2015/04/24 by Andrei Neguţ, Neguţ, Andrei · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Logic #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1504.06525
openalex publication_date 2015/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Uq,t(sln^^), 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 Uq,t(sln^^), and to factor it in terms of pieces that arise from subalgebras isomorphic to quantum affine groups Uq(glm^), 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 Uq,t(sln^^) acting on the stable basis, which provide a wide extension of Murnaghan-Nakayama and Pieri type rules from combinatorics.