Recollement and Tilting Complexes

First, we study recollement of a derived category of unbounded complexes of modules induced by a partial tilting complex. Second, we give equivalent conditions for P^{centerdot} to be a recollement tilting complex, that is, a tilting complex which induces an equivalence between recollements \{\cat{D}_{A/AeA}(A), \cat{D}(A), \cat{D}(eAe)} and \{\cat{D}_{B/BfB}(B), \cat{D}(B), \cat{D}(fBf)}, where e, f are idempotents of A, B, respectively. In this case, there is an unbounded bimodule complex $\varDelta^{\centerdot}_{T}$ which induces an equivalence between \cat{D}_{A/AeA}(A) and \cat{D}_{B/BfB}(B). Third, we apply the above to a symmetric algebra A. We show that a partial tilting complex $P^{\centerdot}$ for A of length 2 extends to a tilting complex, and that $P^{\centerdot}$ is a tilting complex if and only if the number of indecomposable types of $P^{\centerdot}$ is one of A. Finally, we show that for an idempotent e of A, a tilting complex for eAe extends to a recollement tilting complex for A, and that its standard equivalence induces an equivalence between \cat{Mod}A/AeA and \cat{Mod}B/BfB.Comment: 24 page

Constructible characters and canonical bases

We give closed formulas for all vectors of the canonical basis of a level 2 irreducible integrable representation of $U_v(sl_\infty)$. These formulas coincide at v=1 with Lusztig's formulas for the constructible characters of the Iwahori-Hecke algebras of type B and D.Comment: 16 page