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Fri Oct 12

Algebraic Representation Theory Seminar

1:25pm - Vincent Hall 206
A 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 Seminar

1:25pm - Vincent Hall 206
A 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 Seminar

1:25pm - Vincent Hall 206
Topology 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 Seminar

1:25pm - Vincent Hall 206
Topology 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 Seminar

1:25pm - Vincent Hall 206
Partition algebras and their action on tensor space
Weiyan Chen, University of Minnesota

We follow section 2.1 of the paper, On blocks of Deligne’s 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 Seminar

1:25pm - Vincent Hall 206
Partition 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 Deligne’s 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 Seminar

1:25pm - Vincent Hall 206
The 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 Seminar

1:25pm - Vincent Hall 206
The 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 Seminar

4:40pm - Vincent Hall 206
Postponed 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 Seminar

4:40pm - Vincent Hall 206
Indecomposable 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 Seminar

4:40pm - Vincent Hall 206
Indecomposable 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 Seminar

4:40pm - Vincent Hall 206
Semisimplicity 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 Seminar

4:40pm - Vincent Hall 206
Jordan 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 Seminar

4:40pm - Vincent Hall 206
Generic Jordan forms and plane partitions
Sam Hopkins, University of Minnesota

We continue the introduction of the Garver-Patrias-Thomas paper by
exploring what the map from a quiver representations to its generic Jordan
form looks like in the case of a Type A quiver, and explain how this map is
related to bijections and generating functions for plane partitions.

Wed Mar 13

Algebraic Representation Theory Seminar

4:40pm - Vincent Hall 206
Postponed
Dongkwan Kim, University of Minnesota

This talk is postponed to March 27.

Wed Mar 27

Algebraic Representation Theory Seminar

4:40pm - Vincent Hall 206
Reflection 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 Seminar

4:40pm - Vincent Hall 206
Reflection 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 Seminar

4:40pm - Vincent Hall 206
Reflection 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 Seminar

4:40pm - Vincent Hall 206
Heaps 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 Seminar

4:40pm - Vincent Hall 206
Heaps 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 Seminar

4:40pm - Vincent Hall 206
Jordan 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.