We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra Γ of the symmetric functions generated by odd power sums. We give a graphical description of the factorial Schur Q-functions and inhomogeneous power sums as closed diagrams in the twisted Heisenberg category, and show that the bubble generators of the center correspond to two sets of generators of Γ which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the W -algebra W − ⊂ W 1+∞ , on Γ.
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Let CAn=C[S2≀S2≀⋯≀S2] be the group algebra of an n-step iterated wreath product. We prove some structural properties of An such as their centers, centralizers, and right and double cosets. We apply these results to explicitly write down the Mackey theorem for groups An and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of ⨁m≥0(Am,An)−bimodules. A complete description of the category is an open problem.
be the group algebra of n step iterated wreath product. We prove some structural properties of An such as their centers, centralizers, right and double cosets. We apply these results to explicitly write down Mackay theorem for groups An and give a partial description of the natural transformations between induction and restriction functors on the representations of the iterated wreath product tower by computing certain hom spaces of the category of m≥0 (Am, An)−bimodules. A complete description of the category is an open problem. Contents 1. Introduction 1 1.1. Acknowledgements 3 2. Rooted trees and iterated wreath products 3 3. Structural properties of the iterated wreath products of S 2 7 4. Natural transformation between induction and restriction 13 4.1. Mackey Theorem 14 4.2. Morphism spaces as vector spaces 15 4.3. Algebra structure of morphism spaces 17 5. Future directions 19 References 19
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