S. Young. The HTK Book. Cambridge University Press, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....e2, e for K . Let r: 0 r r2 . rm n be integers. The flag manifold Fr has a Bruhat decomposition indexed by those permutations w = ww2. w, in the symmetric group on n letters whose descent set ilwi wi is a subset of r,r2, r . Here, X is a Schubert cell of lFr (see 9 of [3]) Let Gm act on IK by s.ej = sJej for j = 1, n. Suppose K is an (n ri) plane, none of whose Pliicker coordinates vanish. For s G Gin, set K(s) s.K. Let y(K) Gm be the family whose fibre over s G Gm is the simple Schubert variety rI) K(s) Recall that this is the collection of flags ....

....Furthermore, K(s) K(s) ff Q, where Q is the quadric of isotropic points in 1 . It follows that 9(K(s) is the intersection of OG(r) with the simple Schubert variety [2(K(s) of G(r, n) that is 9(K(s) x( 2(K(s) Note that K(s) s.K , as Gm preserves the quadratic form. As in 3. 2 (see [21, 3]) the results proven for the family (K(s) in 3.1 imply the corresponding results for the family y(K) and any collection of such families. Multiplicity freeness follows from a cohomological formula due to Chevalley [2] In this way, we establish Theorem 1.1 for the orthogonal grassmannian ....

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W. FULTON, Young tableaux, Cambridge University Press, Cambridge, 1997.

....integers. The ag manifold F r has a Bruhat decomposition F r = w ; indexed by those permutations w = w 1 w 2 : w n in the symmetric group S n on n letters whose descent set fi j w i w i 1 g is a subset of fr 1 ; r 2 ; r m g. Here, X w is a Schubert cell of F r (see x9 of [3]) e i . Suppose K is an (n r i ) plane, none of whose Pl ucker coordinates vanish. For s 2 G m , set K(s) s:K. Let Y(K) G m be the family whose bre over s 2 G m is the simple Schubert variety i (K(s) The projection F r G(r i ; n) sends a ag E 2 F r to its ith component E i 2 ....

W. Fulton, Young tableaux, Cambridge University Press, Cambridge, 1997.

....techniques formed the backbone of classical elimination theory, as presented in [58, 107] the study of methods for solving systems of polynomials equations by repeated elimination of variables. They also played a major role in the development of invariance theory in the past century [33, 35, 39, 111]. Recent surveys of resultant based methods can be found in [98, 103] An overview of classical elimination methods is given in [49] The reader will nd more details in [47] where an algebraic development of the theory of classical multivariate resultants is presented, and in [56] for forming ....

J.H. Grace and A. Young. The Algebra of Invariants. Cambridge University Press, 1903.

....k : F (L) G k (L) by E q 7 E k . By Corollary 5. 11, Phi k : X j Gamma Y i where X j : k (X u E q X (ju) q ) Schubert varieties Omega of the Grassmannian G k (L) are indexed by ordinary partitions (weakly decreasing sequences) n Gamma k 1 Delta Delta Delta k 0 [20]. We show Phi k (fK 2 Sp 2n C =B j v 2 Kg) Omega ) 15) where (n Gamma k; 1 ) is the hook shaped partition with first row n Gamma k and first column k. It follows from the projection formula that fK j v 2 Kg = deg Omega which is deg(Sw Delta S 0 jw ....

....varieties. Thus a generic K in this intersection determines g(f (K) K L; K ) uniquely. For j 2 [n] and a (not necessarily strict) partition with n Gamma j 1 Delta Delta Delta j 0, let oe 2 H G j (L) be the Schubert class associated to the partition , as in [20]. We show: Lemma 7.7. g f q n = k oe (n Gammak;1 ) k Gamma1 oe (n Gammak 1;1 ) Let i 2 B n be minimal with ffi(i ) 0, L(i) n, and supp(i) n] We construct a minimal permutation j 2 S n with jjjjj = n Gamma 1, a k 2 [n] and w 2 S n with jw C k w, and jw 6C k Gamma1 w. ....

, Young Tableaux, Cambridge University Press, 1996.

....= f( 2 Z : r 1 = r 1 = r 1 = 0g. Even stronger, we have LR r 1 Z = f( 2 Z : r 1 = 0g. Littlewood Richardson semigroups appear naturally in several other contexts: 1. Hall algebra, extensions of abelian p groups: see [M] 2. Schubert calculus on Grassmannians: see [F]. 3. Polynomial matrices and their invariant factors: see [T] 4. Eigenvalues of sums of Hermitian matrices. Let us discuss the last item in more detail. For a Hermitian matrix A of order r, let (A) denote the sequence of eigenvalues of A arranged in a weakly decreasing order (recall that A is ....

W. Fulton, Young tableaux, Cambridge University Press, 1997.

....setting. The primary references for Schubert calculus and Schubert polynomials are the papers of Borel, of Bernstein, Gelfand, and Gelfand, of Demazure, and of Lascoux and Sch utzenberger (see [L] introductory and comprehensive accounts of both the classical theory and newer research are [F,FP] (for the algebraic geometrical side) and [M1,M2,W3] for the algebraic combinatorial side) The purpose of the present paper is to give a short and natural proof of a combinatorial rule for the generation of Schubert polynomials, which was conjectured rst by Kohnert in his Ph. D. dissertation ....

W. Fulton, \Young tableaux", Cambridge University Press, Cambridge, 1997.

....defined hypersurface in G=B. We note that one natural choice of normalization is the following: define p fl as the generalized minor Delta fl; i , in the notation of [5, Section 1. 4] For the type An Gamma1 , the notion of a Plucker coordinate specializes to the ordinary one (see, e.g. [7]) as follows. Let us use the standard numeration of the fundamental weights, so that V 1 = V = C is the defining representation of G = SLn , and V i = V . Plucker weights of level i are naturally identified with subsets I ae [1; n] of cardinality i: under this identification, the weight ....

W. Fulton, Young tableaux, Cambridge University Press, 1997.

....of the Pieri type formula that we complete in Section 3. Also needed is Lemma 2.1, which identifies a particular subspace of H he 1 ; e n i for c . For Lemma 2.1, we work in the (classical) Grassmannian G k (V : he 1 ; e n i. For basic definitions and results see any of [8, 5, 4]. Schubert c of G k (V ) are indexed by partitions oe 2 Y k , that is, integer sequences oe = oe 1 ; oe k ) with n Gamma k oe 1 Delta Delta Delta oe k 0. For oe 2 Y k define j = n Gamma k Gamma oe k 1 Gammaj . For oe; 2 Y k , define : fH 2 G k (V ) j dimH he ....

....formula that we complete in Section A.3. Also needed is Lemma A.2.1, which identifies a particular subspace of H hf 1 ; f n i for H 2 Y c . For Lemma A.2.1, we work in the (classical) Grassmannian G k (W : hf 1 ; f n i. For basic definitions and results see any of [8, 5, 4]. Schubert c of G k (W ) are indexed by partitions oe 2 Y k . For oe 2 Y k define j = n Gamma k Gamma oe k 1 Gammaj . For oe; 2 Y k , define : fH 2 G k (W ) j dimH hf k 1 Gammaj j ; f n i j; 1 j kg c : fH 2 G k (W ) j dimH hf 1 ; f j oe ....

W. Fulton, Young Tableaux, Cambridge University Press, 1996.

....for example) attempts to determine the set of irreducible solutions of MA can be traced back almost 100 years to a paper of Elliott [2] where the author produces generating functions to determine these solutions. Another early attempt at producing this set of minimal solutions can be found in [5]. The development of modern algorithms connected with these solutions has become a popular topic of research in computational algebra (see [4] and [15] There has been some recent progress in the study of the algebraic properties of the monoids MA , and the interested reader is directed to our ....

J.H. Grace and A. Young, The Algebra of Invariants, Cambridge University Press, Cambridge, 1903.

.... Omega Ja K ffl (s) as s approaches t along the rational normal curve. iii) Suppose that ff ffl = ff 1 ; J a ; ff 2 ; ff n are Schubert data. Then d(m; p; ff ffl ) X ff a fi d(m; p; fi; ff 2 ; ff n ) Statement (i) is the usual statement of Pieri s formula [Fulton 1997; Hodge and Pedoe 1952] Statement (ii) is Theorem 8.1 of [Eisenbud and Harris 1983] and (iii) is a direct consequence of (i) Definition (3 1) implies that Omega fi K ffl ae Omega ff K ffl if and only if ff fi coordinatewise. In fact, Omega fi K ffl Omega ff K ffl = Omega fiff K ffl ....

....general 2 planes a; b; c and general 3 planes A; B; C, there are 4 flags X ae Y satisfying the following conditions: 1. X meets a; B, and C nontrivially, and 2. dimY A 2 and Y meets b and c nontrivially. That this number is 4 may be verified using the Schubert calculus for a flag manifold [Fulton 1997] or the equations we give below. Let K ffl (s) be the flag of subspaces osculating the standard rational normal curve. Set a : K 2 (4) b : K 2 (1) c : K 2 ( Gamma5) A : K 3 (0) B : K 3 (3) C : K 3 ( Gamma1) We claim that of the 4 flags X ae Y satisfying conditions 1 and 2 above for ....

W. Fulton, Young tableaux, London Math. Soc. Student Texts 35, Cambridge University Press, Cambridge, 1997.

.... partitions of the elements of H (or, indeed, of a generating set for H) c) Let be any partition of and let G 1 ( be the subgroup of G 1 consisting of all permutations xing all the parts of (the intersection of G with the corresponding Young subgroup of the symmetric group: see Fulton [5]) Now CP(G 1 ( consists of those partitions in CP(G 1 ) lying below in the partition lattice. Hence, if G 2 ( is analogously de ned in G 2 , we have CP(G 1 ( CP(G 2 ( and hence jG 1 ( j = jG 2 ( j, by part (a) of the theorem. Now the conclusion follows by M obius inversion, ....

W. Fulton, Young Tableaux, London Math. Soc. Student Texts 35, Cambridge University Press, Cambridge, 1997.

....A Young tableau of shape and content f1; j jg is a standard Young tableau, see Fig. 4. The bijection of the following Proposition is known as the Robinson Schensted correspondence. This correspondence is the starting point of much combinatorial work on Young tableaux. We refer to [4, 13, 15] for more comprehensive treatments of this topic. 5 Proposition 2. There is a bijection between the permutations of f1; ng and pairs (P; Q) of standard Young tableaux of the same shape and j (P)j = n. A set X of points in R 2 is said to be in general position if no two points have ....

W. Fulton, Young Tableaux, Lond. Math. Soc. Student Texts 35, Cambridge University Press, 1997.

....when the entry which is bumped is larger than all the entries in the next column, in which case it is placed at the end of that column. The result of such a procedure is a tableau which is one box larger than b, call it b x say (Note: this notation is usually reserved for row bumping (e.g. in [5]) but we need it for compatibility with the Japanese word reading) Using this insertion procedure, one can define the product tableaux b 3 c : 1 1 1 ( b c 1 ) c 2 ) 1 1 1) c p ) 1.2) where w(c) c 1 c 2 1 1 1 c p is the Japanese reading of c i.e. reading from top to bottom and ....

....x) K2) These are the usual elementary Knuth relations, but with an extra condition on the pair of letters which swap positions. Just as in the type A case, an ordinary bump can be realized as sequence of K2 transformations followed by a sequence of K1 transformations on the word of the column [13, 5]. Symbolically, let us write ordinary bump K1 r K2 s where the transformations K1 and K2 act on the right. Let us now show that the special bumps I, IIa and IIb can also be realized as a sequence of elementary transformations involving not only K1 and K2, but also y 1 y 1 fi y y fi; y fi ....

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W. Fulton. Young Tableaux. Cambridge University Press, 1st edition, 1997.

.... G k (L) by E q 7 E k . By Corollary 5. 11, Phi k : X j Gamma Y i where X j : k (X u E q T X (ju) E 0 q ) Schubert varieties Omega of the Grassmannian G k (L) are indexed by ordinary partitions (weakly decreasing sequences) n Gamma k 1 Delta Delta Delta k 0 [20]. We show Phi Gamma1 k (fK 2 Sp 2n C =B j v 2 Kg) Omega (n Gammak;1 k Gamma1 ) 15) where (n Gamma k; 1 k Gamma1 ) is the hook shaped partition with first row n Gamma k and first column k. It follows from the projection formula that deg i Y i fK j v 2 Kg j = deg i X j ....

....a generic K in this intersection determines g(f Gamma1 (K) K T L; K T L ) uniquely. For j 2 [n] and a (not necessarily strict) partition with n Gamma j 1 Delta Delta Delta j 0, let oe 2 H G j (L) be the Schubert class associated to the partition , as in [20]. We show: Lemma 7.7. g f q n = k oe (n Gammak;1 k Gamma1 ) k Gamma1 oe (n Gammak 1;1 k Gamma2 ) Let i 2 B n be minimal with ffi(i ) 0, L(i) n, and supp(i) n] We construct a minimal permutation j 2 S n with jjjjj = n Gamma 1, a k 2 [n] and w 2 S n with jw C k w, ....

, Young Tableaux, Cambridge University Press, 1996.

....in Section 3. Also needed is Lemma 2.1, which identifies a particular subspace of H he 1 ; e n i for H 2 X T X 0 c . For Lemma 2.1, we work in the (classical) Grassmannian G k (V ) of k planes in V : he 1 ; e n i. For basic definitions and results see any of [8, 5, 4]. Schubert subvarieties Omega oe ; Omega 0 oe c of G k (V ) are indexed by partitions oe 2 Y k , where n Gamma k oe 1 Delta Delta Delta oe k 0. For oe 2 Y k define oe c 2 Y k by oe c j = n Gamma k Gamma oe k 1 Gammaj . For oe; 2 Y k , define Omega : fH 2 G k (V ) j ....

W. Fulton, Young Tableaux, Cambridge University Press, 1996.

....equals the optimal number in (3) 2.1. An example. We describe the two approaches for the case (m; p) 3; 2) The Grassmannian of 2 planes in C 5 has dimension 6 and is embedded into P 9 . Its degree (3) is five. The Grobner homotopy works directly in the ten Plucker coordinates: 12] [13]; 14] 15] 23] 24] 25] 34] 35] 45] The ideal I 3;2 of the Grassmannian in the Plucker embedding is generated by five quadrics: 14] 23] Gamma [13] 24] 12] 34] 15] 23] Gamma [13] 25] 12] 35] 15] 24] Gamma [14] 25] 12] 45] 15] 34] Gamma [14] 35] 13] 45] 25] 34] ....

....6 and is embedded into P 9 . Its degree (3) is five. The Grobner homotopy works directly in the ten Plucker coordinates: 12] 13] 14] 15] 23] 24] 25] 34] 35] 45] The ideal I 3;2 of the Grassmannian in the Plucker embedding is generated by five quadrics: 14] 23] Gamma [13][24] 12] 34] 15] 23] Gamma [13] 25] 12] 35] 15] 24] Gamma [14] 25] 12] 45] 15] 34] Gamma [14] 35] 13] 45] 25] 34] Gamma [24] 35] 23] 45] 6) This set is the reduced Grobner basis for I 3;2 with respect to any term order which selects the underlined terms as leading ....

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W. Fulton, Young Tableaux, Cambridge University Press, 1996.

....the dimension of each homogeneous component and having a way to construct generators for each component, one can investigate in a second step the construction of a minimal set of generators, via linear algebra. These techniques have already been used for isimplej cases and by hand (see [9] [6]) We plan to extend them to more diOEcult cases with symbolic computation methods. The number of minimal generators of A can roughly be estimated as follows. Let A = P 1 n=1 An . Then the number of independent elements of A=A 2 , which is the number of independent homogeneous elements of A ....

J.H. Grace and A. Young. The Algebra of Invariants. Cambridge University Press, 1903.

....r 1 = r 1 = 0g. Even stronger, we have LR r 1 Z 3r 2 = LR r , where Z 3r 2 = f( 2 Z 3(r 1) r 1 = 0g. Littlewood Richardson semigroups appear naturally in several other contexts: 1. Hall algebra, extensions of abelian p groups: see [M] 2. Schubert calculus on Grassmannians: see [F]. 3. Polynomial matrices and their invariant factors: see [T] 4. Eigenvalues of sums of Hermitian matrices. Let us discuss the last item in more detail. For a Hermitian matrix A of order r, let (A) denote the sequence of eigenvalues of A arranged in a weakly decreasing order (recall that A is ....

W. Fulton, Young tableaux, Cambridge University Press, 1997.

....is real. This has implications for the problem of placing real poles in linear systems theory [1] and is a special case of a far reaching conjecture of Shapiro and Shapiro [14] Special Schubert conditions For background on the Grassmannian, Schubert cycles, and the Schubert calculus, see [7, 6, 5]. Let m; p 1 be integers. Let fl be a rational normal curve in R m p . For k 0 and s 2 fl, let K k (s) be the k plane osculating fl at s. For every integer a 0, let a (s) be the special Schubert cycle consisting of p planes H which meet Km 1 Gammaa (s) nontrivially and let a (s) be ....

....We use the following two results of Eisenbud and Harris [2] who studied such intersections in their theory of limit linear systems. For a Schubert class w, let jwj be the codimension of this conditions and w a be the index of summation in the Pieri formula in the cohomology of the Grassmannian [5] oe w Delta a = X v2w a oe v Proposition 2. 1. Theorem 2.3 of [2] Let s 1 ; s n be distinct points on fl and w 1 ; w n be Schubert conditions. Then the intersection of Schubert cycles oe w1 (s 1 ) oe w2 (s 2 ) Delta Delta Delta oe wn (s n ) is proper in that it ....

, Young Tableaux, Cambridge University Press, 1996.

....and C. Greene [3] Theorem 3.8 is due to A. Berenstein and A. Zelevinsky [1] who also gave reformulations exhibiting other symmetries of the Littlewood Richardson coefficients. References to other versions of the Littlewood Richardson Rule, along with alternative proofs, can be found, e.g. in [3, 4, 8, 15, 17], and in sources cited therein. Generalizations and variations of this rule are numerous, and we do not attempt at reviewing them here. 26 ....

W. Fulton, Young tableaux, Cambridge University Press, 1997.

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S. Young. The HTK Book. Cambridge University Press, 1995.

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Fulton W., Young tableaux (Cambridge University Press) 1997.

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Fulton W., Young tableaux (Cambridge University Press) 1997.

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Grace, J.H. and Young, A., The Algebra of Invariants, Cambridge University Press, 1903.

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W. Fulton, Young tableaux, Cambridge University Press, Cambridge, 1997.

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