# Seminars 2018/19

Speaker: Barry Gardner (University of Tasmania)

Time/Date: Monday, July 15, 2019 at 1:00pm

Room: HH-3017

Title: Generalized monoid rings and other ring extensions

Abstract:

We get a skew polynomial ring over a ring $A$ by defining a skew multiplication by a condition $$xa=b+cx$$ and requiring associativity. Then associating with each $a$ the corresponding $c,b$ defines, respectively, an endomorphism $f$ and an $(f,\mathrm{id})$ derivation $d$ of $A$. Conversely for any such $f,d$ we get an associative multiplication by defining $$xa=d(a)+f(a)x.$$ We generalize this to monoid rings, using a family of self-maps of the coefficient ring to define a new multiplication for which coefficients and monoid elements need not commute. In this context what is usually called a skew monoid ring corresponds to a skew polynomial ring for which the derivation is the zero map.

The connections between our generalized monoid rings and some other types of ring extensions - nornalizing extensions and subnormalizing extensions – will also be discussed.

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Speaker: Mr. Abdallah Shihadeh (MUN)

Time/Date: Wednesday, July 10, 2019 at 1:00pm

Room: HH-3017

Title: The Pauli graded modules $M_\lambda^C\otimes V(2n)$

Abstract:

In my previous talks, I discussed the gradings of the weight $\mathfrak{sl}_2(\mathbb{C})$-modules. Then I explained the gradings of the torsion-free $\mathfrak{sl}_2(\mathbb{C})$-modules of rank 1. Then I focused on some new results about the torsion-free $\mathfrak{sl}_2(\mathbb{C})$-module of rank 2. I talked about the construction of a new family of torsion-free $\mathbb{Z}_2^2$-graded modules of rank 2 and their simplicity. The elements of this family denoted by $M_\lambda^C$, $\lambda\in\mathbb{C}$. In this talk, I will mention some new results about the module $L_{(\lambda,2n)}=M_\lambda^C

\otimes V(2n)$, where $V(2n)$ is the simple finite dimensional $\mathfrak{sl}_2(\mathbb{C})$-module of highest weight $2n$. Hence both $V(2n)$ and $M_\lambda^C$ are Pauli graded, it follows that $L_{(\lambda,2n)}$ is also Pauli graded. First, I will give a full description of the module $L_{(\lambda,2)}=M_\lambda^C\otimes

V(2)$, $\lambda\in\mathbb{C}$. I will show that this module has exactly three Casimir constant, which are distinct if $\lambda\notin\{−2,−1,0\}$. Using some Kostant's theorems, I will explain that this module is isomorphic to a direct sum of modules of the form $M_\alpha^C$ for some $\alpha\in\mathbb{C}$. Finally, I will extend the last result to the module $M_\lambda^C\otimes V(2n)$.

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Speaker: Tom Baird (MUN)

Time/Date: Wednesday, February 20, 2019 at 2:00pm

Room: HH-3017

Title: Representation varieties, point counting, and characters of finite general linear groups (Part 2)

Abstract:

Given a Riemann surface $X$ (i.e., a compact, complex curve), we define the character variety $M_n := \mathrm{Hom}(\pi_1(X), \mathrm{GL}_n(\mathbb{C}))/\mathrm{GL}_n(\mathbb{C})$ to be the set of homomorphisms from the fundamental group $\pi_1(X)$ to the complex general linear group $\mathrm{GL}_n(\mathbb{C})$. This $M_n$ is a very rich geometric object with many applications in topology, representation theory, and mathematical physics.

In 2008, Hausel and Rodriguez-Villegas calculated geometric invariants of $M_n$ (the so-called E-polynomial) by counting homomorphisms from $\pi_1(X)$ into general linear groups $\mathrm{GL}_n(\mathbb{F}_q)$ defined over a finite field $\mathbb{F}_q$. Their method makes use of the character theory of $\mathrm{GL}_n(\mathbb{F}_q)$. In this lecture, I will discuss recent work with Michael Lennox Wong applying their techniques to a “real” analogue of $M_n$.

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Speaker: Tom Baird (MUN)

Time/Date: Wednesday, February 13, 2019 at 1:00pm

Room: HH-3017

Title: Representation varieties, point counting, and characters of finite general linear groups

Abstract:

Given a Riemann surface $X$ (i.e., a compact, complex curve), we define the character variety $M_n := \mathrm{Hom}(\pi_1(X), \mathrm{GL}_n(\mathbb{C}))/\mathrm{GL}_n(\mathbb{C})$ to be the set of homomorphisms from the fundamental group $\pi_1(X)$ to the complex general linear group $\mathrm{GL}_n(\mathbb{C})$. This $M_n$ is a very rich geometric object with many applications in topology, representation theory, and mathematical physics.

In 2008, Hausel and Rodriguez-Villegas calculated geometric invariants of $M_n$ (the so-called E-polynomial) by counting homomorphisms from $\pi_1(X)$ into general linear groups $\mathrm{GL}_n(\mathbb{F}_q)$ defined over a finite field $\mathbb{F}_q$. Their method makes use of the character theory of $\mathrm{GL}_n(\mathbb{F}_q)$. In this lecture, I will discuss recent work with Michael Lennox Wong applying their techniques to a “real” analogue of $M_n$.

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Speaker: Abdallah Shihadeh (MUN)

Time/Date: Thursday, February 7, 2019 at 2:00pm

Room: SN-4040

Title: Torsion-free $\mathfrak{sl}_2(\mathbb{C})$-module of rank 2 (Part 2)

Abstract:

In my previous talk, based on a joint work with Yuri Bahturin, I focused on some new results about torsion-free $\mathfrak{sl}_2(\mathbb{C})$-module of rank 2. I talked about the construction of a new family of torsion-free $\mathbb{Z}_2^2$-graded modules of rank 2. We proved that “almost all” of these modules are simple. In this talk, we will prove that the remaining, reducible, modules in this family contain a unique maximal proper submodule, which is graded simple. Moreover, we will prove rather general result about $\mathbb{Z}$-gradings of the torsion-free $\mathfrak{sl}_2(\mathbb{C})$-module of finite rank.

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Speaker: Cameron Ruether (University of Ottawa)

Time/Date: Wednesday, January 23, 2019 at 1:00pm

Room: HH-3017

Title: Rost Multipliers of Kronecker tensor products

Abstract:

We extend the techniques employed by Garibaldi to construct a map $\mathrm{Sp}_{2n}\times\mathrm{Sp}_{2m}\to\mathrm{Spin}_{4nm}$ for all values of $n$ and $m$. We then show how, depending on the parities of $n$ and $m$, this map induces injections between central quotients, for example $\mathrm{PSp}_{2n}\times\mathrm{PSp}_{2m}\hookrightarrow\mathrm{HSpin}_{4nm}$ when $n$ and $m$ are not both odd. Additionally, we calculate the Rost multipliers for each map we have constructed.

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Speaker: Abdallah Shihadeh (MUN)

Time/Date: Wednesday, November 28, 2018 at 1:00pm

Room: HH-3017

Abstract:

In our previous work with Yuri Bahturin and Mikhail Kotchetov, we studied the torsion-free $\mathfrak{sl}_2(\mathbb{C})$-modules of rank 1, and we found that all $\mathfrak{sl}_2(\mathbb{C})$-modules of rank 1 are not graded. In this talk, we will focus on some new results about the torsion-free $\mathfrak{sl}_2(\mathbb{C})$-module of rank 2. We construct a new family of torsion-free $\mathbb{Z}_2^2$-graded modules of rank 2, and we will prove that some of these modules are simple under specific conditions. This is joint work with Yuri Bahturin.

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Speaker: Yuri Bahturin (MUN)

Time/Date: Wednesday, November 14, 2018 at 1:00pm

Room: HH-3017

Title: Graded fields. II

Abstract:

After significant progress in the study of group gradings on simple algebras over algebraically closed fields and real closed fields, it is time to look at more general fields. Here one could benefit from the rich theory of fields and division algebras over arbitrary fields. One could also expect that, conversely, endowing fields and division algebras over them with gradings can be useful for their theory and applications.

In this talk, I would like to present few simple results in this direction, like the classification of gradings on finite fields and gradings of some global and local fields by elementary abelian 2-groups.

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Speaker: Yuri Bahturin (MUN)

Time/Date: Wednesday, October 31, 2018 at 1:00pm

Room: HH-3017

Title: Graded fields

Abstract:

After significant progress in the study of group gradings on simple algebras over algebraically closed fields and real closed fields, it is time to look at more general fields. Here one could benefit from the rich theory of fields and division algebras over arbitrary fields. One could also expect that, conversely, endowing fields and division algebras over them with gradings can be useful for their theory and applications.

In this talk, I would like to present few simple results in this direction, like the classification of gradings on finite fields and gradings of some global and local fields by elementary abelian 2-groups.

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Speaker: Louisa Catalano, Kent State University (USA)

Time/Date: Wednesday, September 12, 2018 at 10:30am

Room: Arts-1045

Title: Maps Charachterized By Action On Equal Products

Abstract:

Let $D$ be a division ring, and let $R = M_n(D)$. Let $m, k \in R$ be fixed invertible elements. We will describe the form of a map $f : R \to R$ satisfying $f(x)f(y) = m$ whenever $xy = k$. Additionally, let $M$ be the set of all $n \times n$ matrices with complex entries, and let $m, k \in M$ be fixed. We will describe a map $g : M \to M$ satisfying $g(x) \circ g(y) = m$ whenever $x \circ y = k$, where $\circ$ denotes the Jordan product.