring HT (G/P ) of the homogeneous space G/P. The equivariant Schubert classes are parametrized by the set W P of minimal length representatives for W/W P, where W is the Weyl groups of G and W P is the parabolic subgroup associated to P. The set W P also parametrizes the T -fixed points (G/P ) T in G/P. In fact if we put e v = vp (v W P ) then (G/P ) T = {e v } v W P. Let B denote the opposite Borel subgroup such that B B = T. Define the Schubert variety X w associated to the element w W P, to be the closure of B -orbit B e w of e w. Note that the codimension of X w in G/P is l(w), the length of w, and e v X w if and only if w v, where is the partial order on W P induced by the Bruhat-Chevalley ordering of W. Since X w is a T -stable subvariety in G/P, it induces a T -equivariant fundamental class, the equivariant Schubert class, denoted by [X w ] H 2l(w) T (G/P ). Our main goal is to describe [X w ] explicitly. In this paper, we consider G/P in the following list: Type A n 1 : SL(n)/P d (1 d n), Type B n : SO(2n + 1)/P n, Type C n : Sp(2n)/P n, Type D n : SO(2n)/P d (d = n 1, n) where we denote by P d the maximal parabolic subgroup associated to the d-th simple root (the simple roots being indexed as in [6]). It is well known that the space SL(n)/P d can be identified with the Grassmannian G d,n of d-dimensional subspaces in C n. Any other 2000 Mathematics Subject Classification. Primary 05E15, Secondary 14M15. Key words and phrases. Young diagrams, Grassmannians, equivariant Schubert classes, multiplicity. 1

2 2 TAKESHI IKEDA AND HIROSHI NARUSE G/P in the above list is a maximal isotropic Grassmannian with respect to an orthogonal or symplectic form (see Section 3 for details). Our description is based on the ring homomorphism ι : H T (G/P ) H T ((G/P ) T ) = v W P H T (e v ), induced by the inclusion ι : (G/P ) T G/P. This ι is known to be injective and called the localization map. Each summand HT (e v) is canonically isomorphic to the symmetric algebra S = Sym Z ( ˆT ) of the character group ˆT of the torus T. Thus the equivariant Schubert class [X w ] is described by a list {[X w ] v } v W P of polynomials in S, where [X w ] v denote the image of the equivariant Schubert class [X w ] under the homomorphism ι v : HT (G/P ) H T (e v) induced by the inclusion ι v : {e v } G/P. In type A n 1 case (cf. Sec. 5 Theorem 2), Knutson and Tao [20] discovered that [X w ] v can be identified with a suitably specialized factorial Schur function, a multi-parameter deformation of Schur function (see Section 5 for the definition). Their argument uses a remarkable vanishing property of the factorial Schur function (cf. Proposition 5). By a totally different method, Lakshmibai, Raghavan, and Sankaran [22] showed the same result, although they do not state it explicitly in terms of factorial Schur function. In fact, they start from a combinatorial expression for [X w ] v in terms of a set of non-intersecting paths, which comes from a detailed analysis of Gröbner basis of the defining ideal of the Schubert variety due to Kreiman and Lakshmibai [19], and Kodiyalam and Raghavan [12], and then rewrite the expression into a ratio of some determinants, which is a form of factorial Schur function. Type C n, the case of Lagrangian Grassmannian, was studied in a paper [10] by the first named author, where [X w ] v is expressed in terms of factorial Schur Q-function defined by Ivanov [11]. The proof is a comparison of Pieri-Chevalley type recurrence relations for both [X w ] v and the factorial Schur Q-function. This strategy of identification goes well for other G/P in our list above. Actually, we prove in this paper the analogous result for types B n and D n, the orthogonal Grassmannian, i.e., we present a formula for [X w ] v in terms of factorial Schur P -function for these spaces (Sec. 3 Theorem 4). As an application of these formulas, we obtained a Giambelli-type formula (Corollary 2) for isotropic Grassmannians that expresses an arbitrary equivariant Schubert class as a Pfaffian of Schubert classes associated with the two-row (strict) partitions. This formula is an equivariant analogue of the Giambelli formula due to Pragacz [26] in the case of ordinary cohomology. Another type of formulas (Sec.3 Theorem 1, Sec.6 Theorem 3) we discuss in the present paper involve combinatorial objects, which we call excited Young diagrams, EYDs for short. The idea of EYDs was inspired by the work [22] of Lakshmibai, Raghavan, and Sankaran mentioned above. These formulas have a positive nature in the sense that it is expressed as a sum over a set of EYDs with each summand being a products of some positive roots. For an exposition, here we consider G d,n. It is well-known that the set W P d is parametrized by the set of partitions λ = (λ 1,..., λ d ) such that n d λ 1 λ d 0, or equivalently the Young diagrams contained in the rectangle of shape d (n d). Let us denote by D λ the Young diagram of λ. Suppose w, v be elements of W P d such that ev X w. Let λ, µ be the corresponding partitions for w, v respectively. Then we have D λ D µ.

3 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 3 Our formula express [X w ] v as a weighted sum over a set E µ (λ) (see Subsection 3.4 for the definition) of subsets of D µ. For example, let λ = (3, 2), µ = (4, 4, 3, 1). Some typical elements in E µ (λ) are illustrated below. Here we depict the Young diagrams in Russian style. Ground state Two boxes can be excited Excited state No box can be excited Let us imagine that each diagram labels a quantum state. Each box can be excited ( i.e. raised) to the space one unit North if the neighbors in North, North-East, and North-West directions are all unoccupied (see Subsection 3.4 for the precise definition of excitation). In this case, there are nine excited states obtainable by applying successive excitations starting from the ground state i.e. the element D λ. It should be mentioned that the notion of EYDs and its shifted analogue were introduced by Kreiman [17, 18] independently to us. Kreiman [17] proved that the set of non-intersecting paths appeared in [19], [12], [22] is naturally bijective to the set of EYDs. He also presented a combinatorial formula of [X w ] v for type C n case in terms of shifted analogue of EYDs by using a result by Ghorpade and Raghavan [8] analogous to [19], [12]. In this paper, we present a different proof for these results without using the Gröbner machinery mentioned above. An advantage of our method is that we can apply the same argument to isotropic Grassmannians of orthogonal type, for which no explicit description of the Gröbner basis is known. In order to deal with the case of even orthogonal Grassmannian, we introduce another variant of shifted EYDs. Our method begins by identifying the localized classes [X w ] v as ξ-functions defined by Kostant and Kumar ([16]). Then we can make use of a well-known formula (Proposition 3, cf. [4],[1]) that expresses arbitrary ξ-function as a sum over a set of reduced subwords. And then use a theory by Stembridge [30] on fully commutative elements in Coxeter groups. The theory enables us to establish a natural bijection between the set of reduced sub words appearing in the sum formula and a certain set of EYDs. This bijection is a technical heart of our proof of Theorem 1 and Theorem 3. It is known that the multiplicity m v (X w ) at e v in X w is closely related to [X w ] v. Such a multiplicity has been studied in detail by many authors (see [5]). By our combinatorial formula, we can express m v (X w ) as the number of elements in a certain set of excited Young diagrams. And also, we can obtain a closed formula for m v (X w ), which is a specialization of a factorial Schur function. This leads to a Pfaffian formula for the multiplicity of a singular point in a Schubert variety. In Section 2, we explain the relation between the polynomial [X w ] v and the ξ-function. Some fundamental properties of ξ-functions are presented for the later use. We discuss the case of G d,n in Sections 3, 4, 5. We present the combinatorial formula in Section 3 and the proof is given in Section 4. We give the closed formula in Section 5, which can be read independently from the preceding two sections. The isotropic Grassmannians are treated in a parallel manner in Sections 6, 7, 8. In Section 9 we discuss some application

4 4 TAKESHI IKEDA AND HIROSHI NARUSE for m v (X w ). In Section 10 we discuss a relation between two types of formula (Theorem 3 and 4) by using a Gessel-Viennot type argument. Note added. (1) After the first version of the present paper finished, a preprint [28] by Raghavan and Upadhyay appeared. They compute the multiplicity, or more genarally the Hilbert function, of the local ring of a Schubert variety at torus fixed points for type D n case. In particular, they proved a combinatorial formula for the multiplicity equivalent to ours (Corollary 4 (9.3)). Their approach is based on standard monomial theory and quite different from ours. (2) Combinatorics used in [7], [3], [13], [14] are closely connected with EYDs. In fact excitation correspond to possible moves in a diagrams called RC-graph (or reduced pipe dreams). To any permutation, there is a double Schubert polynomial of Lascoux and Schützenberger ([24]). In [14] the double Schubert polynomial associated with vexillary permutations, a class of permutations including the Grassmannian permutations, are expressed as sum over flagged tableaux. Combined with this result and a well-known fact that the double Schubert polynomials indexed by Grassmannian permutations are factorial Schur functions, one has a combinatorial formula for [X w ] v (in type A) quite similar to the one in Theorem 1, but they are different formulas at least in the sense that the cardinalities of the the summands of both formulas are different in general. Future problems. As an application of Theorem 4, one can give a ring presentation for HT (G/P ) in types B and D (see [10] for type C case). We will discuss the details in a separate paper. These description should be related to results in [20] on degeneracy loci. The results on multiplicity can be extended to co-minuscule G/P s. We will discuss the subject elsewhere. Acknowledgments. We benefited from conversations with the participants in Workshop on Contemporary Schubert calculus and Schubert geometry held at Banff in Canada, Mar , We thank the organizers of the workshop, Jim Carrell and Frank Sottile, for kind invitation. We thank Sara Billey, Allen Knutson, Alexander Yong, Harry Tamvakis, and Leonardo Mihalcea, Mark Shimozono, Soichi Okada, and Masao Ishikawa for valuable discussions. T.I. was partially supported by Grant-in-Aids for Young Scientists (B) (No ) from Japan Society of the Promotion of Science. 2. ξ-functions of Kostant and Kumar In this section we introduce the family of functions ξ w for w W defined by Kostant and Kumar. By virtue of the result of Arabia [2], we can identify ξ w (w W P ) with the equivariant Schubert class [X w ] in HT (G/P ). Let R + denote the set of positive roots with respect to B. For β R +, we denote by β its dual coroot and s β W the reflection corresponding to β. Let {α 1,..., α r } be the set of simple roots in R +. The Weyl group W is a Coxeter group generated by the set of simple reflections {s 1,..., s r }, where s i = s αi. In particular, we can talk of the length

5 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 5 l(w) of any element w W. For a simple reflection α we denote by ϖ α the corresponding fundamental weight. Proposition 1. (Kostant and Kumar [16]) There exist a family of functions ξ w : W S for w W with the following properties: (2.1) (1) ξ w (v) equals zero unless w v, (2) ξ w (w) = α R + wr α, (3) ξ e (v) = 1 for all v W, where e is the identity of W, (4) If α is a simple root, then for all v W, (5) If α is a simple root, then ξ sα (v) = ϖ α v(ϖ α ), (ξ sα ξ sα (w))ξ w = w(ϖ α ), β ξ w, w w β where we use the notation w β w to indicate w = s β w for some β R + and l(w ) = l(w) + 1, (6) Each ξ w (v) with v W is homogeneous of degree l(w). Remark. We use notation for ξ-functions in Kumar s book, which is different from the one used in [16]. The function ξ w is directly related to the object of our main interest. Proposition 2. Let w W P. We have for v W P. [X w ] v = ξ w (v) Proof. Arabia [2] proved the result for the flag variety G/B (see Graham s paper [9] for more information). For the parabolic case, the reader can consult Kumar s book [21]. Let P = P d denote the maximal parabolic subgroup associated to α d. Note that we have for w W P (2.2) (ξ sd ξ sd (w))ξ w =. w W P, w β w w(ϖ α ), β ξ w The above relation involves only ξ w (w W P ). This reflects the fact that the Schubert classes [X w ] (w W P ) form an S-basis of HT (G/P ), considered a sub S-algebra of HT (G/B) via the projection G/B G/P. Note that if w W P, ξ w is W P -invariant in the sense that ξ w (vu) = ξ w (v) for all u W P. We can make use of the following formula:

6 6 TAKESHI IKEDA AND HIROSHI NARUSE Proposition 3. ([1], [4]) Let w, v W such that w v. Fix a reduced expression s i1 s ik for v. Put (2.3) Then (2.4) β t = s i1 s it 1 (α it ) for 1 t k. ξ w (v) = j 1,...,j s β j1 β js, where the sum is over all sequences 1 j 1 < < j s k such that s ij1 s ijs is a reduced expression for w. Although the above formula is explicit, it still requires a lot of calculations to get a concrete expression for ξ w (v) in general. If w is an element of W P for classical G and P in our list (cf. Section 1), then we can give a nice combinatorial interpretation for the right hand side of (2.4). 3. Excited Young diagrams We fix positive integers n, d such that 1 d n. In this section, we give a combinatorial formula (Theorem 1) for the restriction of the equivariant Schubert classes in the Grassmannian G d,n to any torus fixed points Schubert variety of G d,n. Let w S n be a Grassmannian permutation, i.e., w(1) < < w(d), w(d + 1) < < w(n). When we identify the space SL(n)/P d with the Grassmannian G d,n of d-dimensional subspaces of C n, the Schubert variety associated with w is given by X w = {V G d,n dim(v F n w(d i+1)+1 ) i for 1 i d}, where F i = span C {e n i+1,..., e n } is the i-plane spanned by the last i vectors in the standard T -basis e 1,..., e n of C n Partitions and Young diagrams. Let λ = (λ 1 λ r 0) be a partition. To every partition λ one associates its Young diagram D λ which is the set of square boxes with coordinate (i, j) Z 2 such that 1 j λ i : D λ = {(i, j) Z 2 1 i r, 1 j λ i }. The boxes in D λ are arranged in a plane with matrix-style coordinates. For example, j i is the Young diagram D λ of λ = (4, 3, 1). Let P d denote the set of all partition with length at most d. If n is an integer with n d, let P d,n be the subset of P d consisting of the elements whose largest part is less than or equal to n d. For any partition µ, let P µ denote the set of all partition λ such that λ µ. In particular, if µ is the partition whose Young diagram is the d (n d) rectangle, then P µ is identical to P d,n.

7 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS Grassmannian permutations and Young diagrams. The set W P d of minimal length coset representatives for W/W Pd with W = S n is identified with the set of all Grassmannian permutations. Let w W P d. We define a partition λ = (λ1,..., λ d ) of n by λ j = w(d j + 1) d + j 1 (1 j d). When considered as a Young diagram, λ is contained in the rectangular shape d (n d). Note that we have l(w) = λ := d j=1 λ i, where l(w) is the length of w. There is a convenient way to recover the Grassmannian permutation from a Young diagram as follows. Given a Young diagram λ contained in the rectangle d (n d), write a path along the boundary of the Young diagram starting from the SW corner to the NE corner of the rectangle. We assign numbers to each arrow from 1 to n. For example, for λ = (4, 3, 1, 0) with n = 9, d = 4 we have the following picture: If the assigned numbers of the vertical arrows are i 1 < < i d and those of the horizontal arrows are j 1 < < j n d, then the corresponding Grassmannian permutation is Explicitly we have w = (i 1,..., i d, j 1,..., j n d ). i k = λ d k+1 + k (1 k d), j k = λ k + k + d (1 k n d), where λ is the conjugate of λ. In the above example, we have w = Excited Young diagrams. Let λ µ be partitions. The Young diagram D λ of λ is a subset of D µ. Take an arbitrary subset C of D µ. Pick up a box x C such that x + (1, 0), x + (0, 1), x + (1, 1) D µ \ C. Then set C = C {x + (1, 1)} \ {x}. The procedure C C of changing C into C is called an elementary excitation occurring at x. If a subset S of D µ is obtained from C by applying elementary excitations successively, i.e., there is a sequence (3.1) C = C 0 C 1 C r 1 C r = S, r 0 of elementary excitations, then we say that S is an excited state of C, or S is obtained from C by excitation. Let E µ (λ) denote the set of all excited states of D λ. Example 1. For example, let λ = (3, 1), µ = (5, 4, 3). Then the set E µ (λ) consists of the following seven elements:

8 8 TAKESHI IKEDA AND HIROSHI NARUSE It is easy to see that the number r in (3.1), the number of elementary excitations, is well-defined for C E µ (λ). In fact, if we define the energy E(C) of C E µ (λ) by E(C) = (i,j) C m(i, j) m(i, j), m(i, j) = 1 (i + j) 2 (i,j) D λ then we have E(C) = r. Let ε 1,..., ε n be a standard basis of the lattice L = Z n. The character group ˆT is identified with a sub lattice of L spanned by ε i ε i+1 (1 i n 1). Now we can state the first combinatorial formula. Theorem 1. ([22],[17]) Let w v W P d, and λ µ Pd,n the corresponding partitions. Then we have [X w ] v = (ε v(d+j) ε v(d i+1) ). C E µ (λ) (i,j) C Proof of the theorem is given in the next section. Example 2. If w = , v = W P d with d = 4, then λ = (3, 1), µ = (5, 4, 3). We fill the boxes of D µ with positive roots as follows: ε 2 ε 9 ε 3 ε 9 ε 4 ε 9 ε 6 ε 9 ε 8 ε 9 7 ε 2 ε 7 ε 3 ε 7 ε 4 ε 7 ε 6 ε 7 5 ε 2 ε 5 ε 3 ε 5 ε 4 ε 5 1 Then our formula reads (cf. Example 1): [X w ] v = (ε 2 ε 9 )(ε 3 ε 9 )(ε 4 ε 9 )(ε 2 ε 7 ) + (ε 2 ε 9 )(ε 3 ε 9 )(ε 6 ε 7 )(ε 2 ε 7 ) + (ε 2 ε 9 )(ε 3 ε 9 )(ε 4 ε 9 )(ε 3 ε 5 ) + (ε 2 ε 9 )(ε 3 ε 9 )(ε 6 ε 7 )(ε 3 ε 5 ) + (ε 2 ε 9 )(ε 4 ε 7 )(ε 6 ε 7 )(ε 2 ε 7 ) + (ε 2 ε 9 )(ε 4 ε 7 )(ε 6 ε 7 )(ε 3 ε 5 ) + (ε 3 ε 7 )(ε 4 ε 7 )(ε 6 ε 7 )(ε 3 ε 5 ). 4. Proof of Theorem Fully commutative elements. Let W be a Coxeter group. An element w in W is fully commutative if any reduced expression for w can be obtained from any other by using only the Coxeter relations that involve commuting generators. It is known that every element w in W P for every (G, P ) in our list (cf. Section 1) is fully commutative ([29], Theorem 6.1).

9 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS Row reading expression for a Grassmannian permutation. Let v be a Grassmannian permutation in W P d and µ Pd,n be the corresponding partition. To each box (i, j) D µ we fill in the simple reflection s d i+j. For example, let v = ( ) S 7 with d = 3. The corresponding partition is µ = (4, 3, 2) and we have the following table: s 3 s 4 s 5 s 6 s 2 s 3 s 4 s 1 s 2 We read the entry of the boxes of the Young diagram D µ from right to left starting from the bottom row to the top row and form a word (4.1) s i1 s ik (k = µ ), which gives a reduced expression for v. For example we have v = = s 2 s 1 s 4 s 3 s 2 s 6 s 5 s 4 s 3. We call the word given by (4.1) the row-reading word of v. The row reading procedure gives a bijective map ϕ : D µ {1,..., k}, k = µ. For example, if µ = (4, 3, 1) then the map ϕ is expressed by the following tableau: Now let C be an arbitrary subset of D µ and ϕ(c) = {j 1,..., j r } with j 1 < < j r be the image of {1,..., k} under the map ϕ. Let s ij be the simple reflection assigned to the box ϕ 1 (j) in D µ for 1 j k. Then we put (4.2) w C = s ij1 s ij r. For example, if C is the subset of D µ indicated by the following gray boxes then w C = s 4 s 2 s 3. In particular, if C = D µ, then w C is nothing but the row-reading word of v. Given another Grassmannian permutation w such that w v, define R v (w) = {C D µ C = l(w), w C = w}. Note that if λ is the partition corresponding to w, then D λ D µ and D λ R v (w). Lemma 1. Let C, C be subsets of a Young diagram D µ such that C is obtained from C by an elementary excitation from C. If C belongs to R v (w) then we have C R v (w). Proof. We may assume that C is obtained from C by an elementary excitation occurred in the 2 2-square of the corner (i, j). Set k := d + j i. Consider the following regions in the diagram D µ : R := {(i, a) j a µ i } {(i + 1, b) 1 b j + 1},

10 10 TAKESHI IKEDA AND HIROSHI NARUSE R := {(i, a) j + 2 a µ i }, R := {(i + 1, b) 1 b j 1}. If we pick up the i-th and the (i + 1)-th rows, they look like i i + 1 R j j + 1 µ i It suffices to compare the sub-words corresponding to C R and C R (given by row reading procedure). In C, we have s k at the position (i, j) indicated above by. The positions indicated by are vacant by definition of the elementary excitation. Then C is obtained by moving s k to the position indicated by. Any simple reflection s l located in C R (resp. C R ) commutes with s k since l j + 2 (resp. l j 2). Hence the subword corresponding to the subset C R can be rewritten into the sub-word corresponding to C R using only the Coxeter relation that involve commuting generators. R Corollary 1. We have E µ (λ) R v (w). Proof. Use induction on energy E(C) of C E µ (λ). The corollary is obvious from Lemma 1. We would like to establish the following. Proposition 4. We have E µ (λ) = R v (w). In order to prove Proposition 4, it suffices to show the following. Lemma 2. Let C R v (w) be such that C D λ. Then there exists an element C R v (w) such that C is obtained from C by a sequence of elementary excitations. Proof. Let s ij1 s ij r be the row-reading word for v, and s ik1 s ik r be the word corresponding to C. Let a be such that j a k a and j t = k t for a < t r. We shall compare the two words s ik1 s ik a, s i j1 s ij a. Note that the element, say w, expressed by the words is fully commutative. Put t = i ja. Since {i k1,..., i ka } is equal to {i j1,..., i ja } as a multiset, we have t {i k1,..., i ka }. Let b be the largest index such that i kb = t. Since w is fully commutative, the simple reflections adjacent to s t, i.e. s t+1, s t 1, can not appear in the subword s ikb+1 s ika. This implies a region R indicated by the picture below is unoccupied in the diagram C : R

11 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 11 So we can move the box corresponding to s kb as illustrated by the above picture to get a subset C of D µ. Clearly C belongs to R v (w). Since C has strictly smaller energy than C, we see that C belongs to E µ (λ) by inductive hypothesis. Now by the construction, C is obtained from C by a sequence of elementary excitation. So Lemma 1 implies C is also a member of E µ (λ) β-sequences. Fix a Grassmannian permutation v, and let µ P d,n be the corresponding partition. Let ϕ : D µ {1, 2,..., k} be the row-reading map of µ. For (i, j) D µ set β i,j = β ϕ(i,j), where β t (1 t k) are defined in (2.3). Lemma 3. For (i, j) D µ, we have β i,j = ε v(d+j) ε v(d i+1). Proof. To prove the lemma, we proceed by induction on µ = l(v). In case l(v) = 0, there is nothing to prove. If l(v) > 0, let v = s i1 s i2 s il be the row-reading expressions for v. Setting v = s i2 s il, which is the row-reading expression for v, and in particular we have l(v ) = l(v) 1. The corresponding shape µ for v is obtained from µ by deleting a box of position (r, µ r ), the left most one in the bottom row, where r is the number of rows in µ. Then we have β r,µr = α i1 = ε d r+µr ε d r+µr+1. By definition of µ r, we have µ r = v(d r + 1) d + r 1, so β r,µr = ε v(d r+1) 1 ε v(d r+1). Then by Lemma 4 below, we have β r,µr = ε v(d+µr ) ε v(d r+1). Let (i, j) µ. By the hypothesis of induction, we have β i,j = ε v (d+j) ε v (d r+i). By definition of β i,j, it is easy to see that β i,j = s i1 (β i,j). Hence we have β i,j = s i1 (β i,j) = s i1 (ε v (d+j) ε v (d r+i)) = ε si1 v (d+j) ε si1 v (d r+i). Since s i1 v = v, we have the lemma. Lemma 4. Let r be the number of rows in µ. Then we have v(d r + 1) = v(d + µ r ) + 1. Proof. For a sequence a 1 < a 2 < < a m of integers, we say a i is a gap if a i a i 1 2. Note that µ r is the largest number such that v(d + 1) < < v(d + µ r ) has no gap. Since v is a Grassmannian permutation, the number v(d + µ r ) + 1 occurs as the smallest gap in 0 < v(1) < < v(d). On the other hand, we have µ d = = µ r+1 = 0 and µ r > 0, which implies that the smallest gap in 0 < v(1) < < v(d) is v(d r + 1). Proof of Theorem 1. By Proposition 3 together with Lemma 3 we have [X w ] v = (ε v(d+j) ε v(d i+1) ). C R v (w) (i,j) C Then the theorem is immediate from Proposition 4.

12 12 TAKESHI IKEDA AND HIROSHI NARUSE 5. Factorial Schur functions Our main goal in this section is to express [X w ] v for G d,n as a specialization of a factorial Schur function. First we recall the definition of the factorial Schur functions Definition of factorial Schur functions. Let x = (x 1,..., x d ) be a finite sequence of variables and let a = (a i ) i=1, be an infinite sequence of parameters. The factorial Schur function for a partition λ of length at most d can be defined as follows. Let for any k 0. Then we put (z a) k = (z a 1 )(z a 2 ) (z a k ) s (d) λ (x a) = det((x j a) λi+d i ) 1 i,j d 1 i<j d (x. i x j ) This function is actually a polynomial in x 1,..., x n and a 1, a 2,..., a λ1 +d 1, homogeneous of degree λ. In particular we have (5.1) s (d) 1 (x a) = x x d a 1 a d. For a partition λ P d, we define a d-tuple a λ = (a λd +1, a λd 1 +2,..., a λ1 +d). Proposition 5. (Vanishing property, cf. [25]) We have s (d) λ (a µ a) = 0 unless µ λ. Let λ P d, and take sufficiently large n such that λ is contained in the d (n d) rectangle. Define an n-tuple (w(1),..., w(n)) by { λ d i+1 + i for 1 i d w(i) = λ i + i for d < i n, then w = (w(1),..., w(n)) is a permutation of {1,..., n}. If we set w(i) = i for all i > n, then the infinite sequence (w(1), w(2),...) does not depend on the choice of n. We also need the following Pieri-type formula: Lemma 5. (cf. [25]) Let λ P d. We have ( ) s (d) 1 (x a) s (d) 1 (a λ a) s (d) λ (x a) = ν s (d) ν (x a), where the summation in the right hand side runs through ν P d such that ν λ and ν = λ Closed formulas. Let λ P n,d. Recall that s (d) λ (x a) is a polynomial in x 1,..., x d and a 1,..., a n 1. A specialization given by a i = ε i (1 i n 1) is important for our geometric application below. For v W P d, we define an n-tuple by Lemma 6. Let v W P d. We have x v = (ε v(1),..., ε v(d) ). [X sd ] v = s (d) 1 (x v ε 1,..., ε n 1 ). Proof. The d-th fundamental weight of SL(n) is given by ϖ d = ε ε d. By Proposition 1, (4) and formula (5.1), the lemma follows.

13 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 13 Theorem 2. (Knutson and Tao [20], Lakshmibai, Raghavan, and Sankaran [22]) Let w v W P d, and λ µ Pd,n be the associated partitions. Then we have (5.2) [X w ] v = ( 1) l(w) s (d) λ (x v ε 1,..., ε n 1 ). Proof. Let P µ denote the set of all partition λ such that λ µ. Consider the following system of equations for the functions F λ = F λ (ε 1,..., ε n ) (λ P µ ) : d (5.3) F ν (λ P µ \ {µ}), (ε λd i+1 +i ε µd i+1 +i) F λ = i=1 ν where the summation in the right hand side runs through ν P µ such that ν λ and ν = λ + 1. This equation together with the initial condition F φ (ε 1,..., ε n ) = 1 for the empty partition λ = φ determine the functions F λ (ε 1,..., ε n ) (λ P µ ) uniquely. Now we can see equation (2.2) for [X w ] v is identical to (5.3) above, in view of the bijection W P d = Pd,n. On the other hand, the right hand side of (5.2) satisfies the same equation from Proposition 5 and Lemmas 5 and 6. In addition, both hand sides of (5.2) satisfy the initial condition. Hence the theorem follows. 6. Lagrangian and Orthogonal Grassmannians Fix a positive integer n. We give here analogous theorems for other classical homogeneous spaces G/P in the following types: Type B n : G = SO(2n + 1, C), P = P n, Type C n : G = Sp(2n, C), P = P n, Type D n : G = SO(2n, C), P = P n 1, P n, where we shall denote by P d the maximal parabolic subgroup associated to the simple root α d (the simple roots being indexed as in [6]). In type B n (resp. type C n ), the variety G/P can be identified with a closed subvariety of G n,2n+1 (resp. G n,2n ) parametrizing the isotropic n-spaces in C 2n+1 (resp. C 2n ) equipped with a non-degenerate symmetric (resp. skew symmetric) form. Our space G/P in type C n is also called the Lagrangian Grassmannian. For the even dimensional space C 2n equipped with a non-degenerate symmetric form, the isotropic n-subspaces constitute a union of two closed subvariety of G n,2n each of which is isomorphic to G/P, P being one of P n 1 and P n. Note that the varieties SO(2n + 1, C)/P n, SO(2n + 2, C)/P d (d = n, n + 1) are all isomorphic (cf. [26]), although they are different as T -spaces. The goal of this section is to present a combinatorial formulas for the restriction of equivariant Schubert classes to any torus fixed points for the classical Grassmannians G/P The set W P. First we fix some notations on a set W P which parametrizes the Schubert classes, and the torus fixed points. The character group ˆT of our torus is a lattice with a standard basis {ε i } r i=1, where r = dim(t ). Type C n (G = Sp(2n, C), P = P n ): We identify W with a subgroup of S 2n acting on the functionals ±ε 1,..., ±ε n. Denote ε i, ε i by i and ī respectively and define a partial order on the set I n = {1,..., n, n,..., 1} by 1 < 2 < < n < n < < 2 < 1.

14 14 TAKESHI IKEDA AND HIROSHI NARUSE Then we have W = {w S 2n w( i ) = w(i)}, where a a is the involution on I n given by ±ε i ε i. Then W P = {w W w(1) < < w(n)}. The simple roots are and the corresponding simple reflections are We have α i = ε i ε i+1 (1 i n 1), α n = 2ε n, s i = (i, i + 1)(i + 1, i) (1 i n 1), s n = (n, n). ϖ n = ε ε n. Type B n (G = SO(2n + 1, C), P = P n ): Since W is identical to the case C n as a Coxeter group, the description of W P is the same as Type C n. The simple roots are given by α i = ε i ε i+1 (1 i n 1), α n = ε n, and we have ϖ n = 1 2 (ε ε n ). Type D n+1 (G = SO(2n + 2, C), P = P n+1 ): In this case, we have { } W = w S 2n+2 w( i ) = w(i), N(w) is even, where we set N(w) = { i 1 i n + 1, w(i) > n + 1}. We have The simple reflections are given by and we have and (6.1) W P = {w W w(1) < < w(n + 1)}. s i = (i, i + 1)(i + 1, i) (1 i n), s n+1 = (n, n + 1)(n + 1, n), α i = ε i ε i+1 (1 i n), α n+1 = ε n + ε n+1, ϖ n+1 = 1 2 (ε ε n+1 ). Remark 1. For type D n+1, the result for P = P n is obtained by simply replacing ε n+1 by ε n+1. So we only consider P n Strict partitions and shifted Young diagrams. Next we need a description of the set W P in terms of strict partitions. Let λ = (λ 1 > > λ r > 0) be a strict partition. Then the shifted Young diagram of λ is the array of boxes with λ i boxes in the i-th row, such that each row is shifted by one position to the right relative to the preceding row. More explicitly, given a strict partition λ 1 > λ 2 > > λ r > 0, the shifted diagram of λ is defined to be For example D λ := {(i, j) Z 2 1 i l, i j < λ i + i}.

15 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 15 j i is the shifted Young diagram of λ = (5, 2, 1). Define a partial order λ µ if and only if λ i µ i for all i. Denote the particular element ρ n = (n,..., 2, 1). Let SP µ denote the set of all strict partitions λ such that λ µ. The cardinality of the set SP ρn is 2 n. Proposition 6. Let G/P be as above. There is a natural order-preserving bijection If λ SP ρn W P = SPρn. corresponds to w W P we have l(w) = λ. Explicitly, the bijections are given as follows: Types B n, C n : Let w W P. Since w is a Grassmannian permutation of 2n letters (with d = n) we have the associated Young diagram, say Λ. It is symmetric and contained in an n n square. Then the associated element λ SP ρn is the upper part of the symmetric Young diagram including the diagonal. Thus if Λ = (Λ 1,..., Λ n ), then λ i = max{λ i i + 1, 0} (1 i n). Type D n+1 : Let w W P. Then the associated Young diagram, say Λ, is symmetric and contained in an (n + 1) (n + 1) square. Then the associated element λ SP ρn is the strictly upper part of the symmetric Young diagram. Thus if Λ = (Λ 1,..., Λ n+1 ), then λ i = max{λ i i, 0} (1 i n). Example 3. Take w 1 = for D 4 and w 2 = for B 3 (or C 3 ). Then the associated strict partitions are the same and given by λ = (3, 1) D 4 B 3, C 3 Note that w 1 = s 3 s 1 s 2 s 4 and w 2 = s 3 s 1 s 2 s 3 and these are the row-reading reduced expressions introduced in Section Excited Young diagrams for shifted cases. The notion of excitation of shifted Young diagrams is defined in the same way as in the case of ordinary Young diagrams if we introduce the idea of elementary excitation. Let µ be a strict partition. Given a subset C of D µ. If a box x C satisfies either of the following conditions: (1) x = (i, j), i < j and x + (1, 0), x + (0, 1), x + (1, 1) D µ \ C, (2) x = (i, i) and x + (0, 1), x + (1, 1) D µ \ C

16 16 TAKESHI IKEDA AND HIROSHI NARUSE then we set C = C {x + (1, 1)} \ {x} and call this procedure C C an elementary excitation of type I occurring at x C. In general if a subset S of D µ is obtained from C by applying elementary excitations of type I successively, then we say that S is an excited state of C. Suppose we are given a strict partition λ such that λ µ. Let us denote by Eµ(λ) I the set of all excited states of D λ. We also define the set Eµ II (λ) consisting of the elements obtained from C by a successive application of elementary excitations of type II defined below. Let C be a subset of D µ and suppose we take a box x C satisfying either of the following conditions: (1) x = (i, j), i < j and x + (1, 0), x + (0, 1), x + (1, 1) D µ \ C, (2) x = (i, i) and x + (0, 1), x + (1, 1), x + (1, 2), x + (2, 2) D µ \ C. In the case of (1) we set C = C {x + (1, 1)} \ {x}, and in the case of (2) C = C {x + (2, 2)} \ {x}. We call the procedure C C an elementary excitation of type II occurring at x C. Clearly we have Eµ II (λ) Eµ(λ). I Type I Type II Example 4. Let λ = (3, 1), µ = ρ 4. The set E I µ(λ) consists of the following ten elements: The five members in the first row form the subset E II µ (λ). The corresponding elements in W P are as follows: for type D 5 : v = , w = , for type C 4 or B 4 : v = , w = Theorem 3. Let w v W P, and λ µ the corresponding strict partitions. We have the following formulas: (1) Type C n : [X w ] v = C Eµ(λ) I (i,j) C (ε v(n+j) ε v(n i+1) ), (2) Type B n : [X w ] v = C Eµ(λ) I (i,j) C 2 δ ij (ε v(n+j) ε v(n i+1) ), (3) Type D n+1 : [X w ] v = C Eµ II (λ) (i,j) C (ε v(n+j+2) ε v(n i+2) ). Example 5. Let n = 4 and µ = ρ 4. We arrange positive roots in D µ as follows.

17 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 17 ε 1 +ε 2 ε 1 +ε 3 ε 1 +ε 4 ε 1 ε 5 ε 2 +ε 3 ε 2 +ε 4 ε 2 ε 5 ε 3 +ε 4 ε 3 ε 5 2ε 1 ε 1 +ε 2 ε 1 +ε 3 ε 1 +ε 4 2ε 2 ε 2 +ε 3 ε 2 +ε 4 2ε 3 ε 3 +ε 4 Type D 5 ε 4 ε 5 Type C 4 2ε 4 Let λ = (2). Then the set E I µ(λ) consists of the following six elements: The first four elements form the subset E II µ (λ). In type D 5 case we have [X w ] v = (ε 1 + ε 2 )(ε 1 + ε 3 ) + (ε 1 + ε 2 )(ε 2 + ε 4 ) + (ε 1 + ε 2 )(ε 3 ε 5 ) + (ε 3 + ε 4 )(ε 3 ε 5 ), in type C 4 case we have [X w ] v = 2ε 1 (ε 1 +ε 2 )+2ε 1 (ε 2 +ε 3 )+2ε 1 (ε 3 +ε 4 )+2ε 3 (ε 3 +ε 4 )+2ε 2 (ε 3 +ε 4 )+2ε 2 (ε 2 +ε 3 ). 7. Proof of Theorem 3 This section is devoted to the proof of Theorem 3. Our strategy is the same as in Section 4. We shall give a notion of row-reading expression for each element of W P. First we fill in the diagram D ρ n the simple reflections. Types B n, C n : We define a map s : D ρ n {s 1,..., s n } by { s n (i = j) s(i, j) =, s n+i j (i < j) where 1 i < j n. Type D n+1 : We define a map s : D ρ n {s 1,..., s n+1 } by s n+1 (i = j, i : odd) s(i, j) = s n (i = j, i : even), s n+i j (i < j) where 1 i < j n. Example 6. Let n = 4. The map s is illustrated as follows: s 1 s 5 s 3 s 2 s 3 s 2 s 4 s 5 s 3 s 4 s 1 s 4 s 3 s 2 s 3 s 2 s 4 s 4 s 3 s 4 D 5 C 4 or B 4

18 18 TAKESHI IKEDA AND HIROSHI NARUSE Let λ be the strict partition associated to w W P. We define the row-reading map ϕ : D λ {1, 2,..., k} as in Section 4. We define the word s i1 s ik, where s ij = s(ϕ 1 (j)). Then we have w = s i1 s ik, which is a reduced expression for w W P. For each subset C of D µ, we define an element w C W in the same way in Section 4. Let w v W P. The set {C D µ C = l(w), w C = w} is denoted by R I v(w) (resp. R II v (w) ) for types C n, B n ( type D n+1 ). The key result to prove Theorem 3 is the following. Proposition 7. Let w v W P and λ µ the associated strict partitions. Then we have Eµ(λ) = R v(w) for = I, II. Lemma 7. Let C, C be subsets of a shifted Young diagram D µ such that C is obtained from C by an elementary excitation of type = I, II. If C belongs to R v(w) then we also have C R v(w). Proof. Suppose C is obtained from C by an elementary excitation of Type II occurring at (i, i). Any simple reflection contained in the region R C are one of s 1,..., s n 2 each of which commutes with s n+1 and s n. R The positions indicated by are vacant by the definition of elementary excitation. Therefore the word w C can obtained from w C by using only commuting relations. Hence we have C R v (w). The arguments for the other cases are simpler than this and we leave them to the reader. Lemma 8. Let C R v(w) be such that D λ C. Then there exists C R v(w) such that C is obtained from C by excitation of type. Proof. Similar to the proof of Lemma 2. Proof of Proposition 7. This is immediate from Lemmas 7, 8. Now we calculate the β-sequence in Proposition 3. Lemma 9. Let µ SP ρn and v the associated element of W P. Denote by ϕ : µ {1, 2,..., k} the row-reading map of µ. For (i, j) D µ set β i,j = β ϕ(i,j), where β t (1 t l) are defined in (2.3). We have the following formulas: (1) Type C n : β i,j = ε v(n+j) ε v(n i+1), (2) Type B n : β i,j = 2 δ ij (ε v(n+j) ε v(n i+1) ), (3) Type D n+1 : β i,j = ε v(n+2+j) ε v(n i+2).

19 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 19 Proof. Similar to the proof of Lemma 3 and 4. Proof of Theorem 3. The same as the proof of Theorem Factorial Q- and P -functions In this section, we first define the factorial Q- and P -functions and present some fundamental properties. Then we use them to express the restriction of the equivariant Schubert classes for the Lagrangian and orthogonal Grassmannians. As an application we give the equivariant Giambelli formula Factorial Q- and P -functions. We first recall the definition of factorial Q- and P -functions due to Ivanov [11]. Let SP(n) denote the set of strict partitions λ = (λ 1 > > λ r > 0) with r n. Definition 1. Let λ SP(n). Put ( P (n) λ (x a) = 1 (8.1) (x 1 a) λ1 (x r a) λr (n r)! w S n w 1 i r, i<j n x i + x j x i x j where w S n acts as a permutation of variables x 1,..., x n. We also put Q (n) λ (x a) = 2 r P (n) λ (x a). The rational expression in the right hand side of (8.1) is actually a polynomial in x 1,..., x n and a 1, a 2,..., a λ1, homogeneous of degree λ. In particular we have { (8.2) P (n) x x n if n is even 1 (x a) = x x n a 1 if n is odd. In [11], the first parameter a 1 is assumed to be zero in the most part of the argument. However, we need a 1 for the later use. In order to generalize Ivanov s results to the case of non-zero a 1 the following is fundamental (cf. [11], Proposition 2.6). Proposition 8. (Stability mod 2) For any λ SP(n), we have (8.3) Proof. We set P (n+2) λ (x 1,..., x n, 0, 0 a) = P (n) λ (x 1,..., x n a). F n = (x 1 a) λ1 (x r a) λ r 1 i r, i<j n x i + x j x i x j. Let w S n+2, and consider the term w(f n+2 ). If w(i) n + 1, n + 2 for all i = 1,..., r, then the rational function w(f n+2 ) has no pole along the hyperplane x n+1 x n+2 = 0. So we can substitute x n+1 = x n+2 = 0 to such a rational function. As the result of the substitution we have w (F n ) for some w S n. In fact, w is obtained by eliminating n + 1, n + 2 from the sequence w(1),..., w(n + 2). By this correspondence, there are (n r + 2)(n r + 1) elements of S n+2 giving the same w S n. Thus our task is to show that all the remaining terms cancel out. Let i = w 1 (n + 1), j = w 1 (n + 2). Suppose i r or j r. We set w = w (i, j). Then both w(f n+2 ) and w (F n+2 ) has simple pole along x n+1 x n+2 = 0, however, the sum w(f n+2 )+w (F n+2 ) has no pole along x n+1 x n+2 = 0. Substituting x n+1 = x n+2 = 0 to the sum, we obtain zero. ),

20 20 TAKESHI IKEDA AND HIROSHI NARUSE 8.2. Vanishing property. Here we present a vanishing property of P (n) λ (x a). For strict partition λ = (λ 1 > > λ r > 0) with r n define an n-tuple { (a λ1 +1,..., a λr +1, 0,..., 0) if n r is even a λ = (a λ1 +1,..., a λr +1, a 1, 0,..., 0) if n r is odd. Proposition 9. We have P (n) λ (a µ a) = 0 unless µ λ. Proof. By Proposition 8, we may assume a µ has no zero entries. Then the proposition follows from definition Pieri s rule and Pfaffian formulas. We collect here some basic facts on the factorial P -Schur functions, which we shall make use of in the next subsection. The proof for a 1 = 0 case is given in [11]. The same proof works for the general a 1 case using the vanishing property. Proposition 10. ([11]) For a strict partition λ SP(n) we have ( ) P (n) 1 (x a) P (n) 1 (a λ a) P (n) λ (x a) = P ν (n) (x a), λ ν where the sum is over ν SP(n) such that ν λ and ν = λ + 1. For any λ = (λ 1 > > λ r > 0) SP(n), set r 0 (λ) = r if r is even and r 0 (λ) = r + 1 if r is odd, and then we put λ r+1 = 0. Lemma 10. ([11]) For a strict partition λ SP(n) we have P (n) λ (n) (x a) = Pf(P λ i,λ j (x a)) 1 i<j r0 (λ) Closed formula for [X w ] v and equivariant Giambelli formula. For each v W P, we shall define an n-tuple x v in the following way: Types C n and B n : Let v W P, and where 1 j 1 < < j k n. Then we put Type D n+1 : Let v W P, and {i 1 i n, v(i) > n} = { j 1,..., j k }, x v = (ε j1,..., ε jk, 0,..., 0). {i 1 i n + 1, v(i) > n + 1} = { j 1,..., j k }, where 1 j 1 < < j k n + 1. Note that k is even. If n is even we put x v = (ε j1,..., ε jk, 0,..., 0). If n is odd, let r = r(λ), where λ is the strict partition corresponding to v, and put { (ε j1,..., ε jr, 0,..., 0) if n r is odd x v = (ε j1,..., ε jr, ε n+1, 0,..., 0) if n r is even. Lemma 11. Let v W P. We have (1) Type C n : [X sn ] v = Q (n) 1 (x v 0, ε n,..., ε 2 ),

21 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 21 (2) Type B n : [X sn ] v = P (n) 1 (x v 0, ε n,..., ε 2 ), (3) Type D n+1 : [X sn+1 ] v = P (n) 1 (x v ( 1) n ε n+1, ε n,..., ε 2 ). Proof. Consider Type D n+1. By Proposition 1, (4), and (6.1), we have [X sn+1 ] v = ϖ n+1 v(ϖ n+1 ) = ε i. 1 i n+1, v(i)>n+1 Now recall the form of P 1 (x a) is given by (8.2). Then it is easy to check our formula. Types C n and B n are similar and much simpler. Now we can state the main result of this section. Theorem 4. Let w v W P, and λ µ the corresponding strict partitions. We have the following formulas: (1) Type C n ([10]): [X w ] v = Q (n) λ (x v 0, ε n,..., ε 2 ), (2) Type B n : [X w ] v = P (n) λ (x v 0, ε n,..., ε 2 ), (3) Type D n+1 : [X w ] v = P (n) λ (x v ( 1) n ε n+1, ε n,..., ε 2 ). Proof. The result for Type C n has been proved in [10]. We consider Type D n+1. Assume n is even for simplicity. The odd case is left for the reader. Fix µ SP ρn. Consider the following system of equations for the functions F λ (ε 1,..., ε n+1 ) (λ SP µ ) : (8.4) r 0 (µ) ε n µi +1 i=1 r 0 (λ) i=1 ε n λi +1 F λ (ε) = ν F ν (ε) for all λ SP µ \ {µ}, where the summation in the right hand side runs through ν SP µ such that ν λ and ν = λ + 1. By the equations (5.3) together with the initial condition F φ (ε 1,..., ε n ) = 1, the set of functions F λ (ε) (λ SP µ ) is characterized. By equation (2.2) for type D n+1, [X w ] v = ξ w (v) (cf. Proposition 2) satisfy equation (8.4) as well as the initial condition (cf. Proposition 1, (1)). Note that the coefficients w(ϖ n+1 ), β, the Chevalley multiplicity, is equal to one in the right hand side of equation (2.2), i.e., G/P in this case is minuscule. On the other hand, the functions on the right hand side of the formula in the theorem also satisfy (8.4). This fact is a consequence of Prpoposition 10 and the vanishing property (Proposition 9) of P (n) λ (x a). The initial condition is also satisfied. Therefore we have the theorem. The type B n case is quite similar to the case of type D n+1. The following result is a direct consequence of Theorem 4 and Lemma 10. Corollary 2. (Equivariant Giambelli) Let G/P be of type C n, B n, D n+1. For any λ SP ρn, we have [X λ ] = Pf ( [X λi,λ j ] ) 1 i<j r 0 (λ), where we denote by X λ the Schubert variety corresponding to λ. Proof. The proof is the same as that given in [10].

22 22 TAKESHI IKEDA AND HIROSHI NARUSE 9. Multiplicity of a singular point in a Schubert variety Another application is to the multiplicity of a singular point in a Schubert variety. We denote the multiplicity of the variety X w at e v by m v (X w ). We will explain the relationship between [X w ] v and m v (X w ). Then we discuss some implication of our result on m v (X w ). Let R uni (P ) be the unipotent radical of P, and let R + P the subset of R+ defined by R + P = {β R U β R uni (P )}, where U β is the root subgroup associated to β. Given v W P. Let Uv be the subgroup of G generated by the root subgroups U β, β v(r + \ R + P ). Under the map G G/P, U v is mapped isomorphically onto its image U v := Uv e v which is a canonical T -stable affine neighborhood of e v with a coordinate system {x β β v(r + \ R + P )}. Let G/P be either of types A n 1, C n, D n. For w, v W P, v w, let us denote Y w,v = X w U v. It is known that in these cases the defining ideal of the affine variety Y w,v is homogeneous in our coordinate system, i.e. Y w,v is a cone in U v. Actually we have a one parameter subgroup φ v Hom(C, T ) that φ v (t) (t C ) acts on U v by dilation. Then there exists h v Lie(T ) such that h v (β) = 1 for all β v(r + \ R + P ). For example, if v W (C n ) P n then h v is given by h v (ε i ) = 1 if v(i) > n and 1 if v(i) n, for 1 i n. 2 2 Note that each [X w ] v S, being a polynomial function on Lie(T ), can be evaluated at h v. For a background of the next proposition we refer to [27]. Proposition 11. Let G/P be either of types A n 1, C n, D n and w v W P. The value of [X w ] v evaluated at h v gives the multiplicity of the variety X w at e v. Proof. Consider the neighborhood U v of e v. The multiplicity m v (X w ) is determined by the Poincaré series of the coordinate ring C[Y w,v ], which is given by restricting the formal character ch C[Y w,v ] as a T -module to the one parameter subgroup φ v associated with h v. Then [X w ] v evaluated at h v is nothing but the classical multiplicity in the sense of Hilbert-Samuel. Remark. There are recurrence relations for m v (X w ) given by Lakshmibai and Weyman [23], which can be obtained by specializing equations (5.3), (8.4) at h v. So we have another proof of the above Proposition. Corollary 3. ([19],[12],[17]) Let G/P = G d,n, w v W P d, and λ µ Pd,n be the corresponding partitions. Then we have (9.1) m v (X w ) = E µ (λ). Proof. This is immediate from Proposition 11 and Theorem 1, since each summand in the right hand side of the formula becomes one, when specialized at h v. Remark 2. This formula can be obtained as a direct consequence of the result in [19], [12] describing the Gröbner basis of the defining ideal of Y w,v, together with a combinatorial argument in [17]. Corollary 4. Let G/P be the Grassmannian of types C n or D n+1. Let w v W P, and λ µ SP ρn be the corresponding strict partitions. Then we have (1) ([8],[18]) Type C n (9.2) m v (X w ) = Eµ(λ), I

23 EXCITED YOUNG DIAGRAMS AND EQUIVARIANT SCHUBERT CALCULUS 23 (2) Type D n+1 (9.3) m v (X w ) = Eµ II (λ). Proof. The same as Corollary 3. Example 7. (cf. Example 4) Let v = , w = (type D 5 ), and v = , w = (type B 4 or C 4 ). Then we have m v (X w ) = 10 for C 4 and m w (X v ) = 5 for B 4, D 5. Remark 3. We should make a remark on (9.2) similar to Remark 2. Ghorpade and Raghavan [8] has given a detailed description of the Gröbner basis of the defining ideal of Y w,v for the Lagrangian Grassmannian. Then (9.2) can be derived from a result in [18]. As for type D n+1, formula (9.3) seems to be new (see note added in Section 1). We close this section by pointing out the following fact. Proposition 12. With notations as in Corollary 4, we have m v (X w ) = Pf(m v (X λi,λ j )) 1 i<j r0 (λ). Proof. This is immediate from Corollary 2 and Proposition Lattice paths method As a supplementary discussion, we will show a direct combinatorial route from the combinatorial formula (Theorem 3) to the equivariant Giambelli formula (Corollary 2). Our argument relies on the lattice path method due to Stembridge. In order to deal with the case of type D n+1 also, we slightly modify the results in [29] Perfect matchings. Let r be a positive even integer and consider a set of r elements U r = {u 1,..., u r }. A perfect matching on U r is a partition of the set U r with each part consisting of two element subset. We denote a perfect matching σ on U r such a way that σ = {(u ik, u jk )} k=1,...,r/2, where i k < j k (1 k r/2), i 1 < < i r/2, and put I σ = {i 1,..., i r/2 }. The signature sgn(σ) of σ is defined as ( 1) c(σ) where c(σ) = #{(i, j, k, l) 1 i < j < k < l r, (u i, u k ), (u j, u l ) σ} is the number of crossings in σ (which depends on the numbering of elements of U r but we fix it once and for all). Denote by M r the set of all perfect matchings on the set U r and let σ 0 be the identical perfect matching {(u 2i 1, u 2i )} r/2 i=1. Lemma 12. There is an involution on M r \ {σ 0 } (σ σ ) such that sgn(σ) = sgn(σ ) and I σ = I σ. Proof. Take the smallest number k such that (u i, u 2k 1 ), (u j, u 2k ) σ. Define a perfect matching σ by replacing (u i, u 2k 1 ), (u j, u 2k ) in σ into (u j, u 2k 1 ), (u i, u 2k ). Then it is easy to see that σ has the desired property Modified version of Stembridge s argument. Let µ = (µ 1,..., µ l ) SP ρn. We define a directed graph as follows. The vertex set is D µ (D µ + a), where a = ( 1 2, 1 2 ). Direct an edge from u to v if (1) v u = b, with b = ( 1 2, 1 2 ) or (2) u D µ and v u = a. If we take µ = ρ n, the directed graph is the following.