Philippe Caldero (Lyon I)

Rational smoothness of varieties of representations for quivers of type A-D-E

The talk is about a recent paper by R. Bedard and R. Schiffler. Let F be an algebraic closure of the finite field F_q, d=(d₁,¼, d_n) Î Z _{³ 0}ⁿ and G_d=Õ_i=1ⁿ GL_di(F). Let Q be a fixed quiver whose underlying orientation is the Dynkin graph of type A-D-E.

The group G_d acts on E_d = Å_{i® j Î Q} Hom(F^d_i,F^d_j). Fix a G_d-orbit O, the problem is to characterize which orbit closures O' are rationnaly smooth. Rational smoothness is a topological property of varieties defined using the local intersection cohomology groups of O'. Let U⁺ be the positive part of the quantized enveloping algebra over Q(v) associated by Drinfeld and Jimbo to the root system of type A-D-E. For each reduced decomposition [w₀] of the longest element of the Weyl group, there is a PBW-basis B_{[w0 ]}. Suppose that [w₀] is the reduced decomposition associated to the quiver Q, then it is well known that the transition matrix between the canonical basis B of U⁺ and B_[w0] can be given in terms of the local intersection cohomology groups. This provides a method to study the rational smoothness of the varieties O'.