Imaginary Schur-Weyl Duality
Alexander Kleshchev, Robert Muth, "Imaginary Schur-Weyl Duality"
2017 | ISBN-10: 1470422492 | 83 pages | PDF | 1 MB
2017 | ISBN-10: 1470422492 | 83 pages | PDF | 1 MB
The authors study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules–one for each real positive root for the corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible imaginary modules–one for each $l$-multiplication. The authors study imaginary modules by means of "imaginary Schur-Weyl duality" and introduce an imaginary analogue of tensor space and the imaginary Schur algebra. They construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra, and construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.