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Conjugacy Decomposition of Canonical and Dual Canonical Monoids

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Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics

Part of the book series: Fields Institute Communications ((FIC,volume 71))

Abstract

Putcha’s theory of conjugacy classes in a reductive monoid culminates in a decomposition of the monoid in terms of these classes, which we call the conjugacy decomposition. With this decomposition, we have a partially ordered set, with partial order analogous to the Bruhat-Chevalley order for the Bruhat-Renner Decomposition of a reductive monoid. We outline the development of the conjugacy decomposition, paying attention to the cases of canonical and dual canonical monoids. These monoids appear in the literature as \(\mathcal{J}\)-irreducible and \(\mathcal{J}\)-coirreducible, respectively, of type ∅. We conclude with a summary of new results, describing the order between classes in the conjugacy decomposition for canonical and dual canonical monoids.

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References

  1. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)

    Google Scholar 

  2. Can, M.B., Renner, L.E.: Bruhat-Chevalley order on the rook monoid. Turk. J. Math. 36, 499–519 (2012)

    MATH  MathSciNet  Google Scholar 

  3. Chevalley, C.: Sur les décompositions cellulaires des espaces G/B, Proc. Symp. Pure Math. Amer. Math. Soc. 56, 1–23 (1991)

    Google Scholar 

  4. Clifford, A.H., Preston, G.B.: Algebraic Theory of Semigroups, vol. 1. American Mathematical Society, Providence (1961)

    Book  MATH  Google Scholar 

  5. De Concini, C., Procesi, C.: Complete symmetric varieties. In: Invariant Theory. Lecture Notes in Mathematics, vol. 131, pp. 1–44. Springer, Berlin/Heidelberg (1983)

    Google Scholar 

  6. Green, J.A.: On the structure of semigroups. Ann. Math. 54, 163–172 (1951)

    Article  MATH  Google Scholar 

  7. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge/New York (1990)

    Book  MATH  Google Scholar 

  8. Pennell, E.A., Putcha, M.S., Renner, L.E.: Analogue of the Bruhat-Chevalley order for reductive monoids. J. Algebra 196, 339–368 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  9. Putcha, M.S.: A semigroup approach to linear algebraic groups. J. Algebra 80, 164–185 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  10. Putcha, M.S.: Regular linear algebraic monoids. Trans. Am. Math. Soc. 290, 615–626 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  11. Putcha, M.S.: Conjugacy classes in algebraic monoids. Trans. Am. Math. Soc. 303, 529–540 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Putcha, M.S.: Linear Algebraic Monoids. London Mathematical Society Lecture Note Series, vol. 133. Cambridge University Press, Cambridge/New York (1988)

    Google Scholar 

  13. Putcha, M.S.: Conjugacy classes in algebraic monoids II. Can. J. Math. 46, 648–661 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Putcha, M.S.: Conjugacy classes and nilpotent variety of a reductive monoid. Can. J. Math. 50, 829–844 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  15. Putcha, M.S.: Shellability in reductive monoids. Trans. Am. Math. Soc. 354, 413–426 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Putcha, M.S.: Bruhat-Chevalley order in reductive monoids. J. Algebr. Comb. 20, 33–53 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Putcha, M.S.: Conjugacy decomposition of reductive monoids. Math. Z. 250, 841–853 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Putcha, M.S.: Nilpotent variety of a reductive monoid. J. Alg. Comb. 27, 275–292 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  19. Putcha, M.S., Renner, L.E.: The canonical compactification of a finite group of Lie type. Trans. Am. Math. Soc. 337, 305–319 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Renner, L.E.: Analogue of the Bruhat decomposition for algebraic monoids. J. Algebra 101, 303–338 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  21. Renner, L.E.: Linear Algebraic Monoids. Encyclopaedia of Mathematical Sciences, vol. 134. Springer, Berlin/New York (2005)

    Google Scholar 

  22. Solomon, L.: An introduction to reductive monoids. In: Fountain, J. (ed.) Semigroups, Formal Languages and Groups. Kluwer Academic, Dordrecht/Boston (1995)

    Google Scholar 

  23. Taylor, D.T.: Bruhat-Chevalley order on \(\mathcal{J}\)-classes of reductive monoids. Semigroup Forum 78, 7–13 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Therkelsen, R.K.: The conjugacy poset of a reductive monoid. Ph.D. Thesis, North Carolina State University, Raleigh (2010)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Bellarmine University, Louisville, KY, 40205, USA

    Ryan K. Therkelsen

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  1. Ryan K. Therkelsen
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Correspondence to Ryan K. Therkelsen .

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Editors and Affiliations

  1. Department of Mathematics, Tulane University, New Orleans, Louisiana, USA

    Mahir Can

  2. Department of Mathematical Sciences, University of South Carolina, Aiken, South Carolina, USA

    Zhenheng Li

  3. Department of Mathematics, City College of New York, New York, New York, USA

    Benjamin Steinberg

  4. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada

    Qiang Wang

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Therkelsen, R.K. (2014). Conjugacy Decomposition of Canonical and Dual Canonical Monoids. In: Can, M., Li, Z., Steinberg, B., Wang, Q. (eds) Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics. Fields Institute Communications, vol 71. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0938-4_8

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