On irreducibility of tensor products of evaluation modules for the quantum affine algebra

Abstract

Every irreducible finite-dimensional representation of the quantized enveloping algebra U q (gl n ) can be extended to the quantum affine algebra U q ( gl n ) via the evaluation homomorphism. We give in explicit form the necessary and sufficient conditions for irreducibility of tensor products of such evaluation modules.

Quantum affine Gelfand–Tsetlin bases and quantum toroidal algebra via K-theory of affine Laumon spaces

“…The construction of the Gelfand-Tsetlin basis for the representations of quantum gl n goes back to M. Jimbo [8]. We will follow the approach of [11]. To a collection According to the Thomason localization theorem, restriction to the T × C * -fixed point set induces an isomorphism…”

“…Let (a kl ) 1≤k,l≤n = A n−1 stand for the Cartan matrix of sl n . The double affine loop algebra U ′ v ( sl n ) is an associative algebra over Q(v) generated by e k,r , f k,r , v ±h k , h k,m (1 ≤ k ≤ n, r ∈ Z, m ∈ Z \ {0}) with the relations (9)(10)(11)(12)(13)(14), where k, l are understood as residues modulo n, so that for instance if k = n then k + 1 = 1.…”

“…2.5. Quantum universal enveloping algebra U v (gl n ) For the quantum universal enveloping algebra U v (gl n ) we follow the notations of section 2 of [11]. Namely, U v (gl n ) has generators t ±1 1 , .…”

Braid group actions and tensor products for Yangians

“…, where the tensor factors may take any order. In the context of quantum affine algebras U q (ĝ), the irreducibility of the tensor product of fundamental representations has been studied by many authors who used a variety of methods [1,10,12,14,19]. This question was generalized by Chari [4], who gave a concrete irreducibility condition for a tensor product of Kirillov-Reshetikhin modules of quantum affine algebras.…”

Dual Canonical Bases for the Quantum Special Linear Group and Invariant Subalgebras

“…In [4], a dual basis of the canonical basis of the modified quantum enveloping algebra of type A was investigated under the name global IC basis. In [12], the dual canonical basis of U q (n + ) was constructed by using the so-called quantum shuffles, and was shown to be related to the representation theories of Hecke algebras and quantum affine algebras [13] (see also [19]). In [1] Berenstein and Zelevinsky conjectured a multiplicative property, which states that two dual canonical basis elements b 1 , b 2 of U q (n + ) q-commute if and only if b 1 b 2 = q m b for some b in the dual canonical basis and some integer m. The conjecture was studied by using Hall algebra techniques in reference [21].…”