Nilpotent variety of a reductive monoid
Mohan S. Putcha
North Carolina State University Department of Mathematics Box 8205 Raleigh NC 27695-8205 USA
DOI: 10.1007/s10801-007-0087-y
Abstract
In this paper we study the variety M nil of nilpotent elements of a reductive monoid M. In general this variety has a completely different structure than the variety G uni of unipotent elements of the unit group G of M. When M has a unique non-trivial minimal or maximal G\times G-orbit, we find a precise description of the irreducible components of M nil via the combinatorics of the Renner monoid of M and the Weyl group of G. In particular for a semisimple monoid M, we find necessary and sufficient conditions for the variety M nil to be irreducible.
Pages: 275–292
Keywords: keywords reductive monoid; nilpotent variety; unipotent variety; Renner monoid; Weyl group
Full Text: PDF
References
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2. Carter, R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. Wiley, New York (1985)
3. Chevalley, C.: Sur les décompositions cellulaires des espaces G/B. In: Algebraic Groups and Their Generalizations. Proc. Symp. Pure Math., vol. 56, pp. 1-23. Am. Math. Soc., Providence (1991)
4. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Univ. Press, Cambridge (1990)
5. Pennell, E.A., Putcha, M.S., Renner, L.E.: Analogue of the Bruhat-Chevalley order for reductive monoids. J. Algebra 196, 339-368 (1997)
6. Putcha, M.S.: A semigroup approach to linear algebraic groups. J. Algebra 80, 164-185 (1983)
7. Putcha, M.S.: Regular linear algebraic monoids. Trans. Am. Math. Soc. 290, 615-626 (1985)
8. Putcha, M.S.: Conjugacy classes in algebraic monoids. Trans. Am. Math. Soc. 303, 529-540 (1987)
9. Putcha, M.S.: Linear Algebraic Monoids. London Math. Soc. Lecture Note Series, vol.
133. Cambridge Univ. Press, Cambridge (1988)
10. Putcha, M.S.: Conjugacy classes and nilpotent variety of a reductive monoid. Can. J. Math. 50, 829- 844 (1998) J Algebr Comb (2008) 27: 275-292