Grothendieck classes of quiver varieties

We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. Our formula is stated in terms of coefficients that are uniquely determined by the geometry and can be co...

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Bibliographic Details
Published in:Duke mathematical journal 2002-10, Vol.115 (1), p.75-103
Main Author: Buch, Anders Skovsted
Format: Article
Language:English
Subjects:
05E10
14C35
19E08
Applications of methods of algebraic $K$-theory [See also 19Exx]
Combinatorial aspects of representation theory [See also 20C30]
K$-theory of schemes [See also 14C35]
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Summary:We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. Our formula is stated in terms of coefficients that are uniquely determined by the geometry and can be computed by an explicit combinatorial algorithm. We conjecture that these coefficients have signs that alternate with degree. The proof of our formula involves K-theoretic generalizations of several useful cohomological tools, including the Thom-Porteous formula, the Jacobi-Trudi formula, and a Gysin formula of P. Pragacz.
ISSN:0012-7094
1547-7398
DOI:10.1215/S0012-7094-02-11513-0