  The purpose of this thesis is to present certain viewpoints on the geometric
representation theory of Nakajima cyclic quiver varieties, in relation to the
Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra,
which arises as the K-theoretic Hall algebra of the double cyclic quiver. We
prove the isomorphism between the shuffle algebra and the quantum toroidal
algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the
shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties,
which were studied by Nakajima and Varagnolo-Vasserot.
  The shuffle algebra viewpoint allows us to construct the universal R-matrix
of the quantum toroidal algebra U_{q,t}(sl_n^^), and to factor it in terms of
pieces that arise from subalgebras isomorphic to quantum affine groups
U_{q}(gl_m^), for various m. This factorization generalizes constructions of
Khoroshkin-Tolstoy to the toroidal case, and matches the factorization that
Maulik-Okounkov produce via the stable basis in the K-theory of Nakajima quiver
varieties. We connect the two pictures by computing formulas for the root
generators of U_{q,t}(sl_n^^) acting on the stable basis, which provide a wide
extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.
