Quivers and Bps Bound States

Exercise 1. Consider the quiver Q1 with 2 nodes and r arrows 1→ 2. (1) Compute using either the Harder-Narasimhan filtrations or the Reineke formula I(~ d) = p(M(~ d, ~ θ))/(s− 1) for ~ θ = (1,−1) for dimension vectors ~ d = (2, 1), (1, 2). (2) Verify that p(M(~ d, ~ θ)) is a polynomial whose degree is given by the dimension formula. (3) Compute the Euler number χ(M(~ d, ~ θ)) = p(M(~ d, ~ θ),−1) (4) The Poincaré polynomial of the Grassmannian Gr(n, k) (the space of n−dimensional hyperplanes in C) is given by