Organized by Fan Zhou, Cailan Li and Alvaro Martinez. Please email Cailan at ccl2166@columbia.edu to join the mailing list.

- Time/date: Thursdays at 4.40pm-6.30pm ET.
- Location: Room 528 at the Columbia Math Department.

This is a learning seminar on representation theory, often with an eye towards categorification. There will be talks relating to quantum groups, Hecke algebras, symmetric functions, diagram algebras, and (in some cases) their categorifications.

Talks will roughly be 45 minutes followed by a 5 minute break followed by 45 more minutes. Please email Cailan at ccl2166@columbia.edu if you'd like to give a talk.

Thu Sep 7 |
Fan Zhou The nilBrauer algebra and semi-highest weight theory Abstract: We discuss a modification of highest weight theory, “stratification theory”, introduced by Brundan-Stroppel, as well as the closely related theory of graded triangular bases, and discuss the nilBrauer algebra as an application of this theory. We also discuss some work-in-progress on the BGG resolution in this context. Notes. |

Thu Sep 14 |
Fan Zhou Stratification and recollement of (stable infinity) categories and the "reconstruction philosophy" of BGG resolutions Abstract: This very general talk, while foreshadowed by the last talk, will be independent. We discuss stratifications of (stable infinity, but one can pretend triangulated more or less) categories first in great generality and then applied to recollements of representation categories. We then discuss how this can be used to give BGG-resolution-type results. Time permitting, we may discuss some more work-in-progress that we didn't get to discussing last time. Notes. |

Thu Sep 21 |
Alvaro Martinez The dg trace Abstract: To a braid, one can associate a complex of certain bimodules over a polynomial ring, called Soergel bimodules. The algbraic counterpart of 'closing up the braid' is taking the 'trace' of this category, a general algebraic construction. In this talk, we will explain how this trace category fails to meet expectations and why, instead, we need a certain dg structure for things to work. Notes. |

Thu Sep 28 |
Alvaro Martinez The dg trace (continued) Abstract: We formally define the dg trace and use it to fix the problem that we encountered last time. Notes. |

Thu Oct 05 |
Nicolas Jaramillo Torres Diagrammatics of Folded Soergel Bimodules Abstract: In this talk we show a diagrammatic presentation for the equivariantization (or 'folding') of the category of Soergel Bimodules of type A1xA1. Even though this is the smallest case of folding, we were surprised to find out there are a lot more relations than we expected. We will also show our future approaches to the subject: getting a light leaves basis, categorically diagonalize the full twist of this category, present other equivariantizations and more. Notes. |

Thu Oct 12 at 12.30pm ET, room 332 in Uris Hall |
Paul Wedrich On skein theory in dimension four The Temperley-Lieb algebra describes the local behaviour of the Jones polynomial and gives rise to the Kauffman bracket skein modules of 3- manifolds. Going up by one dimension, Bar-Natan's dotted cobordisms describe the local behaviour of Khovanov homology and, likewise, give rise to skein modules of 4-manifolds. I will describe how to compute such skein modules via a handle decomposition in terms of link homology in the 3-sphere. Based on joint work with Morrison-Walker and Manolescu-Walker. Notes. |

Thu Oct 19 |
Kevin Chang Shuffle algebras and perverse sheaves Abstract: I will report on work of Kapranov-Schechtman that proves an equivalence of categories between certain Hopf algebras and "factorizable" perverse sheaves on symmetric powers on \mathbb{C}. Along with discussion of the equivalence, I will talk about some notable examples like shuffle algebras and Nichols algebras and their connection with topology and algebraic geometry. Notes. |

Thu Nov 2/td> |
Cailan Li The Temperley Lieb Algebra of Type B Abstract: We give a diagrammatic description of the Temperley Lieb Algebra of type B in terms of decorated tangles following Green. We then talk about the related blob algebra and maybe some of its representation theory. Notes. |

Thu Nov 9 |
Cailan Li A graphical calculus for Lusztig's dual canonical basis Abstract: We go over a part of Khovanov's thesis. Specifically we will show how Khovanov gives a graphical interpretation of Lusztig's dual canonical basis in the tensor product of the standard representation for U_q(sL_2) using the Temperley Lieb algebra. Notes. |

Thu Nov 16 |
Minh-Tam Trinh DAHA dualities and y-ification Abstract: In their original paper introducing y-ification of KhR homology, Gorsky–Hogancamp calculated the y-ified homology of the (n, nk) torus link for any n, k > 0. I will explain how the resulting q, t-symmetric function, which has two sets of symmetric variables, also shows up in a formula of Carlsson–Mellit, and how the latter is related to a DAHA bimodule studied by Boixeda Alvarez–Losev, both arising from the geometry of affine Springer fibers. I will then sketch recent works, joint with Oscar Kivinen and Ting Xue, that respectively generalize these stories. Notes. |

Thu Nov 30 |
Fan Zhou Affine braids, Markov traces and category O Abstract: We roughly follow a paper of Rosa Orellana and Arun Ram of the same title. A functor of taking highest weight vectors of a certain tensor product is constructed to produce, from modules in category O, modules admitting an affine braid/Hecke action. Depending on the precise setting this action may factor through other algebras. This functor is exact and therefore carries BGG resolutions to resolutions which categorify Jacobi-Trudy formulae; however, the stratification/highest-weight theory appears to be carried from category O and not native. Notes. |

Thu Dec 7 |
Pedro Vaz Affine braids, Markov traces and category O Abstract: Decomposing tensor powers of a representation (of a group, say) is a hard task and in general there is no general way of computing the summands explicitely. In this talk I will explain how to count the number of the appearing summands asymptotically in certain monoidal categories with finitely many indecomposable objects. In the last part of the talk I will explain how to generalize the method to categories with infinitely many indecomposable objects. This is joint work with Abel Lacabanne and Daniel Tubbenhauer. Notes. |