C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1. MR 95e:14041 672 NANTEL BERGERON AND FRANK SOTTILE  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rule would express the intersection form in the cohomology of a flag manifold in terms of its basis of Schubert classes. Other than the Littlewood Richardson rule, when the Schubert polynomials are Schur symmetric polynomials, little is known. Using geometry, Monk [28] and more generally Chevalley [7] established a formula for multiplication by linear Schubert polynomials (divisor Schubert classes) A Pieri type formula for multiplication by an elementary or complete homogeneous symmetric polynomial (special Schubert class) was given in [22] There are now several proofs of this result; some ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1. MR 95e:14041 672 NANTEL BERGERON AND FRANK SOTTILE

....H 2 (F l n ; Z) is spanned by the classes p(x i ) oe s i Gamma oe s i Gamma1 , i = 1; n, where, by convention, oe s 0 = 0. There is an explicit rule for multiplying any Schubert class by a 2 dimensional class oe k . Theorem 2.2. 3 (Monk s formula [36] cf. also Chevalley [10]) We have, for any w 2 S n and 1 k n, oe s k oe w = X oe ws ij ; where the sum is over all transpositions s ij such that i r j and (ws ij ) w) 1. 2.2.2 NilHecke ring and Schubert polynomials Bernstein, Gelfand, and Gelfand [5] and Demazure [14] suggested a procedure, based on ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, Proc. Symp. Pure Math. 56, Part 1, 1--23, Amer. Math. Soc., Providence, RI, 1994.

....will also need the following properties of the Schubert polynomials. Corollary 2.7 ( 25, 4.2) Let v; w 2 Sn . Then v Sw = ae S wv Gamma1 if (wv Gamma1 ) w) Gamma (v) 0 otherwise : QUANTUM SCHUBERT POLYNOMIALS 571 Theorem 2. 8 (Monk s formula [26, 25] cf. also Chevalley [6]) We have S sr Sw = X Swt ij ; where the sum is over all transpositions t ij such that i r j and (wt ij ) w) 1. Note that S sr = x 1 Delta Delta Delta x r . The Schubert polynomials have the following orthogonality property (see, e.g. 25, 5.4) For a polynomial f , ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in: Algebraic Groups and Their Generalizations (W. Haboush and B. Parshall, eds.), Proc. Symp. Pure Math., vol. 56, Part 1, Amer. Math. Soc., Providence, RI, 1994, pp. 1--23. MR 95e:14041

....X(w) satisfies A i (X(w) G F Q = 0 for i l(w) 8) see [Ful83, Example 1.7.6] So the conjecture stated above holds at the least for the G F invariant part. 4. RELATING THE CHOW GROUPS OF X(w) TO THOSE OF G=B Chow groups of flag varieties was first described in Chevalley s manuscript [Che94] (unpublished until recently) and later in [Dem74, Dem76] We recall the following facts: 1. The action of G induced on A (X) is trivial. 2. f[X w ] w 2Wg is a basis of A (X) with [X w ] 2 A l(w) X) Setting Y w = X w 0 w we get that f[Y w ] w 2 Wg is a basis of CH (X) Y w ] 2 CH ....

C. Chevalley, Sur les decompositions cellulaires des espaces G=B, Algebraic Groups and Their Generalizations: Classical Methods, Proc. Sympos. Pure Math., vol. 56, Part 1, Amer. Math. Soc., 1994, pp. 1--23.

.... of these rings is known [9] with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 12, 17, 19, 25, 29] Moreover, their multiplicative structure with respect to the Schubert basis is determined by Chevalley s formula [10]. Despite this, it remains an open problem to give a closed or bijective formula for the integral structure constants c w u v defined by the identity S u Delta S v = X w c w u v Sw : These c w u v are non negative as they count the flags in a suitable triple intersection of Schubert ....

....a definition. Definition 1.1. The 0 Bruhat order 0 on B n is defined recursively as follows: ul 0 w is a cover in the 0 Bruhat order if and only if (1) u) 1 = w) and (2) u Gamma1 w is a reflection of the form ( i) or ( j) i) for some 0 i j n. Chevalley s formula [10] may be stated as follows: B u Delta p 1 = X ul0w Bw C u Delta q 1 = X ul0w (u Gamma1 w)Cw ; 1) where (u Gamma1 w) is the number of transpositions in the reflection u Gamma1 w. We enrich the 0 Bruhat order in two complementary ways. Write the two types of covers in the ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1. 46 NANTEL BERGERON AND FRANK SOTTILE

....formula for classical flag varieties [17] was based upon those ideas. Similarly, the ideas here provide a basis for a proof of Pieri type formulas in the cohomology of symplectic flag varieties [1] These Pieri type formulas are due to Hiller and Boe [6] whose proof used the Chevalley formula [2]. Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski [14] These formulas also arise in the theory of projective representations of symmetric groups [15, 9] as product formulas for Schur P and Q functions, and were first ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1.

....flag manifold G=B or the analogues of Schubert polynomials [8, 17, 20, 45] SCHUBERT POLYNOMIALS AND THE BRUHAT ORDER 5 For finite Coxeter groups, this is the basis Delta w in the coinvariant algebra [23] Likewise, Theorem B and the expectation (1.1.3) have analogues. Of the known formulas [11, 24, 40, 42, 43, 44, 52] (see also the survey [41] few [11, 24, 40, 52] have been expressed in a chain theoretic manner. 1.2. Substitutions and the Schubert basis. In xx4.3 and 4.4, we study the c w u v when w(p) u(p) for some p. For w 2 S n 1 and 1 p n 1, let w= p 2 S n be defined by deleting the pth row and ....

....[8, 17, 20, 45] SCHUBERT POLYNOMIALS AND THE BRUHAT ORDER 5 For finite Coxeter groups, this is the basis Delta w in the coinvariant algebra [23] Likewise, Theorem B and the expectation (1.1.3) have analogues. Of the known formulas [11, 24, 40, 42, 43, 44, 52] see also the survey [41] few [11, 24, 40, 52] have been expressed in a chain theoretic manner. 1.2. Substitutions and the Schubert basis. In xx4.3 and 4.4, we study the c w u v when w(p) u(p) for some p. For w 2 S n 1 and 1 p n 1, let w= p 2 S n be defined by deleting the pth row and w(p)th column from the permutation matrix of w. ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1.

....they are known. One important case is Monk s formula [21] which characterizes the algebra of Schubert polynomials. While this is usually attributed to Monk, Chevalley simultaneously established the analogous formula for generalized flag manifolds in a manuscript that was only recently published [6]. Let s k be the transposition interchanging k and k 1. Then S s k = x 1 Delta Delta Delta x k = s 1 (x 1 ; x k ) the first elementary symmetric polynomial. For any permutation w 2 S n , Monk s formula states Sw Delta S s k = Sw Delta s 1 (x 1 ; x k ) X Swt a b ; ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1-- 23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1.

....and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 11, 14, 16, 22, 29] However, these algorithms do not show the non negativity of the c w u v . When S v is a hypersurface Schubert class, the c w u v are either 0, 1, or 2, by Chevalley s formula [10]. It remains an open problem to give a closed or bijective formula for the rest of these constants. The c w u v are expected to count certain chains in the Bruhat order of the Weyl group (see [3] and the references therein) Of particular interest are Pieri type formulas which describe the ....

....class pulled back from a Grassmannian projection (G=P , P maximal parabolic) as these determine the ring structure with respect to the Schubert basis for the cohomology of G=P when P is any parabolic subgroup. When P is a Borel subgroup, the ring structure is determined by Chevalley s formula [10]. Pieri type formulas for Grassmannians of classical groups are known. When G = SL n C , this is the classical Pieri formula, and for other G these formulas are due to Boe and Hiller [8] and to Pragacz and Ratajski [26, 27, 28] A goal of this paper is to begin extending these results to all P . ....

[Article contains additional citation context not shown here]

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, W. Haboush, ed., vol. 56, Part 1 of Proc. Sympos. Pure Math., Amer. Math. Soc., 1994, pp. 1--23.

....Pieri type formula for classical flag varieties [So] was based upon those ideas. Similarly, the ideas here appear in a proof of Pieri type formulas in the cohomology of isotropic flag varieties [BS] These Pieri type formulas are due to Hiller and Boe [BH] whose proof used the Chevalley formula [C]. Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski [PR93] These formulas also arise in the theory of projective representations of symmetric groups [Sc, HH] as product formulas for Schur P and Q functions, and were ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1.

.... of these rings is known [9] with respect to a monomial basis, and there are methods (Schubert polynomials) for expressing the Sw in terms of this basis [6, 7, 12, 17, 19, 25, 29] Moreover, their multiplicative structure with respect to the Schubert basis is determined by Chevalley s formula [10]. Despite this, it remains an open problem to give a closed or bijective formula for the integral structure constants c w u v defined by the identity S u Delta S v = X w c w u v Sw : These c w u v are non negative as they count the flags in a suitable triple intersection of Schubert ....

....a definition. Definition 1.1. The 0 Bruhat order 0 on B n is defined recursively as follows: ul 0 w is a cover in the 0 Bruhat order if and only if (1) u) 1 = w) and (2) u Gamma1 w is a reflection of the form ( i) or ( j) i) for some 0 i j n. Chevalley s formula [10] may be stated as follows: B u Delta p 1 = X ul 0 w Bw C u Delta q 1 = X ul 0 w (u Gamma1 w)Cw ; 1) where (u Gamma1 w) is the number of transpositions in the reflection u Gamma1 w. We enrich the 0 Bruhat order in two complementary ways. Write the two types of covers in the ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces G=B, in Algebraic Groups and their Generalizations: Classical Methods, American Mathematical Society, 1994, pp. 1--23. Proceedings and Symposia in Pure Mathematics, vol. 56, Part 1.

....to the classical Littlewood Richardson rule. An important case is when one of u or v is an adjacent transposition, t k k 1 . This is usually attributed to Monk [17] However, at the same time Chevalley established the analogous formula for generalized flag varieties in an unpublished manuscript [6]. For w 2 S1 , let (w) be the length of w. Monk s rule states: Sw Delta S t k k 1 = X Sw Deltat a b ; 2) the sum over all a k b with (wt a b ) w) 1. We use geometry to prove a similar result, which is the analog for Schubert polynomials of the classical Pieri s rule. This analog ....

C. Chevalley, Sur les d'ecompositions cellulaires des espaces g=b. unpublished manuscript, 1958.

....on G=B by left translation. B wB=B is a locally closed subvariety of X of dimension l(w) in fact B wB=B A l(w) The closure of B wB=B in X is given by Xw = B wB=B = w 0 w B w 0 B=B: 3. 3) The Xw are called Schubert varieties and are generators of the Chow ring of G=B cf. [6, 9]. Let G act diagonally on X Theta X and let O(w) G: eB; wB) f(g 1 B; g 2 B) 2 X Theta X : g Gamma1 1 g 2 2 B wBg (3.4) be the orbit of (eB; wB) under this action. From the decomposition (3.2) it follows that we also have a decomposition of X Theta X, X Theta X = w2W O(w) 3.5) ....

....ring of a non singular variety cf. 13, 8. 1] If we only are interested in the additive structure of CH (Y ) we shall write A (Y ) We will now and then identify (sub)varieties V with their rational equivalence classes [V ] A (X) was first described in Chevalley s (unpublished) manuscript [6] and later in [9, 10] The following facts are sufficient for our purpose: 1. fXw : w 2 Wg is a basis of A (X) with Xw 2 A l(w) X) Setting Yw = Xw0w we get that fYw : w 2 Wg is a basis of CH (X) with Yw 2 CH l(w) X) 2. CH (X) is generated in degree 1: any Schubert variety Xw of ....

C. Chevalley, Sur les decompositions cellulaires des espaces G=B, in Algebraic Groups and Their Generalizations: Classical Methods, vol. 56, Part 1 of Proc. Sympos. Pure Math., Amer. Math. Soc., 1994, pp. 1--23.

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