Abstract
Let $(W,S)$ be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let $J\subseteq S$. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For $w\in W^J$, let $f^{w\smash{,J}}_{i}$ denote the number of elements of length $i$ below $w$ in Bruhat order on $W^J$ (with notation simplified to $f^{w}_{i}$ in the case when $W^J=W$). We show that $$ 0\le i\lt j\le \ell (w)-i \quad\hbox{implies}\quad f^{w\smash{,J}}_{i} \le f^{w\smash{,J}}_{j}.$$ Also, the case of equalities $\smash{f^{w}_{i} = f^{w}_{\ell(w)-i}}$ for $i=1, \ldots,k$ is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial $P_{e,w}(q)$.
We show that if $W$ is finite then the number sequence $\smash{f^{w}_{0}, f^{w}_{1}, \ldots, f^{w}_{\ell (w)}}$ cannot grow too rapidly. Further, in the finite case, for any given $k\ge 1$ and any $w\in W$ of sufficiently great length (with respect to $k$), we show $$ f^{w}_{\ell(w)-k} \ge f^{w}_{\ell(w)-k+1} \ge\cdots \ge f^{w}_{\ell (w)}.$$
The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if $\mskip3mu\overline{\mskip-3mu X} _w$ is a Schubert variety of dimension $d=\ell (w)$, and $\lambda=c_1 (\mathscr L)\in H^2 (\mskip3mu\overline{\mskip-3mu X} _w)$ is the restriction to $\mskip3mu\overline{\mskip-3mu X} _w$ of the Chern class of an ample line bundle, then \[(\lambda^k)\,\cdot \,{} : H^{d-k}(\mskip3mu\overline{\mskip-3mu X} _w) \rightarrow H^{d+k}(\mskip3mu\overline{\mskip-3mu X} _w) \] is injective for all $k\ge 0$.
-
[bjoerner05::combin+coxet] A. Björner and F. Brenti, Combinatorics of Coxeter Groups, New York: Springer-Verlag, 2005.
@book{bjoerner05::combin+coxet, MRKEY = {2133266},
AUTHOR = {Bj{ö}rner, Anders and Brenti, Francesco},
TITLE = {Combinatorics of {C}oxeter Groups},
SERIES = {Grad. Texts in Math.},
FSERIES = {Graduate Texts in Mathematics},
NUMBER = {231},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2005},
PAGES = {xiv+363},
ISBN = {978-3540-442387; 3-540-44238-3},
MRCLASS = {05-01 (05E15 20F55)},
MRNUMBER = {2006d:05001},
MRREVIEWER = {Jian-yi Shi},
ZBLNUMBER = {1110.05001},
} -
[BBD82] A. A. Beuilinson, J. Bernstein, and P. Deligne, "Faisceaux pervers," in Analyse et Topologie sur les Espaces Singuliers, I, Paris: Soc. Math. France, 1982, pp. 5-171.
@incollection{BBD82, MRKEY = {751966},
AUTHOR = {Be{\u\i}linson, A. A. and Bernstein, J. and Deligne, P.},
TITLE = {Faisceaux pervers},
BOOKTITLE = {Analyse et Topologie sur les Espaces Singuliers, {\rm I}},
VENUE = {Luminy, 1981},
SERIES = {Astérisque},
NUMBER = {100},
PAGES = {5--171},
PUBLISHER = {Soc. Math. France},
ADDRESS = {Paris},
YEAR = {1982},
MRCLASS = {32C38},
MRNUMBER = {86g:32015},
ZBLNUMBER = {0536.14011},
MRREVIEWER = {Zoghman Mebkhout},
} -
[bernstein94::equiv] J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, New York: Springer-Verlag, 1994.
@book{bernstein94::equiv, MRKEY = {1299527},
AUTHOR = {Bernstein, Joseph and Lunts, Valery},
TITLE = {Equivariant Sheaves and Functors},
SERIES = {Lecture Notes in Math.},
NUMBER = {1578},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1994},
PAGES = {iv+139},
ISBN = {3-540-58071-9},
MRCLASS = {55N91 (14M25 18E30 54B40 55N30)},
MRNUMBER = {95k:55012},
ZBLNUMBER = {0808.14038},
MRREVIEWER = {Yi Hu},
} -
[braden01::from]
T. Braden and R. MacPherson, "From moment graphs to intersection cohomology," Math. Ann., vol. 321, iss. 3, pp. 533-551, 2001.
@article{braden01::from, MRKEY = {1871967},
AUTHOR = {Braden, Tom and MacPherson, Robert},
TITLE = {From moment graphs to intersection cohomology},
JOURNAL = {Math. Ann.},
FJOURNAL = {Mathematische Annalen},
VOLUME = {321},
YEAR = {2001},
NUMBER = {3},
PAGES = {533--551},
ISSN = {0025-5831},
CODEN = {MAANA},
MRCLASS = {14F43 (14M15 32S60)},
MRNUMBER = {2003g:14030},
ZBLNUMBER = {1077.14522},
DOI = {10.1007/s002080100232},
} -
[brion00::poinc]
M. Brion, "Poincaré duality and equivariant (co)homology," Michigan Math. J., vol. 48, pp. 77-92, 2000.
@article{brion00::poinc, MRKEY = {1786481},
AUTHOR = {Brion, Michel},
TITLE = {Poincaré duality and equivariant (co)homology},
INVNOTE = {Dedicated to William Fulton on the occasion of his 60th birthday},
JOURNAL = {Michigan Math. J.},
FJOURNAL = {The Michigan Mathematical Journal},
VOLUME = {48},
YEAR = {2000},
PAGES = {77--92},
ISSN = {0026-2285},
MRCLASS = {14F43 (14M15 20F55 55N91)},
MRNUMBER = {2001m:14032},
ZBLNUMBER = {1077.14523},
MRREVIEWER = {Dan Edidin},
DOI = {10.1307/mmj/1030132709},
} -
[carrell94::bruhat+coxet+deodh+schub] J. B. Carrell, "The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties," in Algebraic Groups and their Generalizations: Classical Methods, Haboush, W. J. and Parshall, B. J., Eds., Providence, RI: Amer. Math. Soc., 1994, pp. 53-61.
@incollection{carrell94::bruhat+coxet+deodh+schub, MRKEY = {1278700},
AUTHOR = {Carrell, James B.},
EDITOR = {Haboush, William J. and Parshall, Brian J.},
TITLE = {The {B}ruhat graph of a {C}oxeter group, a conjecture of {D}eodhar, and rational smoothness of {S}chubert varieties},
BOOKTITLE = {Algebraic Groups and their Generalizations: {C}lassical Methods},
SERIES = {Proc. Sympos. Pure Math.},
NUMBER = {56},
PAGES = {53--61},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1994},
MRCLASS = {14M15 (14L35)},
MRNUMBER = {95d:14051},
ZBLNUMBER = {0818.14020},
MRREVIEWER = {E. Akyı ldı z},
} -
[carrell95] J. B. Carrell, "On the smooth points of a Schubert variety," in Representations of Groups, Allison, B. N. and Cliff, G. H., Eds., Providence, RI: Amer. Math. Soc., 1995, pp. 15-33.
@incollection{carrell95, MRKEY = {1357193},
AUTHOR = {Carrell, James B.},
EDITOR = {Allison, Bruce N. and Cliff, Gerald H.},
TITLE = {On the smooth points of a {S}chubert variety},
BOOKTITLE = {Representations of Groups},
INVVENUE = {Banff, AB, 1994},
SERIES = {{\rm CMS} Conf. Proc.},
NUMBER = {16},
PAGES = {15--33},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1995},
MRCLASS = {14M15 (14B05)},
MRNUMBER = {96j:14035},
ZBLNUMBER = {0854.14022},
MRREVIEWER = {{\c{S}}erban B{\u{a}}rc{\u{a}}nescu},
} -
[De80]
P. Deligne, "La conjecture de Weil, II," Inst. Hautes Études Sci. Publ. Math., vol. 52, pp. 137-252, 1980.
@article{De80, MRKEY = {601520},
AUTHOR = {Deligne, Pierre},
TITLE = {La conjecture de {W}eil, {II}},
JOURNAL = {Inst. Hautes Études Sci. Publ. Math.},
FJOURNAL = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
VOLUME = {52},
YEAR = {1980},
PAGES = {137--252},
ISSN = {0073-8301},
CODEN = {PMIHA6},
MRCLASS = {14G13 (10H10)},
MRNUMBER = {83c:14017},
ZBLNUMBER = {0456.14014},
MRREVIEWER = {Spencer J. Bloch},
URL = {http://www.numdam.org/item?id=PMIHES_1980__52__137_0},
} -
[dyer91::oen+bruhat+coxet]
M. Dyer, "On the “Bruhat graph” of a Coxeter system," Compositio Math., vol. 78, iss. 2, pp. 185-191, 1991.
@article{dyer91::oen+bruhat+coxet, MRKEY = {1104786},
AUTHOR = {Dyer, Matthew},
TITLE = {On the ``{B}ruhat graph'' of a {C}oxeter system},
JOURNAL = {Compositio Math.},
FJOURNAL = {Compositio Mathematica},
VOLUME = {78},
YEAR = {1991},
NUMBER = {2},
PAGES = {185--191},
ISSN = {0010-437X},
CODEN = {CMPMAF},
MRCLASS = {20F55 (06F15 20F05)},
MRNUMBER = {92c:20076},
ZBLNUMBER = {0784.20019},
MRREVIEWER = {Robert B{é}dard},
URL = {http://www.numdam.org/item?id=CM_1991__78_2_185_0},
} -
[goresky83::inter]
M. Goresky and R. MacPherson, "Intersection homology, II," Invent. Math., vol. 72, iss. 1, pp. 77-129, 1983.
@article{goresky83::inter, MRKEY = {696691},
AUTHOR = {Goresky, Mark and MacPherson, Robert},
TITLE = {Intersection homology, {II}},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {72},
YEAR = {1983},
NUMBER = {1},
PAGES = {77--129},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {57N80 (14F10 32C37 55N30)},
MRNUMBER = {84i:57012},
ZBLNUMBER = {0529.55007},
MRREVIEWER = {Laurence R. Taylor},
DOI = {10.1007/BF01389130},
} -
[goresky81::kazhd+luszt]
M. Goresky, "Kazhdan-Lusztig polynomials for classical groups," Northeastern University, technical report , 1981.
@techreport{goresky81::kazhd+luszt,
author = {Goresky, M.},
TITLE = {Kazhdan-{L}usztig polynomials for classical groups},
INSTITUTION = {Northeastern University},
TYPE = {technical report},
YEAR = {1981},
URL = {http://www.math.ias.edu/~goresky/tables.html},
} -
[hausel02::toric] T. Hausel and B. Sturmfels, "Toric hyperKähler varieties," Doc. Math., vol. 7, pp. 495-534, 2002.
@article{hausel02::toric, MRKEY = {2015052},
AUTHOR = {Hausel, Tam{á}s and Sturmfels, Bernd},
TITLE = {Toric hyper{K}ähler varieties},
JOURNAL = {Doc. Math.},
FJOURNAL = {Documenta Mathematica},
VOLUME = {7},
YEAR = {2002},
PAGES = {495--534},
ISSN = {1431-0635},
MRCLASS = {53C26 (14M25 53D20)},
MRNUMBER = {2004i:53054},
ZBLNUMBER = {1029.53054},
MRREVIEWER = {Andrew Dancer},
} -
[humphreys90::reflec+coxet] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge: Cambridge Univ. Press, 1990.
@book{humphreys90::reflec+coxet, MRKEY = {1066460},
AUTHOR = {Humphreys, James E.},
TITLE = {Reflection Groups and {C}oxeter Groups},
SERIES = {Cambridge Stud. Adv. Math.},
NUMBER = {29},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1990},
PAGES = {xii+204},
ISBN = {0-521-37510-X},
MRCLASS = {20-02 (20F32 20F55 20G15 20H15)},
MRNUMBER = {92h:20002},
ZBLNUMBER = {0725.20028},
MRREVIEWER = {Louis Solomon},
} -
[hultman03::bruhat+weyl]
A. Hultman, "Bruhat intervals of length 4 in Weyl groups," J. Combin. Theory Ser. A, vol. 102, iss. 1, pp. 163-178, 2003.
@article{hultman03::bruhat+weyl, MRKEY = {1970983},
AUTHOR = {Hultman, Axel},
TITLE = {Bruhat intervals of length 4 in {W}eyl groups},
JOURNAL = {J. Combin. Theory Ser. A},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {102},
YEAR = {2003},
NUMBER = {1},
PAGES = {163--178},
ISSN = {0097-3165},
CODEN = {JCBTA7},
MRCLASS = {20F55 (05E15)},
MRNUMBER = {2004c:20066},
ZBLNUMBER = {1060.20036},
MRREVIEWER = {Sait Halı cı o{\u{g}}lu},
DOI = {10.1016/S0097-3165(03)00027-X},
} -
[kazhdan79::repres+coxet+hecke]
D. Kazhdan and G. Lusztig, "Representations of Coxeter groups and Hecke algebras," Invent. Math., vol. 53, iss. 2, pp. 165-184, 1979.
@article{kazhdan79::repres+coxet+hecke, MRKEY = {560412},
AUTHOR = {Kazhdan, David and Lusztig, George},
TITLE = {Representations of {C}oxeter groups and {H}ecke algebras},
JOURNAL = {Invent. Math.},
FJOURNAL = {Inventiones Mathematicae},
VOLUME = {53},
YEAR = {1979},
NUMBER = {2},
PAGES = {165--184},
ISSN = {0020-9910},
CODEN = {INVMBH},
MRCLASS = {20H15 (17B35 20G05 22E47)},
MRNUMBER = {81j:20066},
ZBLNUMBER = {0499.20035},
MRREVIEWER = {Vinay V. Deodhar},
DOI = {10.1007/BF01390031},
} -
[kac83::infin+lie] V. G. Kac, Infinite-Dimensional Lie Algebras: An Introduction, Boston: Birkhäuser, 1983.
@book{kac83::infin+lie, MRKEY = {739850},
AUTHOR = {Kac, Victor G.},
TITLE = {Infinite-Dimensional {L}ie Algebras: {A}n Introduction},
SERIES = {Progr. Math.},
NUMBER = {44},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston},
YEAR = {1983},
PAGES = {xvi+245},
ISBN = {0-8176-3118-6},
MRCLASS = {17B65 (17B10 17B67)},
MRNUMBER = {86h:17015},
ZBLNUMBER = {0537.17001},
MRREVIEWER = {Joseph C. Ferrar},
} -
[kumar02::kac+moody] S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Boston: Birkhäuser, 2002.
@book{kumar02::kac+moody, MRKEY = {1923198},
AUTHOR = {Kumar, Shrawan},
TITLE = {Kac-{M}oody Groups, their Flag Varieties and Representation Theory},
SERIES = {Progr. Math.},
NUMBER = {204},
PUBLISHER = {Birkhäuser},
ADDRESS = {Boston},
YEAR = {2002},
PAGES = {xvi+606},
ISBN = {0-8176-4227-7},
MRCLASS = {22E46 (14M15 17B67 22E65)},
MRNUMBER = {2003k:22022},
ZBLNUMBER = {1026.17030},
MRREVIEWER = {Guy Rousseau},
} -
[SGA4:3] Théorie des Topos et Cohomologie Étale des Schémas, III, Artin, M., Grothendieck, A., Verdier, J. -L., Deligne, P., and Saint-Donat, B., Eds., New York: Springer-Verlag, 1973.
@book{SGA4:3, MRKEY = {0354654},
EDITOR = {Artin, M. and Grothendieck, A. and Verdier, J.-L. and Deligne, P. and Saint-Donat, B.},
TITLE = {Théorie des Topos et Cohomologie Étale des Schémas, {\rm III}},
SERIES = {Lecture Notes in Math.},
NUMBER = {305},
INVNOTE = {Séminaire de Géométrie Algébrique du Bois-Marie 1963--1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de P. Deligne et B. Saint-Donat},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1973},
PAGES = {vi+640},
MRCLASS = {14-06},
MRNUMBER = {50 \#7132},
ZBLNUMBER = {0245.00002},
} -
[saito89::introd+hodge] M. Saito, "Introduction to mixed Hodge modules, Actes du Colloque de Théorie de Hodge, (Luminy, 1987)," Astérisque, pp. 179-180, 1989.
@article{saito89::introd+hodge, MRKEY = {1042805},
AUTHOR = {Saito, Morihiko},
TITLE = {Introduction to mixed {H}odge modules, {\it Actes du Colloque de Théorie de Hodge},
(Luminy, 1987)},
JOURNAL = {Astérisque},
FJOURNAL = {Astérisque},
PAGES = {179-180},
YEAR = {1989},
ISSN = {0303-1179},
MRCLASS = {32S35 (14C05 14C30 32J25)},
MRNUMBER = {91j:32041},
ZBLNUMBER = {0753.32004},
} -
[slodowy86::oen+schub+kac+moody+lie] P. Slodowy, "On the geometry of Schubert varieties attached to Kac-Moody Lie algebras," in Proc. of the 1984 Vancouver Conference in Algebraic Geometry, Carrell, J., Geramita, A. V., and Russell, P., Eds., Providence, RI: Amer. Math. Soc., 1986, pp. 405-442.
@incollection{slodowy86::oen+schub+kac+moody+lie, MRKEY = {846033},
AUTHOR = {Slodowy, Peter},
EDITOR = {Carrell, J. and Geramita, A. V. and Russell, P.},
TITLE = {On the geometry of {S}chubert varieties attached to {K}ac-{M}oody {L}ie algebras},
BOOKTITLE = {Proc. of the 1984 Vancouver Conference in Algebraic Geometry},
SERIES = {{\rm CMS} Conf. Proc.},
NUMBER = {6},
PAGES = {405--442},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {1986},
MRCLASS = {14M15 (17B67 22E65)},
MRNUMBER = {87i:14043},
ZBLNUMBER = {0591.14038},
MRREVIEWER = {Eugene Gutkin},
} -
[springer84] T. A. Springer, "A purity result for fixed point varieties in flag manifolds," J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 31, iss. 2, pp. 271-282, 1984.
@article{springer84, MRKEY = {763421},
AUTHOR = {Springer, T. A.},
TITLE = {A purity result for fixed point varieties in flag manifolds},
JOURNAL = {J. Fac. Sci. Univ. Tokyo Sect. {\rm IA} Math.},
FJOURNAL = {Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics},
VOLUME = {31},
YEAR = {1984},
NUMBER = {2},
PAGES = {271--282},
ISSN = {0040-8980},
CODEN = {JFTMAT},
MRCLASS = {14L30 (14L10 20G05)},
MRNUMBER = {86c:14034},
ZBLNUMBER = {0581.20048},
MRREVIEWER = {Klaus Pommerening},
} -
[stanley78::hilber]
R. P. Stanley, "Hilbert functions of graded algebras," Advances in Math., vol. 28, iss. 1, pp. 57-83, 1978.
@article{stanley78::hilber, MRKEY = {0485835},
AUTHOR = {Stanley, Richard P.},
TITLE = {Hilbert functions of graded algebras},
JOURNAL = {Advances in Math.},
FJOURNAL = {Advances in Mathematics},
VOLUME = {28},
YEAR = {1978},
NUMBER = {1},
PAGES = {57--83},
ISSN = {0001-8708},
MRCLASS = {13D10 (13H10)},
MRNUMBER = {58 \#5637},
ZBLNUMBER = {0384.13012},
MRREVIEWER = {Idun Reiten},
DOI = {10.1016/0001-8708(78)90045-2},
} -
[stanley80::weyl+lefsc+spern]
R. P. Stanley, "Weyl groups, the hard Lefschetz theorem, and the Sperner property," SIAM J. on Algebraic Discrete Methods, vol. 1, iss. 2, pp. 168-184, 1980.
@article{stanley80::weyl+lefsc+spern, MRKEY = {578321},
AUTHOR = {Stanley, Richard P.},
TITLE = {Weyl groups, the hard {L}efschetz theorem, and the {S}perner property},
JOURNAL = {{\rm SIAM} J. on Algebraic Discrete Methods},
FJOURNAL = {Society for Industrial and Applied Mathematics. Journal on Algebraic and Discrete Methods},
VOLUME = {1},
YEAR = {1980},
NUMBER = {2},
PAGES = {168--184},
ISSN = {0196-5212},
CODEN = {SJAMDU},
MRCLASS = {20G05 (05A05 06A10 14M17)},
MRNUMBER = {82j:20083},
ZBLNUMBER = {0502.05004},
DOI = {10.1137/0601021},
} -
[stanton90::uenim]
D. Stanton, "Unimodality and Young’s lattice," J. Combin. Theory Ser. A, vol. 54, iss. 1, pp. 41-53, 1990.
@article{stanton90::uenim, MRKEY = {1051777},
AUTHOR = {Stanton, Dennis},
TITLE = {Unimodality and {Y}oung's lattice},
JOURNAL = {J. Combin. Theory Ser. {\rm A}},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {54},
YEAR = {1990},
NUMBER = {1},
PAGES = {41--53},
ISSN = {0097-3165},
CODEN = {JCBTA7},
MRCLASS = {05A17 (05A30 06A07)},
MRNUMBER = {91d:05009},
ZBLNUMBER = {0736.05009},
MRREVIEWER = {E. Rodney Canfield},
DOI = {10.1016/0097-3165(90)90004-G},
} -
[swartz06::cohen+macaul]
E. Swartz, "$g$-elements, finite buildings and higher Cohen-Macaulay connectivity," J. Combin. Theory Ser. A, vol. 113, iss. 7, pp. 1305-1320, 2006.
@article{swartz06::cohen+macaul, MRKEY = {2259062},
AUTHOR = {Swartz, Ed},
TITLE = {{$g$}-elements, finite buildings and higher {C}ohen-{M}acaulay connectivity},
JOURNAL = {J. Combin. Theory Ser. {\rm A}},
FJOURNAL = {Journal of Combinatorial Theory. Series A},
VOLUME = {113},
YEAR = {2006},
NUMBER = {7},
PAGES = {1305--1320},
ISSN = {0097-3165},
CODEN = {JCBTA7},
MRCLASS = {05E25 (13C14 51D20 51E24)},
MRNUMBER = {2007i:05192},
ZBLNUMBER = {1102.13025},
MRREVIEWER = {Dmitrii V. Pasechnik},
DOI = {10.1016/j.jcta.2005.11.007},
} -
[weber::pure]
A. Weber, "Pure homology of algebraic varieties," Topology, vol. 43, iss. 3, pp. 635-644, 2004.
@article{weber::pure, MRKEY = {2041634},
AUTHOR = {Weber, Andrzej},
TITLE = {Pure homology of algebraic varieties},
JOURNAL = {Topology},
FJOURNAL = {Topology. An International Journal of Mathematics},
VOLUME = {43},
YEAR = {2004},
NUMBER = {3},
PAGES = {635--644},
ISSN = {0040-9383},
CODEN = {TPLGAF},
MRCLASS = {14F43 (14C30 55N33)},
MRNUMBER = {2004m:14036},
ZBLNUMBER = {1072.14023},
MRREVIEWER = {Joost van Hamel},
DOI = {10.1016/j.top.2003.09.001},
}