Bulletin of the American Mathematical Society

MathSciNet review: 1567485
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: 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.

I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Schubert cells, and the cohomology of the spaces $G/P$, Uspehi Mat. Nauk 28 (1973), no. 3(171), 3–26 (Russian). MR 0429933

Anders Björner and Michelle Wachs, Bruhat order of Coxeter groups and shellability, Adv. in Math. 43 (1982), no. 1, 87–100. MR 644668, DOI 10.1016/0001-8708(82)90029-9

Armand Borel, Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207 (French). MR 51508, DOI 10.2307/1969728

Armand Borel, Linear representations of semi-simple algebraic groups, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 421–440. MR 0372054

N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238

Roger W. Carter, Simple groups of Lie type, Pure and Applied Mathematics, Vol. 28, John Wiley & Sons, London-New York-Sydney, 1972. MR 0407163

[Che] C. Chevalley, Sur les décompositions cellulaires des espaces G/B (manuscrit non publié), circa 1958.

C. De Concini and V. Lakshmibai, Arithmetic Cohen-Macaulayness and arithmetic normality for Schubert varieties, Amer. J. Math. 103 (1981), no. 5, 835–850. MR 630769, DOI 10.2307/2374249

[D-P] C. DeConcini and C. Procesi, Complete symmetric varieties (preprint).

Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697

Michel Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. Math. 21 (1973), 287–301 (French). MR 342522, DOI 10.1007/BF01418790

[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).

Leopold Flatto, Invariants of finite reflection groups, Enseign. Math. (2) 24 (1978), no. 3-4, 237–292. MR 519546

Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725

Howard Hiller, Combinatorics and intersections of Schubert varieties, Comment. Math. Helv. 57 (1982), no. 1, 41–59. MR 672845, DOI 10.1007/BF02565846

Howard L. Hiller, Schubert calculus of a Coxeter group, Enseign. Math. (2) 27 (1981), no. 1-2, 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.

James E. Humphreys, Linear algebraic groups, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR 0396773

[H-L] C. Huneke and V. Lakshmibai, Arithmetic Cohen-Macaulayness and normality of the multicones over Schubert varieties in SL(n)/B (preprint).

S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, DOI 10.2307/2317421

Steven L. Kleiman, Problem 15: rigorous foundation of Schubert’s enumerative calculus, Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois Univ., De Kalb, Ill., 1974) Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R.I., 1976, pp. 445–482. MR 0429938

Bertram Kostant, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72–144. MR 142697, DOI 10.2307/1970202

Alain Lascoux, Fonctions de Schur et grassmanniennes. I, C. R. Acad. Sci. Paris Sér. A-B 281 (1975), no. 19, Aii, A813–A815. MR 435093

Alain Lascoux and Marcel-Paul 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. I Math. 295 (1982), no. 11, 629–633 (French, with English summary). 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.

Bernt Lindström, A partition of $L(3,\,n)$ into saturated symmetric chains, European J. Combin. 1 (1980), no. 1, 61–63. MR 576767, DOI 10.1016/S0195-6698(80)80022-9

V. Lakshmibai, C. Musili, and C. S. Seshadri, Geometry of $G/P$, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 2, 432–435. MR 520081, DOI 10.1090/S0273-0979-1979-14631-7

Robert A. Proctor, Classical Bruhat orders and lexicographic shellability, J. Algebra 77 (1982), no. 1, 104–126. MR 665167, DOI 10.1016/0021-8693(82)90280-0

Robert A. Proctor, Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin. 5 (1984), no. 4, 331–350. MR 782055, DOI 10.1016/S0195-6698(84)80037-2

C. S. Seshadri, Geometry of $G/P$. I. Theory of standard monomials for minuscule representations, C. P. Ramanujam—a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Springer, Berlin-New York, 1978, pp. 207–239. MR 541023

Richard P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic Discrete Methods 1 (1980), no. 2, 168–184. MR 578321, DOI 10.1137/0601021

Richard P. Stanley, Some combinatorial aspects of the Schubert calculus, Combinatoire et représentation du groupe symétrique (Actes Table Ronde CNRS, Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976) Lecture Notes in Math., Vol. 579, Springer, Berlin, 1977, pp. 217–251. MR 0465880

Richard P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Combin. 5 (1984), no. 4, 359–372. MR 782057, DOI 10.1016/S0195-6698(84)80039-6

Richard P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 475–511. MR 526968, DOI 10.1090/S0273-0979-1979-14597-X

Wilhelm Stoll, Invariant forms on Grassmann manifolds, Annals of Mathematics Studies, No. 89, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. MR 0481089

Daya-Nand Verma, Möbius inversion for the Bruhat ordering on a Weyl group, Ann. Sci. École Norm. Sup. (4) 4 (1971), 393–398. MR 291045

[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 0429933




[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