Algebra & Number Theory

The biHecke monoid of a finite Coxeter group and its representations

Florent Hivert, Anne Schilling, and Nicolas Thiéry

Full-text: Open access

Abstract

For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each wW a combinatorial module Tw whose support is the interval [1,w]R in right weak order. This module yields an algebra, whose representation theory generalizes that of the Hecke group algebra, with the combinatorics of descents replaced by that of blocks and of the cutting poset.

Article information

Source
Algebra Number Theory, Volume 7, Number 3 (2013), 595-671.

Dates
Received: 8 June 2011
Revised: 20 February 2012
Accepted: 4 April 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513729967

Digital Object Identifier
doi:10.2140/ant.2013.7.595

Mathematical Reviews number (MathSciNet)
MR3095222

Zentralblatt MATH identifier
1305.20071

Subjects
Primary: 20M30: Representation of semigroups; actions of semigroups on sets 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]
Secondary: 06D75: Other generalizations of distributive lattices 16G99: None of the above, but in this section 20C08: Hecke algebras and their representations

Keywords
Coxeter groups Hecke algebras representation theory blocks of permutation matrices

Citation

Hivert, Florent; Schilling, Anne; Thiéry, Nicolas. The biHecke monoid of a finite Coxeter group and its representations. Algebra Number Theory 7 (2013), no. 3, 595--671. doi:10.2140/ant.2013.7.595. https://projecteuclid.org/euclid.ant/1513729967


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References

  • M. H. Albert and M. D. Atkinson, “Simple permutations and pattern restricted permutations”, Discrete Math. 300:1-3 (2005), 1–15.
  • M. H. Albert, M. D. Atkinson, and M. Klazar, “The enumeration of simple permutations”, J. Integer Seq. 6:4 (2003), article 03.4.4.
  • J. Almeida, S. Margolis, B. Steinberg, and M. Volkov, “Representation theory of finite semigroups, semigroup radicals and formal language theory”, Trans. Amer. Math. Soc. 361:3 (2009), 1429–1461.
  • D. J. Benson, Representations and cohomology, I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, 1991.
  • G. Birkhoff, “Rings of sets”, Duke Math. J. 3:3 (1937), 443–454.
  • A. Bj örner and F. Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics 231, Springer, New York, 2005.
  • A. Blass and B. E. Sagan, “Möbius functions of lattices”, Adv. Math. 127:1 (1997), 94–123.
  • K. S. Brown, “Semigroups, rings, and Markov chains”, J. Theoret. Probab. 13:3 (2000), 871–938.
  • R. W. Carter, “Representation theory of the $0$-Hecke algebra”, J. Algebra 104:1 (1986), 89–103.
  • A. H. Clifford, “Matrix representations of completely simple semigroups”, Amer. J. Math. 64 (1942), 327–342.
  • A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, vol. 1, Mathematical Surveys 7, American Mathematical Society, Providence, RI, 1961.
  • C. W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics 11, Interscience Publishers, New York, 1962.
  • T. Denton, “A combinatorial formula for orthogonal idempotents in the $0$-Hecke algebra of $S\sb N$”, pp. 701–711 in 22nd international conference on formal power series and algebraic combinatorics (San Francisco, CA, 2010), Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010.
  • T. Denton, “A combinatorial formula for orthogonal idempotents in the 0-Hecke algebra of the symmetric group”, Electron. J. Combin. 18:1 (2011), paper 28.
  • T. Denton, F. Hivert, A. Schilling, and N. M. Thiéry, “On the representation theory of finite $\mathcal J$-trivial monoids”, Sém. Lothar. Combin. 64 (2010/11), art. B64d.
  • J. Doyen, “Équipotence et unicité de systèmes générateurs minimaux dans certains monoï des”, Semigroup Forum 28:1-3 (1984), 341–346.
  • J. Doyen, “Quelques propriétés des systèmes générateurs minimaux des monoï des”, Semigroup Forum 42:3 (1991), 333–343.
  • G. Duchamp, F. Hivert, and J.-Y. Thibon, “Noncommutative symmetric functions, VI: Free quasi-symmetric functions and related algebras”, Internat. J. Algebra Comput. 12:5 (2002), 671–717.
  • P. H. Edelman, “Abstract convexity and meet-distributive lattices”, pp. 127–150 in Combinatorics and ordered sets (Arcata, CA, 1985), edited by I. Rival, Contemp. Math. 57, American Mathematical Society, Providence, RI, 1986.
  • O. Ganyushkin, V. Mazorchuk, and B. Steinberg, “On the irreducible representations of a finite semigroup”, Proc. Amer. Math. Soc. 137:11 (2009), 3585–3592. http://www.ams.org/mathscinet-getitem?mr=2010h:20150MR 2010h:20150
  • P. Gaucher, “Combinatorics of labelling in higher-dimensional automata”, Theoret. Comput. Sci. 411:11-13 (2010), 1452–1483.
  • I. M. Gessel, “Multipartite $P$-partitions and inner products of skew Schur functions”, pp. 289–317 in Combinatorics and algebra (Boulder, CO, 1983), edited by C. Greene, Contemp. Math. 34, American Mathematical Society, Providence, RI, 1984.
  • J. A. Green, “On the structure of semigroups”, Ann. of Math. $(2)$ 54 (1951), 163–172.
  • F. Hivert and N. M. Thiéry, “The Hecke group algebra of a Coxeter group and its representation theory”, J. Algebra 321:8 (2009), 2230–2258.
  • F. Hivert, A. Schilling, and N. M. Thiéry, “Hecke group algebras as quotients of affine Hecke algebras at level 0”, J. Combin. Theory Ser. A 116:4 (2009), 844–863.
  • F. Hivert, A. Schilling, and N. M. Thiéry, “The biHecke monoid of a finite Coxeter group”, pp. 307–318 in 22nd international conference on formal power series and algebraic combinatorics (San Francisco, CA, 2010), Discrete Math. Theor. Comput. Sci. Proc., Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010.
  • J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, 1990.
  • D. Krob and J.-Y. Thibon, “Noncommutative symmetric functions, IV: Quantum linear groups and Hecke algebras at $q=0$”, J. Algebraic Combin. 6:4 (1997), 339–376.
  • G. Lallement and M. Petrich, “Irreducible matrix representations of finite semigroups”, Trans. Amer. Math. Soc. 139 (1969), 393–412.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995.
  • L. Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs 6, American Mathematical Society, Providence, RI, 2001. ergencystretch=15pt
  • S. Margolis and B. Steinberg, “The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups”, Algebr. Represent. Theory 14:1 (2011), 131–159.
  • P. N. Norton, “$0$-Hecke algebras”, J. Austral. Math. Soc. Ser. A 27:3 (1979), 337–357.
  • A. Nozaki, M. Miyakawa, G. Pogosyan, and I. G. Rosenberg, “The number of orthogonal permutations”, European J. Combin. 16:1 (1995), 71–85. http://www.zentralblatt-math.org/zmath/en/search/?an=0828.05004Zbl 0828.05004
  • OEIS Foundation, The on-line encyclopedia of integer sequences, 2012, http://oeis.org.
  • J.-E. Pin, “Mathematical foundations of automata theory”, course notes, Laboratoire d'Informatique Algorithmique: Fondements et Applications, 2012, http://www.liafa.jussieu.fr/~jep/PDF/MPRI/MPRI.pdf. ergencystretch=15pt
  • C. Le Conte de Poly-Barbut, “Sur les treillis de Coxeter finis”, Math. Inform. Sci. Humaines 125 (1994), 41–57.
  • M. Pouzet, “Théorie de l'ordre: une introduction”, book, to appear on arXiv, 2013.
  • D. Rees, “On semi-groups”, Proc. Cambridge Philos. Soc. 36 (1940), 387–400. http://www.ams.org/mathscinet-getitem?mr=2,127gMR 2,127g
  • J. Rhodes and Y. Zalcstein, “Elementary representation and character theory of finite semigroups and its application”, pp. 334–367 in Monoids and semigroups with applications (Berkeley, CA, 1989), edited by J. Rhodes, World Sci. Publ., River Edge, NJ, 1991.
  • G.-C. Rota, “On the foundations of combinatorial theory, I: Theory of Möbius functions”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368. http://www.zentralblatt-math.org/zmath/en/search/?an=0121.02406Zbl 0121.02406
  • Sage-Combinat community, “Sage-Combinat”, software project, 2008, http://combinat.sagemath.org.
  • F. V. Saliola, “The quiver of the semigroup algebra of a left regular band”, Internat. J. Algebra Comput. 17:8 (2007), 1593–1610.
  • J.-P. Serre, Linear representations of finite groups, Graduate Texts in Mathematics 42, Springer, New York, 1977.
  • R. P. Stanley, Enumerative combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 1997.
  • W. Stein et al., “Sage mathematics software (version 3.3)”, 2009, http://www.sagemath.org. ergencystretch=15pt
  • N. Thiéry, “Cartan invariant matrices for finite monoids”, pp. 887–898 in 24th International Conference on Formal Power Series and Algebraic Combinatorics (Nagoya, 2012), edited by N. Broutin and L. Devroye, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012.
  • “Wikipedia, the free encyclopedia”, web page, Wikipedia, 2010, http://en.wikipedia.org.
  • Y. Zalcstein, “Studies in the representation theory of finite semigroups”, Trans. Amer. Math. Soc. 161 (1971), 71–87.
  • A. V. Zelevinsky, Representations of finite classical groups: A Hopf algebra approach, Lecture Notes in Mathematics 869, Springer, Berlin, 1981.