Advertisement

Partition Algebras and the Invariant Theory of the Symmetric Group

  • Georgia BenkartEmail author
  • Tom Halverson
Chapter
First Online:
Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)

Abstract

The symmetric group \(\mathsf {S}_n\) and the partition algebra \(\mathsf {P}_k(n)\) centralize one another in their actions on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the n-dimensional permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\). The duality afforded by the commuting actions determines an algebra homomorphism \(\varPhi _{k,n}: \mathsf {P}_k(n) \rightarrow \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\) from the partition algebra to the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is a surjection for all  \(k, n \in \mathbb {Z}_{\ge 1}\), and an isomorphism when \(n \ge 2k\).   We present results that can be derived from the duality between \(\mathsf {S}_n\) and \(\mathsf {P}_k(n)\), for example, (i) expressions for the multiplicities of the irreducible \(\mathsf {S}_n\)-summands of \(\mathsf {M}_n^{\otimes k}\), (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra \(\mathsf {P}_k(n)\). When \(2k >n\), the map \(\varPhi _{k,n}\) has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of \(\varPhi _{k,n}\) in terms of the orbit basis of \(\mathsf {P}_k(n)\) and explain how the surjection \(\varPhi _{k,n}\) can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

Keywords

Symmetric group Partition algebra Schur–Weyl duality Invariant theory 

Mathematics Subject Classification (2010)

MSC 05E10 MSC 20C30 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

The authors thank the referee for a careful proofreading and useful suggestions.

References

  1. 1.
    J. Barnes, G. Benkart, T. Halverson, McKay centralizer algebras. Proc. Lond. Math. Soc. 112(2), 375–414 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G. Benkart, M. Chakrabarti, T. Halverson, R. Leduc, C. Lee, J. Stroomer, Tensor product representations of general linear groups and their connections with Brauer algebras. J. Algebra 166(3), 529–567 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. Benkart, T. Halverson, Partition algebras \({\sf P}_k(n)\) with \(2k>n\) and the fundamental theorems of invariant theory for the symmetric group \({\sf S}_n\), J. Lond. Math. Soc. in press. arXiv:1707.1410
  4. 4.
    G. Benkart, T. Halverson, N. Harman, Dimensions of irreducible modules for partition algebras and tensor power multiplicities for symmetric and alternating groups. J. Algebraic Combin. 46(1), 77–108 (2017), arXiv:1605.06543MathSciNetCrossRefGoogle Scholar
  5. 5.
    G. Benkart, D. Moon, Walks on graphs and their connections with tensor invariants and centralizer algebras, J. Algebra 509, 1–39 (2018). arXiv:1610.07837MathSciNetCrossRefGoogle Scholar
  6. 6.
    C. Bowman, M. DeVisscher, R. Orellana, The partition algebra and the Kronecker coefficients. Discrete Math. Theor. Comput. Sci. Proc. AS 321–332 (2013)Google Scholar
  7. 7.
    C. Bowman, M. DeVisscher, R. Orellana, The partition algebra and the Kronecker coefficients. Trans. Amer. Math. Soc. 367, 3647–3667 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    C. Bowman, J. Enyang, and F.W. Goodman, The cellular second fundamental theorem of invariant theory for classical groups, arXiv:1610.09009
  9. 9.
    R. Brauer, On algebras which are connected with the semisimple continuous groups. Ann. of Math. 38(4), 857–872 (1937)MathSciNetCrossRefGoogle Scholar
  10. 10.
    W. Chen, E. Deng, R. Du, R. Stanley, C. Yan, Crossings and nestings of matchings and partitions. Trans. Amer. Math. Soc. 359, 1555–1575 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    J. East, Presentations for (singular) partition monoids: a new approach, in Mathematical Proceedings of the Cambridge Philosophical Society, First View https://doi.org/10.1017/S030500411700069X, Published online: (11 October 2017), pp. 1–14MathSciNetCrossRefGoogle Scholar
  12. 12.
    J. Enyang, Jucys-Murphy elements and a presentation for partition algebras. J. Algebraic Combin. 37, 401–454 (2013)MathSciNetCrossRefGoogle Scholar
  13. 13.
    J. Enyang, A seminormal form for partition algebras. J. Combin. Theory Ser. A 120, 1737–1785 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    J. Farina, T. Halverson, Character orthogonality for the partition algebra and fixed points of permutations. Adv. Appl. Math. 31, 113–131 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    W. Fulton, J. Harris, Representation Theory, A First Course. Graduate Texts in Mathematics, vol. 129 (Springer, New York, 1991)Google Scholar
  16. 16.
    R. Goodman, N.R. Wallach, Representations and Invariants of the Classical Groups. Encyclopedia of Mathematics and Its Applications, vol. 68 (Cambridge University Press, Cambridge, 1998); 3rd corrected printing, Cambridge University Press (2003)Google Scholar
  17. 17.
    T. Halverson, Characters of the partition algebras. J. Algebra 238, 502–533 (2001)MathSciNetCrossRefGoogle Scholar
  18. 18.
    T. Halverson, T. Lewandowski, RSK insertion for set partitions and diagram algebras, Electron. J. Combin. vol. 11, no. 2 (2004/2006), 24 pp (Research Paper 24)Google Scholar
  19. 19.
    T. Halverson, A. Ram, Partition algebras. European J. Combin. 26, 869–921 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    J. Hu, Z. Xiao, On tensor spaces for Birman-Murakami-Wenzl algebras. J. Algebra 324, 2893–2922 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    G. James, A. Kerber, The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and Its Applications, vol. 16 (Addison-Wesley Publishing Co., Reading, Mass, 1981)Google Scholar
  22. 22.
    V.F.R. Jones, The Potts model and the symmetric group, in Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebras (Kyuzeso, 1993) (World Scientific Publishing, River Edge, NJ, 1994), pp. 259–267Google Scholar
  23. 23.
    G. Lehrer, R.B. Zhang, The second fundamental theorem of invariant theory for the orthogonal group. Ann. of Math. 176, 2031–2054 (2012)MathSciNetCrossRefGoogle Scholar
  24. 24.
    G. Lehrer, R.B. Zhang, The Brauer category and invariant theory. J. Eur. Math. Soc. 17, 2311–2351 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    I.G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edn. Oxford Classic Texts in the Physical Sciences (The Clarendon Press, Oxford University Press, New York, 2015)Google Scholar
  26. 26.
    P. Martin, Representations of graph Temperley-Lieb algebras. Publ. Res. Inst. Math. Sci. 26(3), 485–503 (1990)MathSciNetCrossRefGoogle Scholar
  27. 27.
    P. Martin, Temperley-Lieb algebras for non-planar statistical mechanics– the partition algebra construction. J. Knot Theory Ramifications 3, 51–82 (1994)MathSciNetCrossRefGoogle Scholar
  28. 28.
    P. Martin, The structure of the partition algebra. J. Algebra 183, 319–358 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    P. Martin, G. Rollet, The Potts model representation and a Robinson-Schensted correspondence for the partition algebra. Compositio Math. 112, 237–254 (1998)MathSciNetCrossRefGoogle Scholar
  30. 30.
    A. Morales, I. Pak, G. Panova, Hook formulas for skew shapes I. q-analogues and bijections. J. Combin. Theory Ser. A 154, 350–405 (2018)MathSciNetCrossRefGoogle Scholar
  31. 31.
    R. Orellana, M. Zabrocki, Symmetric group characters as symmetric functions, arXiv:1605.06672v3
  32. 32.
    R. Orellana, M. Zabrocki, Products of characters of the symmetric group, arXiv:1709.08098
  33. 33.
    M. Rubey, B. Westbury, A combinatorial approach to classical representation theory, arXiv:1408.3592
  34. 34.
    M. Rubey, B. Westbury, Combinatorics of symplectic invariant tensors, in Proceedings of FPSAC 2015, Discrete Mathematics & Theoretical Computer Science Proceedings of the Association; Discrete Mathematics & Theoretical Computer Science, Nancy (2015), pp. 285–296Google Scholar
  35. 35.
    B.E. Sagan, The symmetric group. Representations, Combinatorial Algorithms, and Symmetric Functions, 2 edn. Graduate Texts in Mathematics, vol. 203 (Springer, New York, 2001), pp. xvi+238Google Scholar
  36. 36.
    R.P. Stanley, Enumerative Combinatorics, vol. 1 (Cambridge, England, 1997)Google Scholar
  37. 37.
    R.P. Stanley, Enumerative Combinatorics, vol. 2 (Cambridge, England, 1999)Google Scholar

Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  1. 1.University of Wisconsin - MadisonMadisonUSA
  2. 2.Macalester CollegeSaint PaulUSA

Personalised recommendations

Cite chapter

Buy options