Browse > Article
http://dx.doi.org/10.4134/JKMS.j180190

COMBINATORIAL AUSLANDER-REITEN QUIVERS AND REDUCED EXPRESSIONS  

Oh, Se-jin (Department of Mathematics Ewha Womans University)
Suh, Uhi Rinn (Department of Mathematical Sciences Research Institute of Mathematics Seoul National University)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.2, 2019 , pp. 353-385 More about this Journal
Abstract
In this paper, we introduce the notion of combinatorial Auslander-Reiten (AR) quivers for commutation classes [${\tilde{w}}]$ of w in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order ${\prec}_{[{\tilde{w}}]}$ on the subset ${\Phi}(w)$ of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection functors, we can apply their properties to the representation theory of KLR algebras and dual PBW-basis associated to any commutation class [${\tilde{w}}_0$] of the longest element $w_0$ of any finite type.
Keywords
combinatorial AR-quiver; reduced expressions;
Citations & Related Records
연도 인용수 순위
  • Reference
1 N. J. A. Sloane, The on-line encyclopedia of integer sequences, published electronically at http://oeis.org.
2 M. Varagnolo and E. Vasserot, Canonical bases and KLR-algebras, J. Reine Angew. Math. 659 (2011), 67-100.
3 D. P. Zhelobenko, Extremal cocycles on Weyl groups, Funktsional. Anal. i Prilozhen. 21 (1987), no. 3, 11-21, 95.   DOI
4 R. Bedard, On commutation classes of reduced words in Weyl groups, European J. Combin. 20 (1999), no. 6, 483-505.   DOI
5 N. Bourbaki, Elements de mathematique. Fasc. XXXIV. Groupes et algebres de Lie. Chapitre IV: Groupes de Coxeter et systemes de Tits. Chapitre V: Groupes engendres par des reflexions. Chapitre VI: systemes de racines, Actualites Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.
6 J. Brundan, A. Kleshchev, and P. J. McNamara, Homological properties of finite-type Khovanov-Lauda-Rouquier algebras, Duke Math. J. 163 (2014), no. 7, 1353-1404.   DOI
7 J. Claxton and P. Tingley, Young tableaux, multisegments, and PBW bases, Sem. Lothar. Combin. 73 (2015), Art. B73c, 21 pp.
8 P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, in Represen- tation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), 1-71, Lecture Notes in Math., 831, Springer, Berlin, 1980.
9 D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. Reine Angew. Math. 701 (2015), 77-126.
10 S.-J. Kang, M. Kashiwara, and M. Kim, Symmetric quiver Hecke algebras and R- matrices of quantum affine algebras. II, Duke Math. J. 164 (2015), no. 8, 1549-1602.   DOI
11 S. Kato, Poincare-Birkhoff-Witt bases and Khovanov-Lauda-Rouquier algebras, Duke Math. J. 163 (2014), no. 3, 619-663.   DOI
12 P. J. McNamara, Finite dimensional representations of Khovanov-Lauda-Rouquier algebras I: Finite type, J. Reine Angew. Math. 707 (2015), 103-124.
13 M. Khovanov and A. D. Lauda, A diagrammatic approach to categorification of quantum groups. I, Represent. Theory 13 (2009), 309-347.   DOI
14 M. Auslander, I. Reiten, and S. O. Smalo, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1995.
15 G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447-498.   DOI
16 G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990), no. 1-3, 89-113.   DOI
17 G. Lusztig, Canonical bases and Hall algebras, in Representation theories and algebraic geometry (Montreal, PQ, 1997), 365-399, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 514, Kluwer Acad. Publ., Dordrecht, 1998.
18 S. Oh, Auslander-Reiten quiver of type D and generalized quantum affine Schur-Weyl duality, J. Algebra 460 (2016), 203-252.   DOI
19 S. Oh, Auslander-Reiten quiver of type A and generalized quantum affine Schur-Weyl duality, Trans. Amer. Math. Soc. 369 (2017), no. 3, 1895-1933.   DOI
20 S. Oh, Auslander-Reiten quiver and representation theories related to KLR-type Schur-Weyl duality, Math. Z. (2018). https://doi.org/10.1007/s00209-018-2093-2.   DOI
21 P. Papi, A characterization of a special ordering in a root system, Proc. Amer. Math. Soc. 120 (1994), no. 3, 661-665.   DOI
22 R. Rouquier, Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq. 19 (2012), no. 2, 359-410.   DOI
23 C. M. Ringel, Tame algebras, Proceedings ICRA 2, Springer LNM 831, (1980), 137-87.
24 C. M. Ringel, PBW-bases of quantum groups, J. Reine Angew. Math. 470 (1996), 51-88.
25 R. Rouquier, 2 Kac-Moody algebras, arXiv:0812.5023 (2008).
26 Y. Saito, PBW basis of quantized universal enveloping algebras, Publ. Res. Inst. Math. Sci. 30 (1994), no. 2, 209-232.   DOI
27 D. Simson and A. Skowronski, Elements of the representation theory of associative algebras. Vol. 2, London Mathematical Society Student Texts, 71, Cambridge University Press, Cambridge, 2007.