Combinatorial Auslander-Reiten quivers and reduced expressions
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In this paper, we introduce the notion of combinatorial Aus\-lander-Reiten (AR) quivers for commutation classes $[\widetilde{w}]$ of $w$ in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order $\prec_{[\widetilde{w}]}$ 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 $[\widetilde{w}_0]$ of the longest element $w_0$ of any finite type.
Abstract
In this paper, we introduce the notion of combinatorial Aus\-lander-Reiten (AR) quivers for commutation classes $[\widetilde{w}]$ of $w$ in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order $\prec_{[\widetilde{w}]}$ 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 $[\widetilde{w}_0]$ of the longest element $w_0$ of any finite type.
참고문헌 (27)
- M. Auslander, "Representation Theory of Artin Algebras" (1995), Cambridge University Press
- R. Bedard, "On commutation classes of reduced words in Weyl groups", European J. Combin, 20(6), pp.483~505 (1999)
- N. Bourbaki, "Actualites Scientifiques et Indus-trielles" (1968), Hermann
- J. Brundan, "Homological properties of nite-type Khovanov-Lauda-Rouquier algebras", Duke Math. J, 163(7), pp.1353~1404 (2014)
- J. Claxton, "Young tableaux, multisegments, and PBW bases", Sem. Lothar. Combin, 73, pp.21~ (2015)
- P. Gabriel, "Auslander-Reiten sequences and representation-finite algebras, in Represen-tation theory, I", Proc. Workshop, Carleton Univ, Lecture Notes in Math, 831, pp.1~71 (1979), Springer
- D. Hernandez, "Quantum Grothendieck rings and derived Hall algebras", J. Reine Angew. Math, 701, pp.77~126 (2015)
- S.-J. Kang, "Symmetric quiver Hecke algebras and R-matrices of quantum ane algebras. II", Duke Math. J, 164(8), pp.1549~1602 (2015)
- S. Kato, "Poincare-Birkhoff-Witt bases and Khovanov-Lauda-Rouquier algebras", Duke Math. J, 163(3), pp.619~663 (2014)
- M. Khovanov, "A diagrammatic approach to categorification of quantum groups. I, Represent", Theory, 13, pp.309~347 (2009)
- G. Lusztig, "Canonical bases arising from quantized enveloping algebras", J. Amer. Math. Soc, 3(2), pp.447~498 (1990)
- G. Lusztig, "Quantum groups at roots of 1", Geom. Dedicata, 35(1-3), pp.89~113 (1990)
- G. Lusztig, "Canonical bases and Hall algebras", Representation theories and algebraic geometry NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci, 514, pp.365~399 (1998), Kluwer Acad. Publ
- P. J. McNamara, "Finite dimensional representations of Khovanov-Lauda-Rouquier al-gebras I : Finite type", J. Reine Angew. Math, 707, pp.103~124 (2015)
- S. Oh, "Auslander-Reiten quiver of type D and generalized quantum ane Schur-Weyl duality", J. Algebra, 460, pp.203~252 (2016)
- S. Oh, "Auslander-Reiten quiver of type A and generalized quantum ane Schur-Weyl duality", Trans. Amer. Math. Soc, 369(3), pp.1895~1933 (2017)
- Se-jin Oh, "Auslander–Reiten quiver and representation theories related to KLR-type Schur–Weyl duality", Mathematische Zeitschrift, 291(1-2), pp.499~554 (2019), Springer Science and Business Media LLC[DOI]
- P. Papi, "A characterization of a special ordering in a root system", Proc. Amer. Math. Soc, 120(3), pp.661~665 (1994)
- C. M. Ringel, "Tame algebras", Proceedings ICRA 2, 831, pp.137~187 (1980), Springer
- C. M. Ringe, "PBW-bases of quantum groups", J. Reine Angew. Math, 470, pp.51~88 (1996)
- R. Rouquier, "2 Kac-Moody algebras", arXiv:0812.5023, ~ (2008)
- R. Rouquier, "Quiver Hecke algebras and 2-Lie algebras", Algebra Colloq, 19(2), pp.359~410 (2012)
- Y. Saito, "PBW basis of quantized universal enveloping algebras", Publ. Res. Inst. Math. Sci, 30(2), pp.209~232 (1994)
- D. Simson, "Elements of the representation theory of associative algebras. Vol. 2" (2007), Cambridge University Press
- "The on-line encyclopedia of integer sequences"[URL]
- M. Varagnolo, "Canonical bases and KLR-algebras", J. Reine Angew. Math, 659, pp.67~100 (2011)
- D. P. Zhelobenko, "Extremal cocycles on Weyl groups", Funktsional. Anal. i Prilozhen, 21(3), pp.11~21 (1987)[DOI]