Rational smoothness of varieties of representations for quivers of type $A$

Authors:
Robert Bédard and Ralf Schiffler

Journal:
Represent. Theory **7** (2003), 481-548

MSC (2000):
Primary 17B37; Secondary 32S60

DOI:
https://doi.org/10.1090/S1088-4165-03-00179-1

Published electronically:
November 14, 2003

MathSciNet review:
2017066

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the authors study when the closure (in the Zariski topology) of orbits of representations of quivers of type $A$ are rationally smooth. This is done by considering the corresponding quantized enveloping algebra ${\mathbf {U}}$ and studying the action of the bar involution on PBW bases. Using Ringel’s Hall algebra approach to quantized enveloping algebras and also Auslander-Reiten quivers, we can describe the commutation relations between root vectors. This way we get explicit formulae for the multiplication of an element of PBW bases adapted to a quiver with a root vector and also recursive formulae to study the bar involution on PBW bases. One of the consequences of our characterization is that if the orbit closure is rationally smooth, then it is smooth.

- Maurice Auslander,
*Relations for Grothendieck groups of Artin algebras*, Proc. Amer. Math. Soc.**91**(1984), no. 3, 336–340. MR**744624**, DOI https://doi.org/10.1090/S0002-9939-1984-0744624-1
2 M. Auslander, I. Reiten and S.O. Smalø, - Sara Billey and V. Lakshmibai,
*Singular loci of Schubert varieties*, Progress in Mathematics, vol. 182, Birkhäuser Boston, Inc., Boston, MA, 2000. MR**1782635** - Klaus Bongartz,
*On degenerations and extensions of finite-dimensional modules*, Adv. Math.**121**(1996), no. 2, 245–287. MR**1402728**, DOI https://doi.org/10.1006/aima.1996.0053
5 P. Caldero and R. Schiffler, - James B. Carrell,
*The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties*, Algebraic groups and their generalizations: classical methods (University Park, PA, 1991) Proc. Sympos. Pure Math., vol. 56, Amer. Math. Soc., Providence, RI, 1994, pp. 53–61. MR**1278700** - Peter Gabriel,
*Auslander-Reiten sequences and representation-finite algebras*, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin, 1980, pp. 1–71. MR**607140** - M. Kashiwara,
*On crystal bases of the $Q$-analogue of universal enveloping algebras*, Duke Math. J.**63**(1991), no. 2, 465–516. MR**1115118**, DOI https://doi.org/10.1215/S0012-7094-91-06321-0 - George Lusztig,
*Finite-dimensional Hopf algebras arising from quantized universal enveloping algebra*, J. Amer. Math. Soc.**3**(1990), no. 1, 257–296. MR**1013053**, DOI https://doi.org/10.1090/S0894-0347-1990-1013053-9 - G. Lusztig,
*Canonical bases arising from quantized enveloping algebras*, J. Amer. Math. Soc.**3**(1990), no. 2, 447–498. MR**1035415**, DOI https://doi.org/10.1090/S0894-0347-1990-1035415-6 - Claus Michael Ringel,
*On algorithms for solving vector space problems. II. Tame algebras*, Representation theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., vol. 831, Springer, Berlin, 1980, pp. 137–287. MR**607143**

*Representation theory of artin algebras*, Cambridge Univ. Press., 1995, p. 425; Corrected reprint, 1997. ;

*Rational smoothness of varieties of representations for quivers of Dynkin type*, preprint.

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Additional Information

**Robert Bédard**

Affiliation:
Département de mathematiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada

Email:
bedard@lacim.uqam.ca

**Ralf Schiffler**

Affiliation:
Département de mathematiques, Université du Québec à Montréal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec, H3C 3P8, Canada

Address at time of publication:
School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottowa, Ontario, K1S 5B6, Canada

MR Author ID:
724459

Email:
ralf@math.uqam.ca, ralf@math.carleton.ca

Keywords:
Quantized enveloping algebras,
local intersection cohomology

Received by editor(s):
October 15, 2002

Received by editor(s) in revised form:
July 25, 2003

Published electronically:
November 14, 2003

Additional Notes:
The first author was supported in part by a NSERC grant

The second author was supported in part by a FCAR scholarships

Article copyright:
© Copyright 2003
American Mathematical Society