Imaginary vectors in the dual canonical basis of Uq(n)

By Bernard Leclerc. 2002

95 citations recorded by the source corpus.

Let n be a maximal nilpotent subalgebra of a simple complex Lie algebra g. We introduce the notion of imaginary vector in the dual canonical basis of Uq(n), and we give examples of such vectors for types An(n ~ 5), Bn(n ~ 3), Cn(n ~ 3), Dn(n ~ 4) , and all exceptional types. This disproves a conjecture of Berenstein and Zelevinsky about q-commuting products of vectors of the dual canonical basis. It also shows the existence of finite-dimensional irreducible representations of quantum affine algebras whose tensor square is not irreducible.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6b5b6a68ff80aede8d77716229bdd2d544913ccc

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Citations in the corpus

Monoidal categorification of cluster algebras (2014, background): “Leclerc conjecture Recall that R is assumed to be a symmetric KLR algebra over a base field k. 3.1.”
Simplicity of heads and socles of tensor products (2014, background): “an automorphism of the set of the isomorphism classes of simple U′ q(g)-modules (Corollary 3.14). The inverse is given by N→ N⋄∗M, where ∗Mis the right dual of M. It is an analogue of Conjecture 2 in [13] originally stated for global bases. Acknowledgements. We would like to thank Bernard Leclerc fo”
Positivity and canonical bases in rank 2 cluster algebras of finite and affine types (2003, background): “This contrast between the “real” generators ym and the “imaginary” generators zn seems to be a special case of a general phenomenon taking place in any cluster algebra of infinite type (cf. [ 14 ]).”
Tensor diagrams and cluster algebras (2012, background): “Leclerc [34]); “fake” exchange relations involving web invariants; and the aforementioned negative structure constants.”
Cluster algebras and representation theory (2010, result): “al simple modules. For g= sl2, all simple Uq(Lg)-modules are real. However for g6= sl2 there exist imaginary simple Uq(Lg)-modules (i.e. simple modules whose tensor square is not simple), as shown in [45]. This is consistent with the expectation that a cluster algebra with inﬁnitely many cluster var”
Semicanonical bases and preprojective algebras (2004, background): “In [31] counterexamples to the BZ-conjecture were found for all types other than 1 2 with @ A .”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, methodology): “Using his quantum shuﬄe algo-rithm from [25], in [26] Leclerc constructs ﬁve counterexamples.”
Representations of Algebras and Related Topics (2011, result): “simple modules. For g = sl 2, all simple Uq(Lg)-modules are real. However for g 6= sl 2 there exist imaginary simple Uq(Lg)-modules (i.e.simple modules whose tensor square is not simple), as shown in [29]. This is consistent with the expectation that a cluster algebra with inﬁnitely many cluster var”
Preprojective algebras and cluster algebras (2008, background): “Keywords. preprojective algebra, cluster algebra, flag variety, rigid module, mutation, Frobenius category, semicanonical basis”