Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-02-06T14:47:09.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Quivers of monoids with basic algebras

Part of: Representation theory of rings and algebras Semigroups Algebraic combinatorics

Published online by Cambridge University Press:  25 July 2012

Stuart Margolis  and
Benjamin Steinberg
Stuart Margolis
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel (email: margolis@math.biu.ac.il) Center for Algorithmic and Interactive Scientific Software, City College of New York, City University of New York, NY 10031, USA
Benjamin Steinberg
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario K1S 5B6, Canada (email: bsteinberg@ccny.cuny.edu) Department of Mathematics, City College of New York, NY 10031, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known as DO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of ℛ-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

Keywords

quiversrepresentation theorymonoidsHochschild–Mitchell cohomologyEI-categories

MSC classification

Secondary: 16G10: Representations of Artinian rings 20M25: Semigroup rings, multiplicative semigroups of rings 05E99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 5 , September 2012 , pp. 1516 - 1560
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[AM06]Aguiar, M. and Mahajan, S., Coxeter groups and Hopf algebras, Fields Institute Monographs, vol. 23 (American Mathematical Society, Providence, RI, 2006), with a foreword by Nantel Bergeron.Google Scholar
[Alm94]Almeida, J., Finite semigroups and universal algebra, Series in Algebra, vol. 3 (World Scientific, River Edge, NJ, 1994), translated from the 1992 Portuguese original and revised by the author.Google Scholar
[AMSV09]Almeida, J., Margolis, S., Steinberg, B. and Volkov, M., Representation theory of finite semigroups, semigroup radicals and formal language theory, Trans. Amer. Math. Soc. 361 (2009), 1429–1461.CrossRefGoogle Scholar
[AS09]Almeida, J. and Steinberg, B., Matrix mortality and the Černý-Pin conjecture, in Developments in language theory, Lecture Notes in Computer Science, vol. 5583 (Springer, Berlin, 2009), 67–80.CrossRefGoogle Scholar
[ASS06]Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).CrossRefGoogle Scholar
[AD10]Athanasiadis, C. A. and Diaconis, P., Functions of random walks on hyperplane arrangements, Adv. Appl. Math. 45 (2010), 410–437.CrossRefGoogle Scholar
[AGGS04]Auinger, K., Gomes, G. M. S., Gould, V. and Steinberg, B., An application of a theorem of Ash to finite covers, Studia Logica 78 (2004), 45–57.CrossRefGoogle Scholar
[AR97]Auslander, M. and Reiten, I., Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36 (Cambridge University Press, Cambridge, 1997), corrected reprint of the 1995 original.Google Scholar
[Ben98]Benson, D. J., Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, second edition (Cambridge University Press, Cambridge, 1998).Google Scholar
[BBBS10]Berg, C., Bergeron, N., Bhargava, S. and Saliola, F., Primitive orthogonal idempotents for R-trivial monoids (2010), http://arxiv.org/abs/1009.4943v1.CrossRefGoogle Scholar
[BHR99]Bidigare, P., Hanlon, P. and Rockmore, D., A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements, Duke Math. J. 99 (1999), 135–174.CrossRefGoogle Scholar
[Bjö08]Björner, A., Random walks, arrangements, cell complexes, greedoids, and self-organizing libraries, in Building bridges, Bolyai Society Mathematical Studies, vol. 19 (Springer, Berlin, 2008), 165–203.CrossRefGoogle Scholar
[Bjo09]Björner, A., Note: Random-to-front shuffles on trees, Electron. Commun. Probab. 14 (2009), 36–41.CrossRefGoogle Scholar
[Bor94]Borceux, F., Handbook of categorical algebra. 1, Encyclopedia of Mathematics and its Applications, vol. 50 (Cambridge University Press, Cambridge, 1994).Google Scholar
[Bri11]Brimacombe, B., The representation theory of the incidence algebra of an inverse semigroup, PhD thesis, Carleton University, Ottawa (2011).Google Scholar
[Bro00]Brown, K. S., Semigroups, rings, and Markov chains, J. Theoret. Probab. 13 (2000), 871–938.CrossRefGoogle Scholar
[Bro04]Brown, K. S., Semigroup and ring theoretical methods in probability, in Representations of finite dimensional algebras and related topics in Lie theory and geometry, Fields Institute Communications, vol. 40 (American Mathematical Society, Providence, RI, 2004), 3–26.Google Scholar
[BD98]Brown, K. S. and Diaconis, P., Random walks and hyperplane arrangements, Ann. Probab. 26 (1998), 1813–1854.CrossRefGoogle Scholar
[CE99]Cartan, H. and Eilenberg, S., Homological algebra, Princeton Landmarks in Mathematics (Princeton University Press, Princeton, NJ, 1999), with an appendix by David A. Buchsbaum, reprint of the 1956 original.Google Scholar
[Car86]Carter, R. W., Representation theory of the 0-Hecke algebra, J. Algebra 104 (1986), 89–103.CrossRefGoogle Scholar
[CKKSTT07]Chattopadhyay, A., Krebs, A., Koucký, M., Szegedy, M., Tesson, P. and Thérien, D., Languages with bounded multiparty communication complexity, in STACS 2007, Lecture Notes in Computer Science, vol. 4393 (Springer, Berlin, 2007), 500–511.CrossRefGoogle Scholar
[Cli41]Clifford, A. H., Semigroups admitting relative inverses, Ann. of Math. (2) 42 (1941), 1037–1049.CrossRefGoogle Scholar
[CP61]Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7 (American Mathematical Society, Providence, RI, 1961).CrossRefGoogle Scholar
[CPS88]Cline, E., Parshall, B. and Scott, L., Finite-dimensional algebras and highest weight categories, J. reine. angrew. Math. 391 (1988), 85–99.Google Scholar
[Coh03]Cohn, P. M., Basic algebra (Springer, London, 2003).CrossRefGoogle Scholar
[CR88]Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (John Wiley & Sons, New York, NY, 1988), reprint of the 1962 original, a Wiley Interscience publication.Google Scholar
[Den11]Denton, T., A combinatorial formula for orthogonal idempotents in the 0-hecke algebra of the symmetric group, Electron. J. Combin. 18 (2011), Research Paper 28, 20 pp.CrossRefGoogle Scholar
[DHST11]Denton, T., Hivert, F., Schilling, A. and Thiéry, N., On the representation theory of finite 𝒥-trivial monoids, Sém. Lothar. Combin. 64 (2011), Art. B64d, 34 pp..Google Scholar
[DS09]Diaconis, P. and Steinberg, B., Colored shuffles and random walks on semigroups (2009), to appear.Google Scholar
[DHT02]Duchamp, G., Hivert, F. and Thibon, J.-Y., Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput. 12 (2002), 671–717.CrossRefGoogle Scholar
[Eil76]Eilenberg, S., Automata, languages, and machines. Vol. B (Academic Press, New York, NY, 1976), with two chapters (‘Depth decomposition theorem’ and ‘Complexity of semigroups and morphisms’) by Bret Tilson, Pure and Applied Mathematics, vol. 59.Google Scholar
[EZ76]Elkins, B. and Zilber, J. A., Categories of actions and Morita equivalence, Rocky Mountain J. Math. 6 (1976), 199–225.CrossRefGoogle Scholar
[Fay05]Fayers, M., 0-Hecke algebras of finite Coxeter groups, J. Pure Appl. Algebra 199 (2005), 27–41.CrossRefGoogle Scholar
[FGG99]Fountain, J., Gomes, G. M. S. and Gould, V., Enlargements, semiabundancy and unipotent monoids, Comm. Algebra 27 (1999), 595–614.CrossRefGoogle Scholar
[GR97]Gabriel, P. and Roiter, A. V., Representations of finite-dimensional algebras (Springer, Berlin, 1997), translated from the Russian, with a chapter by B. Keller, reprint of the 1992 English translation.CrossRefGoogle Scholar
[GM11]Ganyushkin, O. and Mazorchuk, V., On Kiselman quotients of 0-Hecke monoids, Int. Electron. J. Algebra 10 (2011), 174–191.Google Scholar
[GMS09]Ganyushkin, O., Mazorchuk, V. and Steinberg, B., On the irreducible representations of a finite semigroup, Proc. Amer. Math. Soc. 137 (2009), 3585–3592.CrossRefGoogle Scholar
[GHKLMS03]Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S., Continuous lattices and domains, Encyclopedia of Mathematics and its Applications, vol. 93 (Cambridge University Press, Cambridge, 2003).CrossRefGoogle Scholar
[Gre51]Green, J. A., On the structure of semigroups, Ann. of Math. (2) 54 (1951), 163–172.CrossRefGoogle Scholar
[HMS74]Hofmann, K. H., Mislove, M. and Stralka, A., The Pontryagin duality of compact O-dimensional semilattices and its applications, Lecture Notes in Mathematics, vol. 396 (Springer, Berlin, 1974).CrossRefGoogle Scholar
[Hsi09]Hsiao, S. K., A semigroup approach to wreath-product extensions of Solomon’s descent algebras, Electron. J. Combin. 16 (2009), 9, Research Paper 21.CrossRefGoogle Scholar
[JK81]James, G. and Kerber, A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16 (Addison-Wesley, Reading, MA, 1981), with a foreword by P. M. Cohn, with an introduction by Gilbert de B. Robinson.Google Scholar
[Joh86]Johnstone, P. T., Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3 (Cambridge University Press, Cambridge, 1986), reprint of the 1982 edition.Google Scholar
[KPT05]Koucký, M., Pudlák, P. and Thérien, D., Bounded-depth circuits: separating wires from gates [extended abstract], in STOC’05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (ACM, New York, NY, 2005), 257–265.CrossRefGoogle Scholar
[KR68]Krohn, K. and Rhodes, J., Complexity of finite semigroups, Ann. of Math. (2) 88 (1968), 128–160.CrossRefGoogle Scholar
[KRT68]Krohn, K., Rhodes, J. and Tilson, B., Algebraic theory of machines, languages, and semigroups, ed. Arbib, M. A. (Academic Press, New York, NY, 1968), ch. 1, 5–9 with a major contribution by Kenneth Krohn and John L. Rhodes.Google Scholar
[KM08]Kudryavtseva, G. and Mazorchuk, V., Schur-Weyl dualities for symmetric inverse semigroups, J. Pure Appl. Algebra 212 (2008), 1987–1995.CrossRefGoogle Scholar
[Law98]Lawson, M. V., Inverse semigroups (World Scientific, River Edge, NJ, 1998).CrossRefGoogle Scholar
[Lot97]Lothaire, M., Combinatorics on words, Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1997), with a foreword by Roger Lyndon, with a preface by Dominique Perrin, corrected reprint of the 1983 original, with a new preface by Perrin.CrossRefGoogle Scholar
[Lüc89]Lück, W., Transformation groups and algebraic K-theory, Lecture Notes in Mathematics, vol. 1408 (Springer, Berlin, 1989), Mathematica Gottingensis.CrossRefGoogle Scholar
[Mac98]Mac Lane, S., Categories for the working mathematician, Graduate Texts in Mathematics, vol. 5, second edition (Springer, New York, NY, 1998).Google Scholar
[MM94]Mac Lane, S. and Moerdijk, I., Sheaves in geometry and logic, Universitext (Springer, New York, NY, 1994), corrected reprint of the 1992 edition.CrossRefGoogle Scholar
[MR10]Malandro, M. and Rockmore, D., Fast Fourier transforms for the rook monoid, Trans. Amer. Math. Soc. 362 (2010), 1009–1045.CrossRefGoogle Scholar
[Mal10]Malandro, M. E., Fast Fourier transforms for finite inverse semigroups, J. Algebra 324 (2010), 282–312.CrossRefGoogle Scholar
[MR95]Mantaci, R. and Reutenauer, C., A generalization of Solomon’s algebra for hyperoctahedral groups and other wreath products, Comm. Algebra 23 (1995), 27–56.CrossRefGoogle Scholar
[MSS11]Margolis, S. W., Saliola, F. V. and Steinberg, B., A topological approach to the global dimension of left regular band algebras, (2011), to appear.Google Scholar
[MS11]Margolis, S. and Steinberg, B., The quiver of an algebra associated to the Mantaci-Reutenauer descent algebra and the homology of regular semigroups, Algebr. Represent. Theory 14 (2011), 131–159.CrossRefGoogle Scholar
[McA71]McAlister, D. B., Representations of semigroups by linear transformations. I, II, Semigroup Forum 2 (1971), 189–263 ; 2 (1971), 283–320.CrossRefGoogle Scholar
[McA72]McAlister, D. B., Characters of finite semigroups, J. Algebra 22 (1972), 183–200.CrossRefGoogle Scholar
[Mit72]Mitchell, B., Rings with several objects, Adv. Math. 8 (1972), 1–161.CrossRefGoogle Scholar
[Nic71]Nico, W. R., Homological dimension in semigroup algebras, J. Algebra 18 (1971), 404–413.CrossRefGoogle Scholar
[Nic72]Nico, W. R., An improved upper bound for global dimension of semigroup algebras, Proc. Amer. Math. Soc. 35 (1972), 34–36.CrossRefGoogle Scholar
[Nor79]Norton, P. N., 0-Hecke algebras, J. Aust. Math. Soc. Ser. A 27 (1979), 337–357.CrossRefGoogle Scholar
[OP91]Okniński, J. and Putcha, M. S., Complex representations of matrix semigroups, Trans. Amer. Math. Soc. 323 (1991), 563–581.CrossRefGoogle Scholar
[Pfe09]Pfeiffer, G., A quiver presentation for Solomon’s descent algebra, Adv. Math. 220 (2009), 1428–1465.CrossRefGoogle Scholar
[Pin86]Pin, J.-E., Varieties of formal languages, foundations of computer science (Plenum, New York, NY, 1986), with a preface by M.-P. Schützenberger, translated from the French by A. Howie.CrossRefGoogle Scholar
[Put73]Putcha, M. S., Semilattice decompositions of semigroups, Semigroup Forum 6 (1973), 12–34.CrossRefGoogle Scholar
[Put88]Putcha, M. S., Linear algebraic monoids, London Mathematical Society Lecture Note Series, vol. 133 (Cambridge University Press, Cambridge, 1988).CrossRefGoogle Scholar
[Put89]Putcha, M. S., Monoids on groups with BN-pairs, J. Algebra 120 (1989), 139–169.CrossRefGoogle Scholar
[Put91]Putcha, M. S., Monoids of Lie type and group representations, in Monoids and semigroups with applications (Berkeley, CA, 1989) (World Scientific, River Edge, NJ, 1991), 288–305.Google Scholar
[Put94]Putcha, M. S., Classification of monoids of Lie type, J. Algebra 163 (1994), 636–662.CrossRefGoogle Scholar
[Put95]Putcha, M. S., Monoids of Lie type, in Semigroups, formal languages and groups (York, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 466 (Kluwer, Dordrecht, 1995), 353–367.CrossRefGoogle Scholar
[Put96]Putcha, M. S., Complex representations of finite monoids, Proc. Lond. Math. Soc. (3) 73 (1996), 623–641.CrossRefGoogle Scholar
[Put98]Putcha, M. S., Complex representations of finite monoids. II. Highest weight categories and quivers, J. Algebra 205 (1998), 53–76.CrossRefGoogle Scholar
[Put00]Putcha, M. S., Semigroups and weights for group representations, Proc. Amer. Math. Soc. 128 (2000), 2835–2842.CrossRefGoogle Scholar
[PR93]Putcha, M. S. and Renner, L. E., The canonical compactification of a finite group of Lie type, Trans. Amer. Math. Soc. 337 (1993), 305–319.CrossRefGoogle Scholar
[Ren05]Renner, L. E., Linear algebraic monoids, Encyclopaedia of Mathematical Sciences, vol. 134 (Springer, Berlin, 2005), Invariant Theory and Algebraic Transformation Groups, V.Google Scholar
[Rho69]Rhodes, J., Characters and complexity of finite semigroups, J. Combin. Theory 6 (1969), 67–85.CrossRefGoogle Scholar
[RS09]Rhodes, J. and Steinberg, B., The q-theory of finite semigroups, Springer Monographs in Mathematics (Springer, New York, NY, 2009).CrossRefGoogle Scholar
[RZ91]Rhodes, J. and Zalcstein, Y., Elementary representation and character theory of finite semigroups and its application, in Monoids and semigroups with applications (Berkeley, CA, 1989) (World Scientific, River Edge, NJ, 1991), 334–367.CrossRefGoogle Scholar
[RS90]Richardson, R. W. and Springer, T. A., The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), 389–436.CrossRefGoogle Scholar
[Sal07]Saliola, F. V., The quiver of the semigroup algebra of a left regular band, Internat. J. Algebra Comput. 17 (2007), 1593–1610.CrossRefGoogle Scholar
[Sal08]Saliola, F. V., On the quiver of the descent algebra, J. Algebra 320 (2008), 3866–3894.CrossRefGoogle Scholar
[Sal09]Saliola, F. V., The face semigroup algebra of a hyperplane arrangement, Canad. J. Math. 61 (2009), 904–929.CrossRefGoogle Scholar
[Sch06]Schocker, M., The module structure of the Solomon-Tits algebra of the symmetric group, J. Algebra 301 (2006), 554–586.CrossRefGoogle Scholar
[Sch08]Schocker, M., Radical of weakly ordered semigroup algebras, J. Algebraic Combin. 28 (2008), 231–234, with a foreword by Nantel Bergeron.CrossRefGoogle Scholar
[Sch76/77]Schützenberger, M. P., Sur le produit de concaténation non ambigu, Semigroup Forum 13 (1976/77), 47–75.CrossRefGoogle Scholar
[Sim75]Simon, I., Piecewise testable events, in Automata theory and formal languages (Second GI Conference, Kaiserslautern, 1975), Lecture Notes in Computer Science, vol. 33 (Springer, Berlin, 1975), 214–222.CrossRefGoogle Scholar
[Sol76]Solomon, L., A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (1976), 255–264.CrossRefGoogle Scholar
[Sol02]Solomon, L., Representations of the rook monoid, J. Algebra 256 (2002), 309–342.CrossRefGoogle Scholar
[Sta97]Stanley, R. P., Enumerative combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, vol. 49 (Cambridge University Press, Cambridge, 1997), with a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original.CrossRefGoogle Scholar
[Sta99]Stanley, R. P., Enumerative combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999), with a foreword by Gian-Carlo Rota, Appendix 1 by Sergey Fomin.CrossRefGoogle Scholar
[Ste06]Steinberg, B., Möbius functions and semigroup representation theory, J. Combin. Theory Ser. A 113 (2006), 866–881.CrossRefGoogle Scholar
[Ste08]Steinberg, B., Möbius functions and semigroup representation theory. II. Character formulas and multiplicities, Adv. Math. 217 (2008), 1521–1557.CrossRefGoogle Scholar
[Ste10]Steinberg, B., A theory of transformation monoids: combinatorics and representation theory, Electron. J. Combin. 17 (2010), Research Paper 164, 56 pp.CrossRefGoogle Scholar
[Sti73]Stiffler, J. P., Extension of the fundamental theorem of finite semigroups, Adv. in Math. 11 (1973), 159–209.CrossRefGoogle Scholar
[ST88]Straubing, H. and Thérien, D., Partially ordered finite monoids and a theorem of I. Simon, J. Algebra 119 (1988), 393–399.CrossRefGoogle Scholar
[Tit74]Tits, J., Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, vol. 386 (Springer, Berlin, 1974).Google Scholar
[Tit76]Tits, J., Two properties of Coxeter complexes, J. Algebra 41 (1976), 265–268, appendix to [L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 4 (1976), 255–264].CrossRefGoogle Scholar
[Web07]Webb, P., An introduction to the representations and cohomology of categories, in Group representation theory (EPFL Press, Lausanne, 2007), 149–173.Google Scholar
[Web08]Webb, P., Standard stratifications of EI categories and Alperin’s weight conjecture, J. Algebra 320 (2008), 4073–4091.CrossRefGoogle Scholar