Hecke algebras of simply-laced type with independent parameters

Huang, Jia¹

¹ University of Nebraska at Kearney Department of Mathematics and Statistics Kearney, Nebraska 68849, USA

Algebraic Combinatorics, Tome 3 (2020) no. 3, pp. 667-691.

Résumé

We study the (complex) Hecke algebra ${\mathscr{H}}_S\left(\mathbf{q}\right)$ of a finite simply-laced Coxeter system $(W,S)$ with independent parameters $\mathbf{q}\in {\left(ℂ\setminus \left\{\text{roots}\text{of}\text{unity}\right\}\right)}^S$. We construct its irreducible representations and projective indecomposable representations. We obtain the quiver of this algebra and determine when it is of finite representation type. We provide decomposition formulas for induced and restricted representations between the algebra ${\mathscr{H}}_S\left(\mathbf{q}\right)$ and the algebra ${\mathscr{H}}_R\left(\mathbf{q}{|}_R\right)$ with $R\subseteq S$. Our results demonstrate an interesting combination of the representation theory of finite Coxeter groups and their 0-Hecke algebras, including a two-sided duality between the induced and restricted representations.

Détail

Reçu le : 2019-03-11
Révisé le : 2019-12-27
Accepté le : 2019-12-28
Publié le : 2020-06-02
DOI : 10.5802/alco.108

Classification : 16G30, 05E10
Mots-clés : Hecke algebra, independent parameters, simply-laced Coxeter system, induction and restriction, duality.

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Adin, Ron M.; Brenti, Francesco; Roichman, Yuval A construction of Coxeter group representations. II, J. Algebra, Volume 306 (2006) no. 1, pp. 208-226 | Zbl | MR | DOI

[2] Adin, Ron M.; Brenti, Francesco; Roichman, Yuval A unified construction of Coxeter group representations, Adv. Appl. Math., Volume 37 (2006) no. 1, pp. 31-67 | Zbl | MR | DOI

[3] Bergeron, Nantel; Li, Huilan Algebraic structures on Grothendieck groups of a tower of algebras, J. Algebra, Volume 321 (2009) no. 8, pp. 2068-2084 | Zbl | MR | DOI

[4] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Grad. Texts Math., 231, Springer, New York, 2005, xiv+363 pages | Zbl | MR

[5] Björner, Anders; Wachs, Michelle L. Generalized quotients in Coxeter groups, Trans. Am. Math. Soc., Volume 308 (1988) no. 1, pp. 1-37 | Zbl | MR | DOI

[6] Curtis, Charles W.; Reiner, Irving Methods of representation theory. Vol. I, John Wiley & Sons, Inc., New York, 1981, xxi+819 pages (With applications to finite groups and orders, Pure and Applied Mathematics, A Wiley-Interscience Publication) | Zbl | MR

[7] Denton, Tom; Hivert, Florent; Schilling, Anne; Thiéry, Nicolas M. On the representation theory of finite $𝒥$-trivial monoids, Sémin. Lothar. Comb., Volume 64 (2010/11), Art. B64d, 44 pages | Zbl | MR

[8] Dipper, Richard; James, Gordon Representations of Hecke algebras of general linear groups, Proc. Lond. Math. Soc., III. Ser., Volume 52 (1986) no. 1, pp. 20-52 | Zbl | MR | DOI

[9] Duchamp, Gérard; Hivert, Florent; Thibon, Jean-Yves Noncommutative symmetric functions. VI. Free quasi-symmetric functions and related algebras, Int. J. Algebra Comput., Volume 12 (2002) no. 5, pp. 671-717 | Zbl | MR | DOI

[10] Etingof, Pavel; Golberg, Oleg; Hensel, Sebastian; Liu, Tiankai; Schwendner, Alex; Vaintrob, Dmitry; Yudovina, Elena Introduction to representation theory, Stud. Math. Libr., 59, American Mathematical Society, Providence, RI, 2011, viii+228 pages (With historical interludes by Slava Gerovitch) | Zbl | MR | DOI

[11] Geck, Meinolf; Jacon, Nicolas Representations of Hecke algebras at roots of unity, Algebr. Appl., 15, Springer-Verlag London, Ltd., London, 2011, xii+401 pages | Zbl | MR | DOI

[12] Goodman, Frederick M.; Wenzl, Hans Iwahori-Hecke algebras of type $A$ at roots of unity, J. Algebra, Volume 215 (1999) no. 2, pp. 694-734 | Zbl | MR | DOI

[13] Grinberg, Darij; Reiner, Victor Hopf algebras in Combinatorics (2014) (https://arxiv.org/abs/1409.8356)

[14] Huang, Jia 0-Hecke algebra actions on coinvariants and flags, J. Algebr. Comb., Volume 40 (2014) no. 1, pp. 245-278 | Zbl | MR | DOI

[15] Huang, Jia 0-Hecke algebra action on the Stanley–Reisner ring of the Boolean algebra, Ann. Comb., Volume 19 (2015) no. 2, pp. 293-323 | Zbl | MR | DOI

[16] Huang, Jia Hecke algebras with independent parameters, J. Algebr. Comb., Volume 43 (2016) no. 3, pp. 521-551 | Zbl | MR | DOI

[17] Huang, Jia A tableau approach to the representation theory of 0-Hecke algebras, Ann. Comb., Volume 20 (2016) no. 4, pp. 831-868 | Zbl | MR | DOI

[18] Huang, Jia A uniform generalization of some combinatorial Hopf algebras, Algebr. Represent. Theory, Volume 20 (2017) no. 2, pp. 379-431 | Zbl | MR | DOI

[19] Humphreys, James E. Reflection groups and Coxeter groups, Camb. Stud. Adv. Math., 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | Zbl | MR | DOI

[20] König, Sebastian The decomposition of 0-Hecke modules associated to quasisymmetric Schur functions, Algebr. Comb., Volume 2 (2019) no. 5, pp. 735-751 | Zbl | MR | DOI

[21] Krob, Daniel; Thibon, Jean-Yves Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at $q=0$, J. Algebr. Comb., Volume 6 (1997) no. 4, pp. 339-376 | Zbl | MR | DOI

[22] Li, Fang; Chen, Lili The natural quiver of an Artinian algebra, Algebr. Represent. Theory, Volume 13 (2010) no. 5, pp. 623-636 | Zbl | MR | DOI

[23] Lusztig, George On a theorem of Benson and Curtis, J. Algebra, Volume 71 (1981) no. 2, pp. 490-498 | Zbl | MR | DOI

[24] Lusztig, George Hecke algebras with unequal parameters, CRM Monogr. Ser., 18, American Mathematical Society, Providence, RI, 2003, vi+136 pages | Zbl | MR

[25] Norton, P. N. $0$-Hecke algebras, J. Aust. Math. Soc., Ser. A, Volume 27 (1979) no. 3, pp. 337-357 | DOI | MR

[26] Simson, Daniel; Skowroński, Andrzej Elements of the representation theory of associative algebras. Vol. 3, Lond. Math. Soc. Stud. Texts, 72, Cambridge University Press, Cambridge, 2007, xii+456 pages (Representation-infinite tilted algebras) | Zbl | MR

[27] Sloane, Neil J. A. The on-line encyclopedia of integer sequences, Notices Am. Math. Soc., Volume 65 (2018) no. 9, pp. 1062-1074 | Zbl | MR

[28] Solomon, Louis A decomposition of the group algebra of a finite Coxeter group, J. Algebra, Volume 9 (1968), pp. 220-239 | MR | DOI

[29] Steinberg, Benjamin Representation theory of finite monoids, Universitext, Springer, Cham, 2016, xxiv+317 pages | Zbl | MR | DOI

[30] Stembridge, John R. A short derivation of the Möbius function for the Bruhat order, J. Algebr. Comb., Volume 25 (2007) no. 2, pp. 141-148 | Zbl | MR | DOI

[31] Tewari, Vasu V.; van Willigenburg, Stephanie J. Modules of the 0-Hecke algebra and quasisymmetric Schur functions, Adv. Math., Volume 285 (2015), pp. 1025-1065 | Zbl | MR | DOI

[32] Tewari, Vasu V.; van Willigenburg, Stephanie J. Permuted composition tableaux, 0-Hecke algebra and labeled binary trees, J. Comb. Theory, Ser. A, Volume 161 (2019), pp. 420-452 | Zbl | MR | DOI