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This proceedings volume covers a range of research topics in algebra from the Southern Regional Algebra Conference (SRAC) that took place in March 2017. Presenting theory as well as computational methods, featured survey articles and research papers focus on ongoing research in algebraic geometry, ring theory, group theory, and associative algebras. Topics include algebraic groups, combinatorial commutative algebra, computational methods for representations of groups and algebras, group theory, Hopf-Galois theory, hypergroups, Lie superalgebras, matrix analysis, spherical and algebraic spaces, and tropical algebraic geometry.

Since 1988, SRAC has been an important event for the algebra research community in the Gulf Coast Region and surrounding states, building a strong network of algebraists that fosters collaboration in research and education. This volume is suitable for graduate students and researchers interested in recent findings in computational and theoretical methods in algebra and representation theory.

Inhaltsverzeichnis

Frontmatter

Hattori-Torsion-Freeness and Endomorphism Rings

Abstract
This paper introduces the notions of \(K^r\)-faithfulness and quasi-flatness. They are used to discuss non-singularity and Hattori-torsion-freeness in the context of endomorphism rings. Several additional examples are given.
Ulrich Albrecht, Bradley McQuaig

Regularity of Edge Ideals and Their Powers

Abstract
We survey recent studies on the Castelnuovo–Mumford regularity of edge ideals of graphs and their powers. Our focus is on bounds and exact values of \(\text {reg}I(G)\) and the asymptotic linear function \(\text {reg}I(G)^q\), for \(q \ge 1\), in terms of combinatorial data of the given graph G.
Arindam Banerjee, Selvi Kara Beyarslan, Hà Huy Tài

A Survey of Rings Satisfying Annihilator or Extending Conditions on Projection Invariant Ideals

Abstract
In this paper, we survey results involving the projection invariant condition on one-sided ideals of rings. We focus on rings satisfying the right projection invariant extending condition (denoted right \(\pi \)-extending) or the projection invariant Baer condition (denoted \(\pi \)-Baer). Examples are provided to illustrate and delimit the results.
Gary F. Birkenmeier, Yeliz Kara, Adnan Tercan

Classification of Reductive Monoid Spaces over an Arbitrary Field

Abstract
In this semi-expository paper, we review the notion of a spherical space. In particular, we present some recent results of Wedhorn on the classification of spherical spaces over arbitrary fields. As an application, we introduce and classify reductive monoid spaces over an arbitrary field.
Mahir Bilen Can

Computing with Matrix and Basic Algebras

Abstract
This is a survey of my lecture delivered at the Southern Regional Algebra Conference in March 2017. It is meant to demonstrate some of the methods and technology that can be used to investigate examples in the theory of representations of groups and algebras. The concentration is on using computational methods to determine the structure of a matrix algebra as in the theorems of Wedderburn and investigating homological properties of the algebra by way of its basic algebra. For every split finite-dimensional associative algebra, there is a basic algebra, one whose simple modules have dimension one, that has the same representation theory. In the computational setting, it is usually much more practical to first find the basic algebra when studying representation theory.
Jon F. Carlson

A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes

Abstract
The free sum is a basic geometric operation among convex polytopes. This note focuses on the relationship between the normalized volume of the free sum and that of the summands. In particular, we show that the normalized volume of the free sum of full-dimensional polytopes is precisely the product of the normalized volumes of the summands.
Tianran Chen, Robert Davis

On the Cohomological Spectrum and Support Varieties for Infinitesimal Unipotent Supergroup Schemes

Abstract
We show that if G is an infinitesimal elementary supergroup scheme of height \(\le r\), then the cohomological spectrum \(\left| G \right| \) of G is naturally homeomorphic to the variety \(\mathscr {N}_r(G)\) of supergroup homomorphisms \(\rho : \mathbb {M}_r\rightarrow G\) from a certain (non-algebraic) affine supergroup scheme \(\mathbb {M}_r\) into G. In the case \(r=1\), we further identify the cohomological support variety of a finite-dimensional G-supermodule M as a subset of \(\mathscr {N}_1(G)\). We then discuss how our methods, when combined with recently announced results by Benson, Iyengar, Krause, and Pevtsova, can be applied to extend the homeomorphism \(\mathscr {N}_r(G)\simeq \left| G \right| \) to arbitrary infinitesimal unipotent supergroup schemes.
Christopher M. Drupieski, Jonathan R. Kujawa

A Survey of the Marcus–de Oliveira Conjecture

Abstract
We classify and survey the progress on the famous Marcus–de Oliveira determinantal conjecture (MOC) and related problems. The MOC claims that for two normal matrices A and B with eigenvalues \(a_1,\ldots , a_n\) and \(b_1,\ldots ,b_n\), respectively, the set \(\varDelta (A,B)=\{\det (A+UBU^*): U \text { is unitary}\}\) is in the convex hull of the set \(\{\prod _{i=1}^{n} (a_i+b_{\sigma (i)}):\sigma \in S_n\}\). We review the origin and the motivations of this conjecture from M. Fiedler’s work in the Hermitian case to Marcus and de Oliveira’s independent questions. Then, we survey the major positive cases of the MOC in terms of matrix degree, eigenvalues, and other things. We also describe some known properties of the set \(\varDelta (A,B)\), including compactness, connectivity, simply connectivity, convexity, star shapedness, and boundary and corner properties. Finally, we list some extended results that are related to the MOC. The main goal of this article is to provide a fairly comprehensive and brief overview of the progress of the MOC to interested readers and researchers for further explorations of the subject.
Huajun Huang

Schubert Polynomial Analogues for Degenerate Involutions

Abstract
We survey the recent study of involution Schubert polynomials and a modest generalization that we call degenerate involution Schubert polynomials. We cite several conditions when (degenerate) involution Schubert polynomials have simple factorization formulae. Such polynomials can be computed by traversing through chains in certain weak order posets, and we provide explicit descriptions of such chains in weak order for involutions and degenerate involutions. As an application, we give several examples of how certain multiplicity-free sums of Schubert polynomials factor completely into very simple linear factors.
Michael Joyce

The Structure of Hopf Algebras Acting on Dihedral Extensions

Abstract
We discuss isomorphism questions concerning the Hopf algebras that yield Hopf–Galois structures for a fixed separable field extension L/K. We study in detail the case where L/K is Galois with dihedral group \(D_p\), \(p\ge 3\) prime and give explicit descriptions of the Hopf algebras which act on L/K. We also determine when two such Hopf algebras are isomorphic, either as Hopf algebras or as algebras. For the case \(p=3\) and a chosen L/K, we give the Wedderburn–Artin decompositions of the Hopf algebras.
Alan Koch, Timothy Kohl, Paul J. Truman, Robert Underwood

Semi-extraspecial Groups

Abstract
We survey the results regarding semi-extraspecial p-groups. Semi-extraspecial groups can be viewed as generalizations of extraspecial groups. We present the connections between semi-extraspecial groups, Camina groups, and VZ-groups, and give upper bounds on the order of the center and the orders of abelian normal subgroups. We define ultraspecial groups to be semi-extraspecial groups where the center is as large as possible, and demonstrate a connection between ultraspecial groups that have at least two abelian subgroups whose order is the maximum and semifields.
Mark L. Lewis

The 3-Modular Character Table of the Automorphism Group of the Sporadic Simple O’Nan Group

Abstract
We compute the 3-modular character table of the group \(\mathrm{O'N}.2\). Much of the table is deduced character theoretically from the known 3-modular character table of the sporadic simple O’Nan group \(\text {O}'\text {N}\). We finish the remaining questions module theoretically with an application of condensation.
Klaus Lux, Alexander Ryba

Low Degree Cohomology of Frobenius Kernels

Abstract
Let G be a simple algebraic group defined over an algebraically closed field of characteristic \(p>0\). For a positive integer r, let \(G_r\) be the r-th Frobenius kernel of G. We determine in this paper a number m such that the cohomology \(\text {H}^n(G_r,k)\) is isomorphic to \(\text {H}^n(G_1,k)\) for all \(n\le m\) where m depends on p and the type of G.
Nham V. Ngo

On the Finite W-Algebra for the Queer Lie Superalgebra

Abstract
In this paper, we study the finite W-algebra \(W_{\chi }\) for the queer Lie superalgebra \({\mathfrak {g}= Q(N)}\) associated with an arbitrary even nilpotent element \(\chi \) in the coadjoint representation. We describe the annihilator \({\mathfrak {g}^{\chi }}\) and construct a set of elements in the generalized Whittaker module \({U(\mathfrak {g})/I_{\chi }}\) which under certain map project onto a homogeneous basis in \({\mathfrak {g}^{\chi }}\). In the case when the corresponding nilpotent element has Jordan blocks of equal size, these elements form a set of generators for \(W_{\chi }\).
Elena Poletaeva

An Alternate Proof to Derek Robinson’s 1968 Local Characterization Theorem on T-Groups

Abstract
Define \(\mathcal {T}\) to be the class of groups in which normality is a transitive relation. Define \({\mathscr {C}}_p\) to be the class of finite groups G for which each subgroup of a Sylow p-subgroup of G is normal in the corresponding Sylow normalizer. In [5] Robinson proved that a finite group satisfies \({\mathscr {C}}_p\) for all primes p if and only if it is a finite solvable \(\mathcal {T}\)-group. Here a new proof of this classic and influential result is presented.
Matthew F. Ragland

Some Tables of Right Set Properties in Affine Weyl Groups of Type A

Abstract
The tables of the title are a first attempt to understand empirically the sizes of certain distinguished sets, introduced by Hankyung Ko, of elements in affine Weyl groups. The distinguished sets themselves each have a largest element w, and all other elements are constructible combinatorially from that largest element. The combinatorics are given in the language of right sets, in the sense of Kazhdan–Lusztig. Collectively, the elements in a given distinguished set parameterize highest weights of possible modular composition factors of the “reduction modulo p” of a pth root of unity irreducible characteristic 0 quantum group module. Here, p is a prime, subject to conditions discussed below, in some cases known to be quite mild. Thus, the sizes of the distinguished sets in question are relevant to estimating how much time might be saved in any future direct approach to computing irreducible modular characters of algebraic groups from larger irreducible characters of quantum groups. Actually, Ko has described two methods for obtaining potentially effective systems of such sets. She has proved one method to work at least for all primes p as large as the Coxeter number h, in a context she indicates largely generalizes to smaller p. The other method, which produces smaller distinguished sets, is known for primes \(p\ge h \) for which the Lusztig character formula holds, but is currently unknown to be valid without the latter condition. In the tables of this paper, we calculate, for all w indexing a (p-)regular highest weight in the (p-)restricted parallelotope, distinguished set sizes for both methods, for affine types A\(_3\), A\(_4\), and A\(_5\). To keep the printed version of this paper sufficiently small, we only use those w indexing actual restricted weights in the A\(_5\) case. The sizes corresponding to the two methods of Ko are listed in columns (6) and (5), respectively, of the tables. We also make calculations in column (7) for a third, more “obvious” system of distinguished sets (see part (1) of Proposition 1), to indicate how much of an improvement each of the first two systems provides. Finally, all calculations have been recently completed for affine type A\(_6\), and the restricted cases are listed in this paper as a final table.
Leonard L. Scott, Ethan C. Zell

Hypergroups All Nonidentity Elements of Which Are Involutions

Abstract
The notion of a hypergroup generalizes the notion of a group. We introduce a hypergroup-theoretic generalization of the group-theoretic notion of an involution and characterize the hypergroups all nonidentity elements of which are involutions. Our characterization sheds new light on previous investigations in which a corresponding condition was considered within the theory of association schemes [10, 12] and table algebras [2]. We also show in how far hypergroups all nonidentity elements of which are involutions play a role in the investigation [11] of constrained hypergroups and their relationship to Coxeter hypergroups.
Paul-Hermann Zieschang
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