Quiver varieties and $t$-analogs of $q$-characters of quantum affine algebras

By H. Nakajima. 2001

193 citations recorded by the source corpus.

We consider a specialization of an untwisted quantum affine algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ?computable? polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ?compute? q-characters for all simple modules. The result is based on ?computations? of Betti numbers of graded/cyclic quiver varieties. (The reason why we use ? ? will be explained at the end of the introduction.)

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3a050d0ac99242c43e66d269b2f0b1be6d014525

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Citations in the corpus

Simple tensor products (2009, methodology): “Let us explain this proof since it is related to a nice symmetry property of the corresponding Kazhdan-Lusztig polynomials (a priori, this method cannot be extended to the general case since quiver varieties used in [ 27 ] are not known to exist for the non-simplylaced cases).”
Quantum Geometry and Quiver Gauge Theories (2013, methodology): “[133] H. Nakajima, “t-analogs of q-characters of quantum affine algebras of type E6, E7, E8,”. http://dx.doi.org/10.1007/978-0-8176-4697-4_10.”
Representations of Quantum Affinizations and Fusion Product (2003, background): “The finite dimensional representations of quantum affine algebras are the subject of intense research (see, among the others, [AK], [CP1], [CP3], [CP4], [EM], [FR], [FM], [N1], [ N2 ], [VV2] and references therein).”
BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters (2015, unknown): “Nakajima in [75, 78–80].”
The Kirillov-Reshetikhin conjecture and solutions of T-systems (2005, unknown): “In particular Nakajima [31, 32] made a remarkable advance by proving the Kirillov-Reshetikhin conjecture and T -systems in simply-laced cases : using the main result of [31] (the algorithm) he proved that in simply-laced cases the Kirillov-Reshetikhin modules are special and noticed that this proper”
Quiver varieties and cluster algebras (2009, methodology): “Using these morphisms, we define a homomorphism from R to the cluster algebra.”
Kac’s conjecture from Nakajima quiver varieties (2008, background): “Nakajima [13, 14] constructs, as holomorphic symplectic quotient, a complex variety M(v, w) of dimension 2dv,w, which we call a Nakajima quiver variety .I n [ 15 ] Nakajima found a combinatorial algorithm to determine the Betti numbers of these varieties.”