Notes from Seminar on Deligne Categories, MIT, Spring 2013


Organizer: Pavel Etingof

Useful links:

P. Deligne's paper on Rep(S_t) (definitions of Rep(GL_t), Rep(O_t) given in the end)
Paper by J. Comes and V. Ostrik on Rep(S_t)
Paper by J. Comes and B. Wilson on Rep(GL_t)
Paper by J. Comes and J. Kujawa on modified traces on Rep(S_t) and an invariant of framed knots
Paper by J. Comes on ideals in Rep(GL_t)
Paper by A. Mathew on Deligne categories and extrapolation of dAHA
Paper by I. Entova-Aizenbud on extrapolation of rational Cherednik algebras of type A
J. Comes and V. Ostrik's paper on the abelian envelope of Rep(S_t)
Papers by F. Knop: [1] , [2]
Paper by M. Mori

More papers on Deligne categories (added later):

Papers by P. Etingof: "Representation Theory in Complex Rank", Part 1, Part 2 (includes many open problems)
Paper by I. Entova-Aizenbud on Schur-Weyl duality for Deligne categories (and its sequel)
An application of Deligne categories to the study of Kronecker coefficients, by I. Entova-Aizenbud
Paper by N. Harman on generators for the Grothendieck ring of Rep(S_t) and some wreath products
Paper by L. Sciarappa on simple commutative algebras in Rep(S_t)
Applications of Rep(GL_t) to the study of representations of superalgebras by T. Heidersdorf
Paper by J. Comes and T. Heidersdorf on ideals in Rep(O_t)
Paper by A. Del Padrone on integral objects and Deligne's category Rep(S_t)



  • February 12th, 2013, 4:30-5:30, Room 2-136
  • Pavel Etingof (MIT)
  • Deligne Categories
  • I will give an introduction to the works of Deligne, who constructed extrapolations $O(n,C)$ and $Sp(2n,C)$) to complex values of $n$. Then I will describe how to generalize this to $GL_n(F_q)$, real groups (i.e., Harish-Chandra modules), Lie superalgebras, degenerate affine and double affine Hecke algebras, current algebras, Yangians, and other representation theoretical settings. Finally, I will describe some open problems. This will be a beginning of a student seminar on this subject, which will operate on Tuesdays.

    Links:
    Notes by P. Etingof
    Similar talk by P. Etingof on video (in Newton Institute)
    Other relevant material:
    P. Deligne's paper on Tannakian categories
    Notes from a lecture by V. Ostrik on tensor categories (after P. Deligne's work)

  • February 26th, 2013
  • Inna Entova-Aizenbud (MIT)
  • Construction of Rep(S_t)
  • We will give the explicit construction of Deligne's categories Rep(S_t), which are extrapolations to complex t of the categories of finite dimensional representations of the symmetric groups.

    Links:
    Notes from the lecture, with some remarks from the lecture incorporated (slightly messy, but hopefully readable).
    The exposition followed the paper by J. Comes and V. Ostrik on Rep(S_t)
    Paper by T. Halverson and A. Ram on partition algebras


  • March 5th, 2013
  • No meeting


  • March 12th, 2013
  • Inna Entova-Aizenbud (MIT)
  • Indecomposable objects in Deligne's category Rep(S_t)
  • We will continue with the study of Deligne's category Rep(S_t), t being a complex number. In this talk, we will discuss the structure of indecomposable objects in Rep(S_t), focusing on the blocks of Rep(S_{t=n}) for non-negative integer n, and the connection between Rep(S_t) and Rep(S_T), the latter being Deligne's category with a formal parameter T (defined over the field of Laurent series in (T-t) over the field of complex numbers). The exposition will follow the paper by Comes and Ostrik.

    Lecture notes will be posted together with the notes for next lecture.


  • March 19th, 2013
  • Inna Entova-Aizenbud (MIT)
  • Blocks in Deligne's category Rep(S_t)
  • The talk will focus on the blocks of Deligne's category Rep(S_t), t being a complex number. We will pick up where we left off last time, describe explicitly the lifting of indecomposable objects from Rep(S_t) to Rep(S_T), and derive a description of the blocks of Rep(S_{t=d}) for non-negative integer d using the notion of a d-core of a partition. The exposition will follow the paper by Comes and Ostrik.

    Links:
    Notes from the lecture.




  • April 2nd, 2013
  • Inna Entova-Aizenbud (MIT)
  • Construction of Deligne's category Rep(GL_t)
  • We will construct Deligne's category Rep(GL_t(C)), and describe the connection between Rep(GL_d(C)) and representations of superalgebras gl(m|n) with m-n=d. The talk will follow the works of P. Deligne, J. Comes and B. Wilson.
  • Note that this lecture does not have any prerequisites and is independent of the previous ones.

    Links:
    Notes from the lecture.



  • April 9th, 2013
  • Inna Entova-Aizenbud (MIT)
  • Indecomposable objects in Deligne's category Rep(GL_t)
  • We will discuss the indecomposable objects in Deligne's category Rep(GL_t(C)). These indecomposable objects are classified using idempotents in walled Brauer algebras, which arise in the construction of Rep(GL_t(C)) as certain Hom-spaces. We will also expalin how to lift indecomposable objects from Rep(GL_t) to Rep(GL_T), the latter being Deligne's category with a formal parameter T (defined over the field of Laurent series in (T-t) with complex coefficients).
    The talk will follow the work of J. Comes and B. Wilson.

    Links:
    Notes from the lecture.




  • April 23rd, 2013
  • Inna Entova-Aizenbud (MIT)
  • Abelian envelope of Deligne's category Rep(S_t)
  • We will define the abelian envelope Rep^{ab}(S_{t=d}) of the pseudo-abelian (Karoubian) category Rep(S_{t=d}), and give an explicit construction of this category, along with its universal property. We will follow a paper by Comes and Ostrik.

    Links:
    J. Comes and V. Ostrik's paper
    Notes from the lecture.




  • April 30th, 2013
  • Akhil Mathew (Harvard)
  • Wreath products and Deligne categories
  • In this talk, we will describe Knop's generalization of Deligne's construction, which produces a tensor category out of a regular category along with some combinatorial data. Knop's tensor categories can be used to "interpolate" representation categories of wreath products S_n \rtimes G^n. We will follow the papers of Knop, "Tensor envelopes of regular categories" and "A construction of semisimple tensor categories."

    Links:
    Notes from the lecture.
    See related papers by F. Knop: [1] , [2] and M. Mori




  • Friday May 3rd, 4-6 p.m., Room 1-142 (note room and time change!)
  • Inna Entova-Aizenbud (MIT)
  • Representations of the rational Cherednik algebra of type A in complex rank
  • We study a family of abelian categories $O_{c, t}$ depending on complex parameters $c, t$ which are interpolations of the $O$-category for the rational Cherednik algebra $H_c(t)$ of type $A$, where $t$ is a positive integer. This interpolation is based on Deligne's construction of the category Rep(S_t), which was discussed in Deligne Categories Seminar.

    Links:
    Notes from the lecture.