|
Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
Retrieve article in:
PDF
Book Information
Author(s):
H. Hiller
Title:
The geometry of Coxeter groups
Additional book information:
Research Notes in Mathematics, Vol. 54, Pitman Advanced Publishing Program, Boston, 1982, 213 pp., $21.95. ISBN 0-2730-8517-4
References:
- [BGG] I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Schubert cells, and the cohomology of the spaces G/P, Russian Math. Surveys 28 (1973), no. 3, 1-26. MR 429933
- [B-W] A. Björner and M. Wachs, Bruhat order of Coxeter groups and shellability, Adv. in Math. 43 (1982), 423-435. MR 644668
- [Bo1] A. Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes des groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115-207. MR 51508
- [Bo2] A. Borel, Linear representations of semi-simple algebraic groups, Algebraic Geometry, Arcata, 1974 (R. Hartshorne, ed.), Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, R.I., 1975, pp. 421-441. MR 372054
- [Bou] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4-6, Hermann, Paris, 1968. MR 240238
- [Car] R. W. Carter, Simple groups of Lie type, Wiley, New York, 1972. MR 407163
- [Che] C. Chevalley, Sur les décompositions cellulaires des espaces G/B (manuscrit non publié), circa 1958.
- [D-L] C. DeConcini and V. Lakshmibai, Arithmetical Cohen-Macaulayness and arithmetical normality for Schubert varieties, Amer. J. Math. 103 (1981), 835-850. MR 630769
- [D-P] C. DeConcini and C. Procesi, Complete symmetric varieties (preprint).
- [De1] M. Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53-88. MR 354697
- [De2] M. Demazure, Invariants symétriques des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287-301. MR 342522
- [E-G] P. Edelman and C. Greene, Combinatorial correspondences for Young tableaux, balanced tableaux, and maximal chains in the Bruhat order of S (research announcement).
- [Fla] L. Flatto, Invariants of finite reflection groups, Énseign. Math. (2) 24 (1978), 237-292. MR 519546
- [G-H] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York, 1978. MR 507725
- [Hi1] H. Hiller, Combinatorics and intersections of Schubert varieties, Comment. Math. Helv. 57 (1982), 41-59. MR 672845
- [Hi2] H. Hiller, Schubert calculus of a Coxeter group, Énseign. Math. (2) 27 (1982), 57-84. MR 630960
- [H-B] H. Hiller and B. Boe, Pieri formula for SO (preprint).
- [Hir] F. Hirzebruch, Characteristic numbers of homogeneous domains, Seminars on Analytic Functions, Vol. 2, Institute for Advanced Study, Princeton, N.J., 1957.
- [Hum] J. Humphreys, Linear algebraic groups, Springer-Verlag, New York, 1975. MR 396773
- [H-L] C. Huneke and V. Lakshmibai, Arithmetic Cohen-Macaulayness and normality of the multicones over Schubert varieties in SL(n)/B (preprint).
- [K-L] S. L. Kleiman and D. Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061-1082. MR 323796
- [Kle] S. L. Kleiman, Problem 15: Rigorous foundation of Schubert's enumerative calculus, Mathematical Developments Arising from Hilbert Problems (F. Browder, ed.), Proc. Sympos. Pure Math., vol. 28, Amer. Math. Soc., Providence, R.I., 1976, pp. 445-482. MR 429938
- [Kos] B. Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72-144. MR 142697
- [Las] A. Lascoux, Fonctions de Schur et grassmanniennes, C. R. Acad. Sci. Paris Sér. A 281 (1975), 813-815; 851-854. MR 435093
- [L-S] A. Lascoux and M.-P. Schützenberger, Structure de Hopf de l'anneau de cohomologie et de l'anneau de Grothendieck d'une variété de drapeaux, C. R. Acad. Sci. Paris Sér. A 295 (1982), 629-633. MR 686357
- [Li1] B. Lindström, Conjecture on a theorem similar to Sperner's, Combinatorial Structures and their Applications (R. Guy, ed.), Gordon and Breach, New York, 1970.
- [Li2] B. Lindström, A partition of L(3, n) into saturated symmetric chains, European J. Combin. 1 (1980), 61-63. MR 576767
- [LMS] V. Lakshmibai, C. Musili and C. S. Seshadri, Geometry of G/P, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 432-435. MR 520081
- [Pr1] R. Proctor, Classical Bruhat orders and lexicographic shellability, J. Algebra 77 (1982), 104-126. MR 665167
- [Pr2] R. Proctor, Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin. (to appear). MR 782055
- [Ses] C. S. Seshadri, Geometry of G/P. I: Theory of standard monomials for minuscule representations, C. P. Ramanujan--A tribute, Tata Inst. Fund. Res. Studies in Math., 8, Springer, Berlin, 1978, pp. 207-239. MR 541023
- [St1] R. Stanley, Weyl groups, the hard Lefschetz theorem and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), 168-184. MR 578321
- [St2] R. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et Representation du Groupe Symetrique (D. Foata, ed.), Lecture Notes in Math., No. 579, Springer-Verlag, New York, 1977. MR 465880
- [St3] R. Stanley, On the number of reduced decompositions of elements of Coxeter groups (preprint). MR 782057
- [St4] R. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Soc. (N.S.) 1 (1979), 475-511. MR 526968
- [Sto] W. Stoll, Invariant forms on Grassmann manifolds, Ann. of Math. Studies, No. 89, Princeton Univ. Press, Princeton, N.J., 1977. MR 481089
- [Ver] D.-N. Verma, Möbius inversion for the Bruhat ordering on a Weyl group, Ann. Sci. École Norm. Sup. (4) 4 (1971), 393-398. MR 291045
Additional Information:
Reviewer(s):
Robert A.
Proctor
Review Information:
Journal:
Bull. Amer. Math. Soc.
10
(1984),
142-150.
DOI:
10.1090/S0273-0979-1984-15223-6
PII:
S 0273-0979(1984)15223-6
|
