## Seminar Categories

## Current Series

Fri Oct 12 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206A theorem of Nakaoka on the homology of symmetric groups Peter Webb, University of Minnesota A theorem of Nakaoka states that the inclusion of symmetric groups induces an inclusion of group homology provided that the degrees of the symmetric groups are large enough. A proof of this statement in homological stability can be given using the equivariant cohomology of the complex of injective words. I will start describing the machinery needed to give this proof, trying to assume as little as I can. |

Fri Oct 19 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206A theorem of Nakaoka on the homology of symmetric groups - continuation Peter Webb, University of Minnesota A theorem of Nakaoka states that the inclusion of symmetric groups induces an isomorphism of group homology provided that the degrees of the symmetric groups are large enough. This week I will continue with the proof of this, explaining the equivariant cohomology spectral sequence and the specific calculations that arise, trying to assume as little as I can. |

Fri Oct 26 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206Topology and combinatorics of the complex of injective words Victor Reiner, University of Minnesota According to a result of Farmer, the homology of this complex vanishes except in the bottom and top dimensions. This property is a consequence of shellability of the complex. We will discuss these matters, used in a proof of Nakaoka's theorem on stability of the corestriction map in the group homology of the symmetric groups. |

Fri Nov 02 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206Topology and combinatorics of the complex of injective words - continuation Victor Reiner, University of Minnesota I will continue the elementary treatment of properties of this complex, indicating situations where it arises, including a context considered by myself and Peter Webb. |

Fri Nov 09 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206Partition algebras and their action on tensor space Weiyan Chen, University of Minnesota We follow section 2.1 of the paper, On blocks of Delignes category Rep(S_t), by Comes and Ostrik. The longer term goal is to describe Deligne's category. In section 2.1 the partition algebras are introduced. They arose in work of Martin in a context of statistical mechanics, and were later shown by Jones to be in Schur-Weyl duality with the group algebras of symmetric groups. We introduce this theory. |

Fri Nov 16 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206Partition algebras and their action on tensor space - continuation Weiyan Chen, University of Minnesota We continue with section 2.1 of the paper, On blocks of Delignes category Rep(S_t), by Comes and Ostrik. The longer term goal is to describe Deligne's category. In section 2.1 the partition algebras are introduced. They arose in work of Martin in a context of statistical mechanics, and were later shown by Jones to be in Schur-Weyl duality with the group algebras of symmetric groups. The exposition is introductory. |

Fri Nov 30 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206The definition of Deligne's category Rep(S_t) Cecily Santiago, University of Minnesota Following section 2.2 of the paper, On blocks of Deligne's category Rep(S_t), by Comes and Ostrik, we give the definition of this category and explore some of its first properties. |

Fri Dec 07 |
## Algebraic Representation Theory Seminar1:25pm - Vincent Hall 206The definition of Deligne's category Rep(S_t) - continuation Cecily Santiago, University of Minnesota We will complete the steps in the definition of Deligne's category described in section 2.2 of the paper, On blocks of Deligne's category Rep(S_t), by Comes and Ostrik. |

Wed Jan 30 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Postponed to next week Peter Webb, University of Minnesota We will start on Section 3 of the paper by Comes and Ostrik, On blocks of Deligne's category Rep(S_t). This will be preceded by a review of the set-up in this category, which is built up out of partition algebras. Section 3 begins with the result that the indecomposable objects are parametrized by partitions of all sizes. |

Wed Feb 06 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Indecomposable objects in Deligne's category Rep(S_t) Peter Webb, University of Minnesota We will start on Section 3 of the paper by Comes and Ostrik, On blocks of Deligne's category Rep(S_t). This will be preceded by a review of the set-up in this category, which is built up out of partition algebras. Section 3 begins with the result that the indecomposable objects are parametrized by partitions of all sizes. |

Wed Feb 13 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Indecomposable objects in Deligne's category Rep(S_t) - continuation Peter Webb, University of Minnesota We continue with Section 3 of the paper by Comes and Ostrik, On blocks of Deligne's category Rep(S_t). Section 3 begins with the result that the indecomposable objects are parametrized by partitions of all sizes, and this has to do with a description of the idempotents in the partition algebras. We use a fundamental relationship that goes back to Green, between idempotents an algebra, and idempotents in a factor algebra by an idempotent, and in the algebra obtained by cutting by that idempotent. |

Wed Feb 20 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Semisimplicity of Deligne's category Rep(S_t) Peter Webb, University of Minnesota Continuing to follow the paper of Comes and Ostrik, On blocks of Deligne's category Rep(S_t), we prove the result in Section 3 that the category is semisimple except for countably. many values of t. The proof is an interesting use of the characterization of semisimple algebras by non-degeneracy of the trace form. |

Wed Feb 27 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Jordan form data of quiver representations Sam Hopkins, University of Minnesota We introduce the paper of Garver, Patrias and Thomas: Minuscule reverse plane partitions via quiver representations, starting with a review of aspects of quiver representations. |

Wed Mar 06 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Generic Jordan forms and plane partitions Sam Hopkins, University of Minnesota We continue the introduction of the Garver-Patrias-Thomas paper by |

Wed Mar 13 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Postponed Dongkwan Kim, University of Minnesota This talk is postponed to March 27. |

Wed Mar 27 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Reflection functors Dongkwan Kim, University of Minnesota We define the notion of reflection functors between the categories of quiver representations and discuss some of their basic properties. After this, we explain how these functors are related to (generic) Jordan types and corresponding representations of quivers. |

Wed Apr 03 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Reflection functors and nilpotent endomorphisms Dongkwan Kim, University of Minnesota We describe how the Jordan form data of quiver representations interact with reflection functors, continuing with the paper of Garver, Patrias and Thomas: Minuscule reverse plane partitions via quiver representations. |

Wed Apr 10 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Reflection functors and nilpotent endomorphisms - continuation Dongkwan Kim, University of Minnesota We describe how the Jordan form data of quiver representations interact with reflection functors, continuing with the paper of Garver, Patrias and Thomas: Minuscule reverse plane partitions via quiver representations. |

Wed Apr 17 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Heaps and minuscule posets Victor Reiner, University of Minnesota We review minuscule weights, their weight posets, and minuscule heaps, with the goal of eventually understanding Section 4 of the paper by Garver, Patrias and Thomas. |

Wed Apr 24 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Heaps and minuscule posets - coontinuation Victor Reiner, University of Minnesota We continue reviewing minuscule weights, their weight posets, and minuscule heaps, with the goal of eventually understanding Section 4 of the paper by Garver, Patrias and Thomas |

Wed May 01 |
## Algebraic Representation Theory Seminar4:40pm - Vincent Hall 206Jordan form data from the perspective of the derived category Peter Webb, University of Minnesota We show that certain sets of representations of Dynkin quivers are determined by their Jordan form data. The description is phrased in terms of the bounded derived category of representations of the quiver. This continues the study of the paper of Garver, Patrias and Thomas: Minuscule reverse plane partitions via quiver representations. |