Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 052, 28 pages      math.AG/0510455      https://doi.org/10.3842/SIGMA.2017.052

A Combinatorial Study on Quiver Varieties

Shigeyuki Fujii a and Satoshi Minabe b
a) Accenture Strategy, 107-8672 Tokyo, Japan
b) Department of Mathematics, Tokyo Denki University, 120-8551 Tokyo, Japan

Received January 13, 2017, in final form June 30, 2017; Published online July 06, 2017

Abstract
This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating function of Poincaré polynomials of quiver varieties in rank 1 cases. Our main tools are cores and quotients of Young diagrams. In the second part, we give a brief survey of instanton counting in physics, where quiver varieties appear as moduli spaces of instantons, focusing on its combinatorial aspects.

Key words: Young diagram; core; quotient; quiver variety; instanton.

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