The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph

Journal of Algebraic Combinatorics volume 52, pages469–504(2020)Cite this article

Abstract

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra \(\Gamma \) 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 \(\Gamma \) 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 \(W^-\subset W_{1+\infty }\), on \(\Gamma \).

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References

  1. 1.

    Beliakova, A., Guliyev, Z., Habiro, K., Lauda, A.D.: Trace as an alternative decategorification functor. Acta Math. Vietnam 39(4), 425–480 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Biane, P.: Representations of symmetric groups and free probability. Adv. Math. 138(1), 126–181 (1998)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Borodin, A., Olshanski, G.: Infinite-dimensional diffusions as limits of random walks on partitions. Probab. Theory Related Fields 144(1–2), 281–318 (2009)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Borodin, A.M.: Multiplicative central measures on the Schur graph, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 240 (Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 2), 44–52, 290–291 (1997)

  5. 5.

    Cautis, S., Lauda, A.D., Licata, A.M., Sussan, J.: W-algebras from Heisenberg categories. J. Inst. Math. Jussieu 17(5), 981–1017 (2018)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cautis, S., Sussan, J.: On a categorical Boson–Fermion correspondence. Comm. Math. Phys. 336(2), 649–669 (2015)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Hill, D., Kujawa, J.R., Sussan, J.: Degenerate affine Hecke–Clifford algebras and type \(Q\) Lie superalgebras. Math. Z. 268(3–4), 1091–1158 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Ivanov, V.N.: The Gaussian limit for projective characters of large symmetric groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 283 no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 6, 73-97, 259 (2001)

  9. 9.

    Ivanov, V.N.: Interpolation analogues of Schur \(Q\)-functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 307 no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 10, 99–119, 281–282 (2004)

  10. 10.

    Ivanov, V.N., Kerov, S.: The algebra of conjugacy classes in symmetric groups, and partial permutations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 256 no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 3, 95–120, 265 (1999)

  11. 11.

    Kac, V.G., Wang, W., Yan, C.H.: Quasifinite representations of classical Lie subalgebras of \(\cal{W}_{1+\infty }\). Adv. Math. 139(1), 56–140 (1998)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Kerov, S.: Transition probabilities of continual Young diagrams and the Markov moment problem. Funktsional. Anal. i Prilozhen. 27(2), 32–49 (1993)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Kerov, S.: Anisotropic Young diagrams and symmetric Jack functions. Funktsional. Anal. i Prilozhen. 34(1), 51–64 (2000)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Khovanov, M.: Heisenberg algebra and a graphical calculus. Fund. Math. 225(1), 169–210 (2014)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Kleshchev, A.: Linear and Projective Representations of Symmetric Groups. Cambridge Tracts in Mathematics, 163. Cambridge University Press, Cambridge (2005)

  16. 16.

    Kvinge, H., Licata, A.M., Mitchell, S.: Khovanov’s Heisenberg category, moments in free probability, and shifted symmetric functions. Algebr. Comb. 2(1), 49–74 (2019)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Lascoux, A., Thibon, J.: Vertex operators and the class algebras of symmetric groups, , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 283 no. Teor. Predst. Din. Sist. Komb. i Algoritm. Metody. 6, 156–177, 261 (2001)

  18. 18.

    Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Classic Texts in the Physical Sciences, vol. 2. The Clarendon Press, Oxford University Press (2015)

    Google Scholar 

  19. 19.

    Nazarov, M.: Young’s symmetrizers for projective representations of the symmetric group. Adv. Math. 127(2), 190–257 (1997)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Okounkov, A., Olshanski, G.: Shifted Schur functions. Algebra i Analiz 9(2), 73–146 (1997)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Ozan Oğuz, C., Reeks, M.: Trace of the twisted Heisenberg category. Comm. Math. Phys. 356(3), 1117–1154 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Petrov, L.: Random walks on strict partitions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 373 no. Teoriya Predstavleniĭ, Dinamicheskie Sistemy, Kombinatornye Metody. XVII, 226–272, 351

  23. 23.

    Read, E.W.: The \(\alpha \)-regular classes of the generalized symmetric group. Glasg. Math. J. 17(2), 144–150 (1976)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Reeks, M.: Cocenters of Hecke–Clifford and spin Hecke algebras. J. Algebra 476, 85–112 (2017)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Sergeev, A.N.: Tensor algebra of the identity representation as a module over the Lie superalgebras \({\rm Gl}(n,\, m)\) and \(Q(n)\). Mat. Sb. (N. S.) 123(165)(3), 422–430 (1984)

    MathSciNet  Google Scholar 

  26. 26.

    Vershik, A.M., Sergeev, A.N.: A new approach to the representation theory of the symmetric groups. IV. \(Z_2\)-graded groups and algebras: projective representations of the group \(S_n\). Mosc. Math. J. 8(4), 813–842 (2008)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Wan, J.: Completely splittable representations of affine Hecke–Clifford algebras. J. Algebraic Combin. 32(1), 15–58 (2010)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Wan, J., Wang, W.: Lectures on spin representation theory of symmetric groups. In: Proceedings for Taipei Winter School 2010. Bulletin of Institute of Mathematics Academia Sinica, vol. 7, pp. 91–164 (2012)

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Affiliations

  1. Pacific Northwest National Laboratory, Seattle, WA, USA

    Henry Kvinge

  2. Galatasaray University, Istanbul, Turkey

    Can Ozan Oğuz

  3. Bucknell University, Lewisburg, PA, USA

    Michael Reeks

Authors
  1. Henry Kvinge
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  2. Can Ozan Oğuz
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  3. Michael Reeks
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Correspondence to Michael Reeks.

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Kvinge, H., Oğuz, C.O. & Reeks, M. The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph. J Algebr Comb 52, 469–504 (2020). https://doi.org/10.1007/s10801-019-00910-w

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Keywords

  • Hecke algebras
  • Spin representation theory
  • Schur Q-functions
  • Schur graph