Betti Numbers of Toric Varieties and Eulerian Polynomials

REFERENCES

• Bialynicki-Birula , A. ( 1973 ). Some theorems on actions of algebraic groups . Ann. Math. 98 : 480 – 497 .
   Web of Science ®Google Scholar
• Bjorner , A. , Brenti , F. ( 2005 ). Combinatorics of Coxeter Groups . Graduate Texts in Mathematics Vol. 231. New York : Springer Verlag .
   Google Scholar
• Brenti , F. ( 1994 ). q-Eulerian polynomials arising from Coxeter groups . European J. Combin. 15 : 417 – 441 .
   Web of Science ®Google Scholar
• Brion , M. ( 2008 ). Local structure of algebraic monoids . Moscow Math. J. 8 : 647 – 666 .
   Web of Science ®Google Scholar
• Brion , M. ( 1999 ). Rational smoothness and fixed points of torus actions . Transformation Groups 4 : 127 – 156 .
   Web of Science ®Google Scholar
• Cohen , A. M. ( 2008 ). Eulerian polynomials of spherical type . Münster J. Math. 1 : 1 – 8 .
   Google Scholar
• Danilov , V. I. ( 1978 ). The Geometry of Toric Varieties . Russian Math. Surveys 33 : 97 – 154 .
   Google Scholar
• Dolgachev , I. , Lunts , V. ( 1994 ). A character formula for the representation of a Weyl group on the cohomology of the associated toric variety . J. Algebra 168 : 741 – 772 .
   Web of Science ®Google Scholar
• Fulton , W. ( 1980 ). Introduction to toric varieties. Ann. Math. 131 .
   Google Scholar
• Hirzebruch , F. ( 2008 ). Eulerian polynomial . Münster J. Math. 1 : 9 – 14 .
   Google Scholar
• Li , Z. ( 2007 ). Poincaré polynomial of certain toric varieties. Unpublished Manuscript.
   Google Scholar
• Kempf , G. , Knudsen , F. , Mumford , D. , Saint-Donat , B. ( 1973 ). Toroidal Embeddings. Lecture Notes in Mathematics, Vol. 339. Springer-Verlag .
   Google Scholar
• Procesi , C. ( 1990 ). The toric variety associated to Weyl chambers . In: Lothaire , M. , ed. “Mots” . Paris : Hermés , pp. 153 – 161 .
   Google Scholar
• Putcha , M. S. (1998). Linear Algebraic Monoids. LNM, Cambridge University Press.
   Google Scholar
• Putcha , M. S. ( 1989 ). Monoids on group with BN-pair . J. Algebra 120 : 139 – 169 .
   Web of Science ®Google Scholar
• Putcha , M. S. , Renner , L. E. ( 1988 ). The system of idempotents and lattice of 𝒥-classes of reductive algebraic monoids . J. Algebra 116 : 385 – 399 .
   Web of Science ®Google Scholar
• Renner , L. E. ( 1986 ). Analogue of the Bruhat decomposition for algebraic monoids . J. Algebra 101 : 303 – 338 .
   Web of Science ®Google Scholar
• Renner , L. E. ( 1989 ). Classification of semisimple varieties . J. Algebra 122 : 275 – 287 .
   Web of Science ®Google Scholar
• Renner , L. E. ( 2005 ). Linear Algebraic Monoids. Encyclopedia of Mathematical Sciences. Invariant Theory and Algebraic transformation Groups, Vol. 134. Springer-Verlag .
   Google Scholar
• Renner , L. E. ( 2008 ). The H-polynomial of a Semisimple Monoid . J. Algebra. 319 : 360 – 376 .
   Web of Science ®Google Scholar
• Renner , L. E. ( 2009 ). Descent Systems for Bruhat Posets. J. Algebr. Comb. 29:413–435. Online version http://www.math.uwo.ca/lex/cv/renner-files.html
   Google Scholar
• Renner , L. E. Weyl groups, descent systems and betti numbers. To appear in Rocky Mountain J. Math.
   Google Scholar
• Renner , L. E. ( 2009 ). Rationally Smooth Algebraic Monoids . Semigroup Forum 78 : 384 – 395 .
   Web of Science ®Google Scholar
• Renner , L. E. ( 2010 ). Canonical Cellular Decomposition for 𝒥-irreducible monoids . Comm. Algebra 38 : 181 – 198 .
   Web of Science ®Google Scholar
• Renner , L. E. ( 2008 ). Betti numbers and Permutations. Online manuscript. Available at http://www.math.uwo.ca/lex/cv/renner-files.html .
   Google Scholar
• Solomon , L. ( 1995 ). An introduction to reductive monoids. Semigroups, formal languages and groups. In: Fountain, J. ed. Kluwer Academic Publishers, pp. 295–352 .
   Google Scholar
• Stembridge , J. R. ( 1994 ). Some permutation representations of Weyl groups associated with the cohomology of toric varieties . Adv. Math. 106 : 244 – 301 .
   Web of Science ®Google Scholar
• Stanley , R. P. ( 1987 ). Generalized h-vectors, intersection cohomology of toric varieties, and related results . Advanced Studies in Pure Mathematics 11 : 187 – 213 .
   Google Scholar
• Stanley , R. P. ( 2004 ). Recent development in algebraic combinatorics . Israel J. Math 143 : 317 – 340 .
   Web of Science ®Google Scholar
• Stanley , R. P. ( 1997 ). Enumerative Combinatorics. Cambridge University Press, Vol. 1. Online edition available http://www.math.mit.edu/rstan/ec1.pdf
   Google Scholar
• Communicated by V. Gould.
   Google Scholar