Abstract
Let G=GL n (F q ) be the finite general linear group and let M=M n (F q ) be the monoid of all n×n matrices over F q . Let B be a Borel subgroup of G, let W be the subgroup of permutation matrices, and let ℛ⊃W be the monoid of all zero-one matrices which have at most one non-zero entry in each row and each column. The monoid ℛ plays the same role for M that the Weyl group W does for G. In particular there is a length function on ℛ which extends the length function on W and a C-algebra H C (M, B) which includes Iwahori's ‘Hecke algebra’ H C (G, B) and shares many of its properties.
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References
Borel, A. and Tits, J., ‘Groupes réductifs’, Publ. Math. I.H.E.S. 27 (1965), 55–151.
Bourbaki, N., Groupes et Algèbres de Lie, Chapitres IV, V, VI, Hermann, Paris, 1968.
Bruhat, F., ‘Représentations Induites des Groupes de Lie Semi-Simples Connexes’, C.R. Acad. Sci. Paris 238 (1954), 437–439.
Carter, R. W., Simple Groups of Lie Type, Wiley, Interscience, 1972.
Carter, R. W., Finite Groups of Lie Type — Conjugacy Classes and Complex Characters, Wiley, Interscience, 1985.
Chevalley, C., ‘Sur certains groupes simples’, Tôhoku Math. J. 7 (1955), 14–66.
Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Vol. I, Math. Surveys 7, American Math. Soc., 1961.
Curtis, C. W. and Reiner, I., Methods of Representation Theory, with Applications to Finite Groups and Orders, Vol. II, Wiley, 1987.
Grigor'ev, D. Ju., ‘An analogue of the Bruhat decomposition for the closure of the cone of a Chevalley group of the classical series’, Soviet Math. Dokl. 23 (1981), 393–397.
Iwahori, N., ‘On the structure of a Hecke ring of a Chevalley group over a finite field’, J. Fac. Sci. Univ. Tokyo, Sec. I, 10 (1964), 215–236.
Iwahori, N., ‘Generalized Tits system (Bruhat decomposition) on p-adic semisimple groups’, in Proc. Symp. Pure Math. 9 (1966), Algebraic Groups and Discontinuous Subgroups, pp. 71–83.
Iwahori, N., ‘On some properties of groups with BN-pairs’, in Theory of Finite Groups, a Symposium (eds R. Brauer and C.-H. Sah), W. A. Benjamin, 1969, pp. 203–212.
Lehrer, G. I., ‘A survey of Hecke algebras and the Artin braid groups’, Contemp. Math. 74 (1988), 365–385.
Macdonald, I. G., Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
Munn, W. D., ‘Matrix representations of semigroups’, Proc. Camb. Phil. Soc. 53 (1957), 5–12.
Munn, W. D., ‘The characters of the symmetric inverse semigroup’, Proc. Camb. Phil. Soc. 53 (1957), 13–18.
Putcha, M. S., ‘Linear algebraic monoids’, London Math. Soc. Lecture Notes 133, Cambridge Univ. Press, 1988.
Putcha, M. S., ‘Monoids on groups with BN-pairs’, J. Algebra 120 (1989), 139–169.
Renner, L., ‘Classification of semisimple algebraic monoids’, Trans. Amer. Math. Soc. 292 (1985), 193–223.
Renner, L., ‘Analogue of the Bruhat decomposition for algebraic monoids’, J. Algebra 101 (1986), 303–338.
Rodrigues, O., ‘Note sur les inversions, ou dérangements produits dans les permutations’, J. de Math., 1 Série, 4 (1839), 236–239.
Siegel, E., ‘On the representations of a Hecke ring of the affine group over a finite field’, thesis, Univ. of Wisconsin, Madison, 1990.
Solomon, L., ‘The orders of the finite Chevalley groups’, J. Algebra 3 (1966), 376–393.
Solomon, L., ‘The affine group I. Bruhat decomposition’, J. Algebra 20 (1972), 512–539.
Steinberg, R., ‘Lectures on Chevalley groups’, Yale University, 1967, mimeographed notes.
Tits, J., ‘Théorème de Bruhat et sous-groupes paraboliques’, C.R. Acad. Sci. Paris 254 (1962), 2910–2912.
Tits, J., ‘Algebraic and abstract simple groups’, Ann. of Math. 80 (1964), 313–329.
Waterhouse, W. C., ‘The unit groups of affine algebraic monoids’, Proc. Amer. Math. Soc. 85 (1982), 506–508.
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For Jacques Tits on his sixtieth birthday
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Solomon, L. The Bruhat decomposition, Tits system and Iwahori ring for the monoid of matrices over a finite field. Geom Dedicata 36, 15–49 (1990). https://doi.org/10.1007/BF00181463
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DOI: https://doi.org/10.1007/BF00181463