Heisenberg algebra, wedges and crystals
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Heisenberg algebra, wedges and crystals Thomas Gerber1
Received: 6 March 2017 / Accepted: 8 March 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract We explain how the action of the Heisenberg algebra on the space of qdeformed wedges yields the Heisenberg crystal structure on charged multipartitions, by using the Boson–Fermion correspondence and looking at the action of the Schur functions at q = 0. In addition, we give the explicit formula for computing this crystal in full generality. Keywords Fock space · Categorification · Quantum groups · Heisenberg algebra · Crystals · Symmetric functions · Combinatorics
1 Introduction Categorification of representations of affine quantum groups has proved to be an important tool for understanding many classic objects arising from modular group representation theory, among which Hecke algebras and rational Cherednik algebras of cyclotomic type, and finite classical groups. More precisely, the study of crystals e ) gives and canonical bases of the level Fock space representations Fs of Uq (sl answers to several classical problems in combinatorial terms. In particular, we know e )-crystal graph of Fs can be categorified in the following ways: that the Uq (sl – by the parabolic branching rule for modular cyclotomic Hecke algebras [1], when restricting to the connected component containing the empty -partition, – by Bezrukavnikov and Etingof’s parabolic branching rule for cyclotomic rational Cherednik algebras [23],
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Thomas Gerber [email protected] Lehrstuhl D für Mathematik, RWTH Aachen University, 52062 Aachen, Germany
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– by the weak Harish-Chandra modular branching rule on unipotent representations for finite unitary groups [3,10], for = 2 and s varying. In each case, the branching rule depends on some parameters that are explicitly determined by the parameters e and s of the Fock space. Recently, there has been some important developments when Shan and Vasserot [24] categorified the action of the Heisenberg algebra on a certain direct sum of Fock spaces, in order to prove a conjecture by Etingof [5]. Losev gave in [19] a formulation of Shan and Vasserot’s results in terms of crystals, as well as an explicit formula for computing it in the asymptotic case (see Definition 3.5). Independently and using different methods, the author defined a notion of Heisenberg crystal for higher level Fock spaces [9], that turns out to coincide with Losev’s crystal. An explicit formula was e )-crystal, also given in another particular case, using level-rank duality. Like the Uq (sl the Heisenberg crystal gives some information at the categorical level. In particular, it yields a characterisation of – the finite-dimensional irreducible modules in the cyclotomic Cherednik category O by [24] and [9], – the usual cuspidal irreducible unipotent modular representations of finite unitary groups [4]. This paper solves two remaining problems about the Heisenberg crystal. Firstly, even though it originally arises from the s
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