Bruhat-Chevalley Order in Reductive Monoids
Mohan S. Putcha
DOI: 10.1023/B:JACO.0000047291.42015.a6
Abstract
Let M be a reductive monoid with unit group G. Let 
denote the idempotent cross-section of the G \times G-orbits on M. If W is the Weyl group of G and e, f 
with e 
f, we introduce a projection map from WeW to WfW. We use these projection maps to obtain a new description of the Bruhat-Chevalley order on the Renner monoid of M. For the canonical compactification X of a semisimple group G 0 with Borel subgroup B 0 of G 0, we show that the poset of B 0 \times B 0-orbits of X (with respect to Zariski closure inclusion) is Eulerian.
denote the idempotent cross-section of the G \times G-orbits on M. If W is the Weyl group of G and e, f with e f, we introduce a projection map from WeW to WfW. We use these projection maps to obtain a new description of the Bruhat-Chevalley order on the Renner monoid of M. For the canonical compactification X of a semisimple group G 0 with Borel subgroup B 0 of G 0, we show that the poset of B 0 \times B 0-orbits of X (with respect to Zariski closure inclusion) is Eulerian.Pages: 33–53
Keywords: reductive monoid; Renner monoid; Bruhat-Chevalley order; projections
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References
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4. E.A. Pennell, M.S. Putcha, and L.E. Renner, “Analogue of the Bruhat-Chevalley order for reductive monoids,” J. Algebra 196 (1997), 339-368.
5. M.S. Putcha, “A semigroup approach to linear algebraic groups,” J. Algebra 80 (1983), 164-185.
6. M.S. Putcha, Linear Algebraic Monoids, London Math. Soc. Lecture Note Series 133, Cambridge Univ. Press, 1988.
7. M.S. Putcha, “Shellability in reductive monoids,” Trans. Amer. Math. Soc. 354 (2001), 413-426.
8. M.S. Putcha and L.E. Renner, “The system of idempotents and the lattice of J -classes of reductive algebraic monoids,” J. Algebra 116 (1988), 385-399.
9. M.S. Putcha and L.E. Renner, “The canonical compactification of a finite group of Lie type,” Trans. Amer. Math. Soc. 337 (1993), 305-319.
10. L.E. Renner, “Analogue of the Bruhat decomposition for reductive algebraic monoids,” J. Algebra 101 (1986), 303-338.
11. L.E. Renner, “Analogue of the Bruhat decomposition for reductive algebraic monoids II. The length function and trichotomy,” J. Algebra 175 (1995), 695-714.
12. L.E. Renner, “Classification of simisimple varieties,” J. Algebra 122 (1989), 275-287.
13. L.E. Renner, “An explicit cell decomposition of the canonical compactification of an algebraic group,” Can. Math. Bull 46 (2003), 140-148.
14. L. Solomon, “An introduction to reductive monoids,” in Semigroups, Formal Languages and Groups (J. Fountain, ed.), Kluwer (1995) 293-352.
15. T.A. Springer, “Intersection cohomology of B \times B orbit closures in group compactifications,” J. Algebra 258 (2002), 71-111.
16. R.P. Stanley, “Some aspects of groups acting on finite posets,” J. Comb. Theory A 32 (1982), 132-161.
17. R.P. Stanley, Enumerative Combinatorics vol. 1, Cambridge Studies in Advanced Math 49, Cambridge University Press, 1997.
18. D.-N. Verma, “M\ddot obius inversion for the Bruhat ordering on a Weyl group,” Ann. Sci. École Norm. Sup. 4 (1971), 393-398.
19. E.B. Vinberg, “ On reductive algebraic semigroups,” Amer. Math. Soc. Transl. Series 2 169 (1994), 145-182.
2. C. DeConcini, “Equivariant embeddings of homogenous spaces,” Proc. Internat. Congress of Mathematicians (1986), 369-377.
3. V.V. Deodhar, “A splitting criterion for the Bruhat orderings on Coxeter groups,” Comm. Algebra 15 (1987), 1889-1894.
4. E.A. Pennell, M.S. Putcha, and L.E. Renner, “Analogue of the Bruhat-Chevalley order for reductive monoids,” J. Algebra 196 (1997), 339-368.
5. M.S. Putcha, “A semigroup approach to linear algebraic groups,” J. Algebra 80 (1983), 164-185.
6. M.S. Putcha, Linear Algebraic Monoids, London Math. Soc. Lecture Note Series 133, Cambridge Univ. Press, 1988.
7. M.S. Putcha, “Shellability in reductive monoids,” Trans. Amer. Math. Soc. 354 (2001), 413-426.
8. M.S. Putcha and L.E. Renner, “The system of idempotents and the lattice of J -classes of reductive algebraic monoids,” J. Algebra 116 (1988), 385-399.
9. M.S. Putcha and L.E. Renner, “The canonical compactification of a finite group of Lie type,” Trans. Amer. Math. Soc. 337 (1993), 305-319.
10. L.E. Renner, “Analogue of the Bruhat decomposition for reductive algebraic monoids,” J. Algebra 101 (1986), 303-338.
11. L.E. Renner, “Analogue of the Bruhat decomposition for reductive algebraic monoids II. The length function and trichotomy,” J. Algebra 175 (1995), 695-714.
12. L.E. Renner, “Classification of simisimple varieties,” J. Algebra 122 (1989), 275-287.
13. L.E. Renner, “An explicit cell decomposition of the canonical compactification of an algebraic group,” Can. Math. Bull 46 (2003), 140-148.
14. L. Solomon, “An introduction to reductive monoids,” in Semigroups, Formal Languages and Groups (J. Fountain, ed.), Kluwer (1995) 293-352.
15. T.A. Springer, “Intersection cohomology of B \times B orbit closures in group compactifications,” J. Algebra 258 (2002), 71-111.
16. R.P. Stanley, “Some aspects of groups acting on finite posets,” J. Comb. Theory A 32 (1982), 132-161.
17. R.P. Stanley, Enumerative Combinatorics vol. 1, Cambridge Studies in Advanced Math 49, Cambridge University Press, 1997.
18. D.-N. Verma, “M\ddot obius inversion for the Bruhat ordering on a Weyl group,” Ann. Sci. École Norm. Sup. 4 (1971), 393-398.
19. E.B. Vinberg, “ On reductive algebraic semigroups,” Amer. Math. Soc. Transl. Series 2 169 (1994), 145-182.