The center of the twisted Heisenberg category, factorial Schur Q -functions, and transition functions on the Schur graph
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The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph Henry Kvinge1 · Can Ozan Oguz ˘ 2 · Michael Reeks3 Received: 28 June 2018 / Accepted: 4 October 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra 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 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 − ⊂ W1+∞ , on . Keywords Hecke algebras · Spin representation theory · Schur Q-functions · Schur graph
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The Schur graph and Sergeev algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The subalgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The twisted Heisenberg category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Michael Reeks [email protected]
1
Pacific Northwest National Laboratory, Seattle, WA, USA
2
Galatasaray University, Istanbul, Turkey
3
Bucknell University, Lewisburg, PA, USA
123
Journal of Algebraic Combinatorics
1 Introduction In [14], Khovanov describes a linear monoidal category H which conjecturally categorifies the Heisenberg algebra. The morphisms of H are governed by a graphical calculus of planar diagrams. This category has connections to many interesting areas of representation theory and combinatorics. The trace of H, which can be defined diagrammatically as the algebra of diagrams on the annulus, is shown in [5] to be isomorphic to the W -algebra W1+∞ at level one. The center of H, which is the algebra EndH (1) of endomorphisms of the monoidal identity, is shown in [16] to be isomorphic to the algebra of shifted symmetric functions ∗ of Okounkov and Olshanski [20]. A twisted version of Khovanov’s Heisenberg category was introduced by Cautis and Sussan [6]. The twisted Heisenberg category Htw is a C-linear additive monoidal category, with an additional Z/2Z-grading. It conjecturally categorifies the twisted Heisenberg algebra. The center of Htw , EndHtw (1), was studied in [21] where it was shown that as a commutative C-algebra, EndHtw (1) ∼ = C[d0 , d2 , d4 , . . . ] ∼ = C[d¯2 , d¯4 , d¯6 , . . . ], where d2k and d¯2k are certain clockwise and counterclockwise bubble generators,
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