You are here

Algebra Seminar

The Algebra Seminar provides graduate students, advanced undergraduate students, and faculty a forum to discuss and/or present Research, Problems, Applications, or Exposition in Algebra. For more information contact Arturo Magigin.

Spring 2020

During the Spring 2020 semester we will meet on Tuesdays from 3:30-4:20 in Maxim Doucet Hall, room 203.

  • 11 February 2020
    The Chermak-Delgado Lattice and generalizations I: An introduction
    Arturo Magidin (UL Lafayette)
    In 1989, Andrew Chermak and Alberto Delgado introduced a "measure" for subgroups of a group $G$ being acted on by a group $H$: given a subgroup $A\leq G$, $m_r(A) = |A|^r|C_H(A)|$. The set of subgroups for which the measure achieves its largest value has interesting properties: for example, it forms a sublattice of the lattice of subgroups of $G$, is self-dual, and all its elements are equal to their double centralizers.
    L-C Kappe, Elizabeth Wilcox, William Cocke and are currently exploring some generalizations, replacing the centralizer with some other constructions. As a lead-up to discussing some of those results, I will discuss the properties of the measure and some of the results about the lattice.
  • 18 February 2020
    The Chermak-Delgado Lattice and generalizations II: A digression into words and margins
    Arturo Magidin (UL Lafayette)
    The center of a group and the commutator subgroup play roles that are "dual" to each other in a way that was formalized by Philip Hall in the 1940s, as part of a proposed scheme to classify finite $p$-groups. He also indicated how these notions could be generalized to other subgroups, including but not restricted to terms of the upper and lower central series. These are the marginal and verbal subgroups; the latter were well known at the time, but the former and the connection between them was new.
    One can view centralizer subgroups as a kind of "relative center"; the point of departure for our attempts at generalizing the Chermak-Delgado construction lattice is trying to take this view and apply it to the general setting of marginal and verbal subgroups, by defining a "relative marginal subgroup".
    To put this in context, I will give an introduction to verbal and marginal subgroups.
  • 3 March 2020
    The Chermak-Delgado Lattice and generalizations III
    Arturo Magidin (UL Lafayette)
  • 17 March 2020 (CANCELLED)
    Lower Finite Modules over Commutative Rings
    Luke Harmon (University of Colorado Colorado Springs)
    Abstract: A partially ordered set (P,lt) is lower finite provided P is infinite and for each x in P, there are but finitely many elements y in P such that y lt x. We will call a module M lower finite if the set of proper submodules of M, partially ordered by set-theoretic containment, is lower finite. We will use the (well-studied) class of Jonsson modules (along with other classical results) to classify the lower finite modules over a commutative ring with identity.

    Fall 2019

    During the Fall 2019 semester we will meet on Mondays from 4:00-5:00 in Maxim Doucet Hall, room 201.

    • 16 September 2019
      A Partial Order on Subsets of Baer Bimodules with Applications to C*-Modules
      Gary Birkenmeier
      Abstract: In a series of seminars, the concept of Baer (p, q)-sets is introduced. Using this notion, we define Rickart, Baer, quasi-Baer and pi-Baer (S, R)-bimodules, respectively. We show how these conditions relate to each other. We also develop new properties of the minus partial order \leq-, we extend the relation \leq- to (S, R)-bimodules, and use it to characterize the aforementioned Rickart, Baer, quasi-Baer, and pi-Baer (S, R)-bimodules. Note that \leq- generalizes the natural order on the projections of a C*-algebra. Moreover we specify subsets K of the power set of a (S, R)-bimodule for which \leq- is a partial order and for which \leq- determines a lattice. We analyze the relation \leq- by examining the associated Baer (p, q)- sets. Finally, we apply our results to C*-modules. Examples are provided to illustrate and delimit our results.
    • 23 September 2019
      Gary Birkenmeier
    • 30 September 2019
      Gary Birkenmeier
    • 7 October 2019
      Gary Birkenmeier
    • 14 October 2019
      Normal subgroups of invertibles and of unitaries in C^*-algebras
      Leonel Robert Gonzalez
      Abstract: I will talk about my work on the normal subgroups of the group of invertibles in the connected component of the identity in a C*-algebra.
    • 4 November 2019
      Algebraic properties of modular tensor categories
      Yilong Wang
      Louisiana State University
      Abstract: Modular tensor categories (MTC) are categorical generalizations of finite abelian groups with non-degenerate quadratic forms. They give rise to invariants of links and 3-manifolds that has similar "quantum" nature as the Jones polynomial. They also provide representations of SL(2, Z/NZ) and mapping class groups of higher genus surfaces which can be viewed as derived from (2+1)-topological quantum field theories (TQFTs) associated to the MTCs. MTCs are also indispensable ingredients in the setup of the mathematical foundations for topological quantum computation.
      In this talk, we will introduce the notion of modular tensor categories with an emphsis on its rich connection with representation theory, (2+1)-TQFTs and topological quantum computing. Then we will present our recent work on the algebraic properties of MTCs and the TQFTs associated to them.
    • 11 November 2019
      Principal Series Representations of GL(2) Over Finite Fields
      Regina Poderzay
      UL Lafayette
      Abstract: The goal of this talk is to construct the principal series representations of GL(2). To begin, we observe rudimentary results and examples using prerequisite knowledge from linear algebra and group theory. This is followed by inducing new representations from old ones. A key component in the classification of induced representations is Mackey’s theorem. At last is where we construct the principal series representations. This construction is motivated by the Bruhat decomposition of GL(2) into the Borel subgroup and is achieved by counting the conjugacy classes of GL(2,q).
    • 22 November 2019 (Friday, 3:00-4:00)
      Linearly Presented Toric Edge Ideals of Bipartite Graphs
      Jason McCullough
      Iowa State University
      Abstract: Let G = (V,E) be a finite simple graph and let k be a field. Denote by k[E] the polynomial ring with edges of G identified with variables and k[V] the polynomial ring with vertices of G identified with variables. The toric edge ring of G is the k-subalgebra of k[V] generated by v_i*v_j for all {i,j} in E. If we take the canonical surjection from k[E] to k[G], the kernel defines a (prime) toric ideal called the toric edge ideal I_G of G. In the case when G is a complete bipartite graph K_m,n, I_G is the defining ideal of the Segre embedding of P^(m-1) x P^(n-1) in P^(mn - 1). It is well-known that generators of I_G correspond to even closed walks in G. Hibi and Herzog previously characterized those bipartite graphs G such that I_G is generated by quadratic polynomials. We present a combinatorial description of those ideals whose ideals are linearly presented. This result has connections to polyomino ideals and Hibi rings that I will explain. This is joint work with Zach Greif.

    Spring 2019

    During the Spring 2019 semester we will meet on Fridays from 2:10-3:10 in Maxim Doucet Hall, room 210.

    • 1 February 2019
      The structure of a minimal counterexample to the Feit-Thompson theorem
      George Glauberman
      University of Chicago
      Abstract: This talk supplements my colloquium talk by describing in more detail the proof of the Feit-Thompson Theorem.
    • 15 February 2019
      Minimal bisets for fusion systems and blocks of finite group algebras, I
      Justin Lynd
      Abstract: I will discuss the background for ongoing work with M. Gelvin (Bilkent) in which we investigate the question: which finite groups are minimal bisets for their fusion systems? As it turns out, this question is closely related to the following question in the modular representation theory of finite groups: when k is an algebraically closed field of characteristic p, for which finite groups G is the group algebra kG indecomposable as a k-algebra?
      A saturated fusion system F over a finite p-group S is a finite category with objects the subgroups of S and with morphisms between subgroups which are "conjugation-like" group homomorphisms. Here, S plays the role of a Sylow p-subgroup of the category. In the first talk, I will give the definition of a saturated fusion system F. Although F need not be the localization of any finite group G, one can define an S-S biset associated with F that plays the same role as the S-S biset G, whenever F is the localization of G at p.
    • 8 March 2019
      Minimal bisets for fusion systems and blocks of finite group algebras, II
      Justin Lynd
    • 15 March 2019
      Minimal bisets for fusion systems and blocks of finite group algebras, III
      Justin Lynd
    • 22 March 2019
      Minimal bisets for fusion systems and blocks of finite group algebras, IV
      Justin Lynd
      Abstract: In the first three talks, we have defined the notion of a saturated fusion system \F on a finite p-group S, and we have associated to it an S-S biset X. When \F is the fusion system of a finite group G on its Sylow p-subgroup S, then one may take X = G. But such a characteristic biset is not uniquely determined by \F. In the remaining talks, we will look at the unique up to isomorphism minimal such biset under inclusion, and we will try to bring out the connection between this minimal biset, when \F = \F_S(G), and the blocks of the group algebra kG.
    • 25 April 2019
      Minimal bisets for fusion systems and blocks of finite group algebras, V
      Justin Lynd
      Abstract: In the first three talks, we have defined the notion of a saturated fusion system \F on a finite p-group S, and we have associated to it an S-S biset X. When \F is the fusion system of a finite group G on its Sylow p-subgroup S, then one may take X = G. But such a characteristic biset is not uniquely determined by \F. In the remaining talks, we will look at the unique up to isomorphism minimal such biset under inclusion, and we will try to bring out the connection between this minimal biset, when \F = \F_S(G), and the blocks of the group algebra kG.

    Fall 2018

    During the Fall 2018 semester we will meet on Fridays from 2:10-3:00 in Maxim Doucet Hall, room 201.

    • 14 September 2018
      Closure operators on subgroup and other lattices
      Arturo Magidin
       
      Abstract: I will talk about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure opeartors on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
    • 21 September 2018
      Closure operators on subgroup and other lattices II
      Arturo Magidin
       
      Abstract: I will continue the series of talks about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure operators on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
      I will present our main "exclusion result", which provides that certain lattices L do not have the desired property, as well as a reduction result that allows us to invoke inductive arguments. If time permits, we will complete the determination of all finite lattices L for which co(L) is isomorphic to sub(G) for some group G.
    • 28 September 2018
      Closure operators on subgroup and other lattices III
      Arturo Magidin
       
      Abstract: I will continue the series of talks about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure operators on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
      The next step is relate some downsets on the lattice of closure operators co(L) with the closure operators on certain subsets of L; this will allow us to show that if co(sub(G)) is isomorphic to a subgroup lattice, and N is a normal subgroup of G, then co(sub(G/N)) is also a subgroup lattice. This allows for inductive arguments on |G|, since we have already reduced the problem to groups whose order is of the form p^a*q^b.
    • 12 October 2018
      Closure operators on subgroup and other lattices IV
      Arturo Magidin
       
      Abstract: I will finish my series of talks about joint work with Martha Kilpack, in which we determined all finite lattices L for which the lattice of closure operators on L is isomorphic to the lattice of subgroups of a group; and also the same problem but starting from the lattice of subgroups of an infinite group.
      To finish off we will consider the case where G is an infinite group, leading to closure operators on an infinite lattice. We will discuss some technical issues that arise from this, and how the notion of "compact element" of a lattice and "algebraic closure operator" come in to cover the breach. Finally, we determine all infinite groups G for which the lattice of algebraic closure operators on Sub(G) form a subgroup lattice.
    • 19 October 2018
      π-Baer Rings
      Yeliz Kara
       
      Abstract: I will talk about joint work with Gary F. Birkenmeier and Adnan Tercan, in which we introduced and investigated the concept of π-Baer rings. I will present connections between the π-Baer condition and the related conditions such as the Baer and quasi-Baer conditions.
    • 19 October 2018
      π-Baer Rings II
      Yeliz Kara
       
      Abstract: I will complete the series of talks about joint work with Gary F. Birkenmeier and Adnan Tercan, in which we introduced and investigated the concept of π-Baer rings. I will explain the π-Baer ring results on polynomial rings and 2-by-2 generalized upper triangular matrix rings.
    • 2 November 2018
      Update on the Classification of Indecomposable Quasi-Frobenius Rings
      Gary Birkenmeier
       
      Abstract: This talk will review and update the nilary quasi-Frobenius rings. Recall that a ring R with 1 is "nilary" if AB = 0 implies that either A or B is nilpotent, for all ideals A and B of R. A ring is "quasi-Frobenius", denoted QF, if it is right selfinjective and right Noetherian.
    • 9 November 2018
      Update on the Classification of Indecomposable Quasi-Frobenius Rings II
      Gary Birkenmeier
       
      Abstract: This talk will review and update the nilary quasi-Frobenius rings. Recall that a ring R with 1 is "nilary" if AB = 0 implies that either A or B is nilpotent, for all ideals A and B of R. A ring is "quasi-Frobenius", denoted QF, if it is right selfinjective and right Noetherian.
    • 30 November 2018
      Update on the Classification of Indecomposable Quasi-Frobenius Rings II
      Gary Birkenmeier
       
      Abstract: This talk will review and update the nilary quasi-Frobenius rings. Recall that a ring R with 1 is "nilary" if AB = 0 implies that either A or B is nilpotent, for all ideals A and B of R. A ring is "quasi-Frobenius", denoted QF, if it is right selfinjective and right Noetherian.

    Spring 2018

    During the Spring 2018 semester we will meet on Tuesdays from 4:00-4:50 in Maxim Doucet Hall room 210.

    • 23 January 2018
      A Description of indecomposable QF-rings, part 1
      Gary Birkenmeier
       
      Abstract: In this series of talks, I will present a classification of indecomposable QF-rings in terms of the essentiality of the ideals generated by primitive idempotents.
    • 23 January 2018 (4:30-5:30 TIME CHANGE)
      A Description of indecomposable QF-rings, part 2
      Gary Birkenmeier
    • 13 February 2018
      No seminar this week
    • 20 February 2018
      No seminar this week
    • 27 February 2018
      no seminar this week
    • 6 March 2018
      No seminar this week
    • 13 March 2018
      No seminar this week
    • 27 March 2018
      A Description of indecomposable QF-rings, part 3
      Gary Birkenmeier
    • 3 April 2018
      no seminar this week (Spring Break)
    • 10 April 2018
      A Description of indecomposable QF-rings, part 4
      Gary Birkenmeier
    • 17 April 2018
      Support Varieties for Algebraic Groups and the Humphreys Conjecture
      William Hardesty
      Louisiana State University  
      Abstract: I will begin by introducing the notion of the support variety of a module over a finite group scheme. This will be followed by a brief overview of classical results and calculations for the case when the finite group scheme is the first Frobenius kernel of a reductive algebraic group G. In 1997, J. Humphreys conjectured that the support varieties of indecomposable tilting modules for G (a very important class of modules) is controlled by a combinatorial bijection, due to G. Lusztig, between nilpotent orbits and a certain collection of subsets of the affine Weyl group called "canonical cells". This later became known as the "Humphreys conjecture". I will discuss some recent developments concerning this conjecture, including its complete verification for G = GL(n) (appearing in my thesis) as well as some additional results in other types appearing in joint work with P. Achar and S. Riche.
    • 9 May 2018 (4:00-5:00 WEDNESDAY room 209)
      An application of algebra to topology
      George Glauberman
      University of Chicago
      Abstract: The theory of fusion systems is a new branch of mathematics with applications to finite group theory and algebraic topology. In particular, it is involved in a result in topology known as the Martino-Priddy Conjecture, which was proved recently by assuming the classification of finite simple groups. New research with J. Lynd has removed the classification from the proof by using group theory. I plan to describe these topics and some open problems.

    Fall 2017

    During the Fall 2017 semester we will meet on Tuesdays from 4:00-4:50 in Maxim Doucet Hall room 210.

    • 19 September 2017
      Some ideas and methods from the classification of the finite simple groups
      Justin Lynd
       
      Abstract: I intend to give a historical and high-level overview of the original program for the classification of the finite simple groups. Depending on one's point of view, this is a quest that began either in the late 1890s or in 1954, and it ended either in the early 1980s or in 2004. We will begin at the first beginning and end roughly at the first end. In the middle, the focus will be on the ideas behind the program, which partitions the collection of finite simple groups into three classes: the groups of low 2-rank (small), the groups of component type (odd), and the groups of characteristic 2-type (even). Inasmuch as there will be any details (unlikely, but this depends on one's definition of "details"), we will focus on the identification of groups in the small and odd cases.
    • 26 September 2017
      Some ideas and methods from the classification of the finite simple groups, part 2
      Justin Lynd
    • 10 October 2017
      Some ideas and methods from the classification of the finite simple groups, part 3
      Justin Lynd
    • 24 October 2017
      Some ideas and methods from the classification of the finite simple groups, part 4
      Justin Lynd
    • 31 October 2017
      Some ideas and methods from the classification of the finite simple groups, part 5
      Justin Lynd

    Spring 2017

    During the Spring 2017 semester we will meet on Fridays from noon-12:50 in Maxim Doucet Hall room 214.

    • 27 January 2017
      When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice?
      Arturo Magidin
       
      Abstract: (This is joint work with Martha Kilpack of Brigham Young University.)
      A lattice is a partially ordered set in which every pair of elements have a least upper bound and a greatest lower bound; it can also be viewed as an algebra with two binary operators satisfying certain identities. A typical example of a lattice is the lattice of all subgroups of a given group, ordered by inclusion (or more generally, of all substructures of a given structure).
      A closure operator on a partially ordered set P is a function f mapping P into P such that f satisfies three conditions:
      (i) f is increasing: x less than or equal to f(x) for all x in P;
      (ii) f is isotone: if x less than or equal to y, then f(x) less than or equal to f(y) for all x,y in P;
      (iii) f is idempotent: f(f(x)) = f(x).
      We can partially order all closure operators on P, by letting f less than or equal to g if and only if f(x) less than or equal to g(x) for all x in P. If P is a lattice, then this makes the set of all closure operators on P into a lattice.
      It is a theorem of Birkhoff that every complete lattice is the lattice of subalgebras of some (possibly infinitary algebra); and a theorem of Whitman that every lattice can be embedded as a sublattice of a subgroup lattice. This leads to the question of which lattices of closure operators are isomorphic to the subgroup lattice of a group.
      Previously, we had shown that if we look at the closure operators on a subgroup lattice Sub(G), then this resulting lattice, c.o.(Sub(G)) is itself a subgroup lattice if and only if G is cyclic of prime power order. We will extend the investigation first to the case of closure operators on an arbitrary finite lattice; and later, to the lattice of subgroups of an infinite group.
    • 3 February 2017
      When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 2)
      Arturo Magidin
    • 10 February 2017
      When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 3)
      Arturo Magidin
    • 17 February 2017
      When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (part 4)
      Arturo Magidin
    • 24 February 2017
      If G is an infinite group, when is the lattice of algebraic closure operators on Sub(G) isomorphic to the subgroup lattice of a group K?
      Arturo Magidin
    • 3 March 2017
      When is the lattice of closure operators on a finite lattice isomorphic to a subgroup lattice? (last part)
      Arturo Magidin
    • 10 March 2017 (SPECIAL TIME 12:20)
      A description of indecomposable quasi-Frobenius rings
      Gary Birkenmeier
      Abstract In this talk, we characterize a nilary QF-ring R in terms of the essentiality of the ideals ReR where e is a primitive idempotent of R. Recall that a ring R is nilary if AB = 0 implies either A or B is nilpotent for all ideals A, B of R. Note that nilary rings are indecomposable rings. This is a preliminary report on joint research with Omar A. Al-Mallah andHafedh M. Al-Noghashi.
    • 24 March 2017
      A description of indecomposable quasi-Frobenius rings (part 2)
      Gary Birkenmeier
    • 31 March 2017
      A description of indecomposable quasi-Frobenius rings (part 3)
      Gary Birkenmeier
    • 7 April 2017
      A description of indecomposable quasi-Frobenius rings (part 4)
      Gary  Birkenmeier