Jan Schröer (Leeds)

Module theoretic interpretation of quantum minors

Let L be a the preprojective algebra of type An, and let B* be the dual canonical basis of the associated quantized algebra Uv-. The elements in B* are indexed by multisegments m. To each quantum minor bm* in B* we associate a L-module Lm (this is a laminated module in the sense of Ringel [3]). Our main result is the following:

Theorem.  Let b*m and b*n be quantum flag minors. Then the following are equivalent:
(1) b*m and b*n are multiplicative, i.e. b*mb*n is in vZB*;
(2) ExtL1(Lm,Ln) = 0.

The proof of this theorem uses a combinatorial criterion due to Leclerc, Nazarov and Thibon [2] for two quantum flag minors to be multiplicative. For all missing definitions we refer to [1], [2] and [3].

References

[1]
A. Berenstein, A. Zelevinsky, String bases for quantum groups of type Ar. I.M. Gelfand Seminar, 51-89, Adv. Soviet Math. 16, Part 1, Amer. Math. Soc., Providence, RI (1993).
[2]
B. Leclerc, M. Nazarov, J.-Y. Thibon, Induced representations of affine Hecke algebras and canonical bases of quantum groups. Preprint arXiv:math.QA/0011074 (2000), 1-33.
[3]
C.M. Ringel, The multisegment duality and the preprojective algebras of type A. AMA Algebra Montp. Announc. 1999, Paper 2, 6pp.