Combinatorial supersymmetry: Supergroups, superquasigroups, and their multiplication groups

임복희,Jonathan D. H. Smith 대한수학회 2024 대한수학회지 Vol.61 No.1

The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely set-theoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues --- quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

HOMOGENEOUS CONDITIONS FOR STOCHASTIC TENSORS

Im, Bokhee,Smith, Jonathan D.H. Korean Mathematical Society 2022 대한수학회논문집 Vol.37 No.2

Fix an integer n ≥ 1. Then the simplex Π_n, Birkhoff polytope Ω_n, and Latin square polytope Λ_n each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ω_n are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.

Trilinear products and comtrans algebra representations

Im, Bokhee,Smith, Jonathan D.H. Elsevier 2009 Linear Algebra and its Applications Vol.430 No.1

<P><B>Abstract</B></P><P>Comtrans algebras are modules over a commutative ring <I>R</I> equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called <I>comtrans identity</I>. The standard construction of a comtrans algebra uses the ternary commutator and translator of a trilinear product. If 6 is invertible in <I>R</I>, then each comtrans algebra arises in this standard way from the so-called <I>bogus product</I>.</P><P>Consider a vector space <I>E</I> of dimension <I>n</I> over a field <I>R</I>. While the dimension of the space of all trilinear products on <I>E</I> is <SUP>n4</SUP>, the dimension of the space of all comtrans algebras on <I>E</I> is less, namely 56<SUP>n4</SUP>-12<SUP>n3</SUP>-13<SUP>n2</SUP>. The paper determines which trilinear products may be represented as linear combinations of the commutator and translator of a comtrans algebra. For <I>R</I> not of characteristic 3, the necessary and sufficient condition for such a representation is the <I>strong alternativity</I> xxy+xyx+yxx=0 of the trilinear product <I>xyz</I>. For <I>R</I> also not of characteristic 2, it is shown that the representation may be given by the bogus product. A suitable representation for the characteristic 2 case is also obtained.</P>

CROSS-INTERCALATES AND GEOMETRY OF SHORT EXTREME POINTS IN THE LATIN POLYTOPE OF DEGREE 3

Bokhee Im,Jonathan D. H. Smith Korean Mathematical Society 2023 대한수학회지 Vol.60 No.1

The polytope of tristochastic tensors of degree three, the Latin polytope, has two kinds of extreme points. Those that are at a maximum distance from the barycenter of the polytope correspond to Latin squares. The remaining extreme points are said to be short. The aim of the paper is to determine the geometry of these short extreme points, as they relate to the Latin squares. The paper adapts the Latin square notion of an intercalate to yield the new concept of a cross-intercalate between two Latin squares. Cross-intercalates of pairs of orthogonal Latin squares of degree three are used to produce the short extreme points of the degree three Latin polytope. The pairs of orthogonal Latin squares fall into two classes, described as parallel and reversed, each forming an orbit under the isotopy group. In the inverse direction, we show that each short extreme point of the Latin polytope determines four pairs of orthogonal Latin squares, two parallel and two reversed.

TRILINEAR FORMS AND THE SPACE OF COMTRANS ALGEBRAS

IM, BOKHEE,SMITH, JONATHAN D.H. The Honam Mathematical Society 2005 호남수학학술지 Vol.27 No.4

Comtrans algebras are modules equipped with two trilinear operations: a left alternative commutator and a translator satisfying the Jacobi identity, the commutator and translator being connected by the so-called comtrans identity. These identities have analogues for trilinear forms. On a given vector space, the set of all comtrans algebra structures itself forms a vector space. In this paper, the dimension of the space of comtrans algebra structures on a finite-dimensional vector space is determined.

ON HOPF ALGEBRAS IN ENTROPIC J´ONSSON-TARSKI VARIETIES

Anna B. Romanowska,Jonathan D. H. Smith 대한수학회 2015 대한수학회보 Vol.52 No.5

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic J´onsson-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying J´onsson-Tarski monoid of the generating algebra is cancellative. The problem of determining when the J´onsson-Tarski monoid forms a group is open.

ON HOPF ALGEBRAS IN ENTROPIC JÓNSSON-TARSKI VARIETIES

ROMANOWSKA, ANNA B.,SMITH, JONATHAN D.H. Korean Mathematical Society 2015 대한수학회보 Vol.52 No.5

Comonoid, bi-algebra, and Hopf algebra structures are studied within the universal-algebraic context of entropic varieties. Attention focuses on the behavior of setlike and primitive elements. It is shown that entropic $J{\acute{o}}nsson$-Tarski varieties provide a natural universal-algebraic setting for primitive elements and group quantum couples (generalizations of the group quantum double). Here, the set of primitive elements of a Hopf algebra forms a Lie algebra, and the tensor algebra on any algebra is a bi-algebra. If the tensor algebra is a Hopf algebra, then the underlying $J{\acute{o}}nsson$-Tarski monoid of the generating algebra is cancellative. The problem of determining when the $J{\acute{o}}nsson$-Tarski monoid forms a group is open.

RICCI CURVATURE, CIRCULANTS, AND EXTENDED MATCHING CONDITIONS

Dagli, Mehmet,Olmez, Oktay,Smith, Jonathan D.H. Korean Mathematical Society 2019 대한수학회보 Vol.56 No.1

Ricci curvature for locally finite graphs, as proposed by Lin, Lu and Yau, provides a useful isomorphism invariant. A Matching Condition was introduced as a key tool for computation of this Ricci curvature. The scope of the Matching Condition is quite broad, but it does not cover all cases. Thus the current paper introduces extended versions of the Matching Condition, and applies them to the computation of the Ricci curvature of a class of circulants determined by certain number-theoretic data. The classical Matching Condition is also applied to determine the Ricci curvature for other families of circulants, along with Cayley graphs of abelian groups that are generated by the complements of (unions of) subgroups.

Ricci curvature, circulants, and extended matching conditions

Mehmet Dagli,Oktay Olmez,Jonathan D. H. Smith 대한수학회 2019 대한수학회보 Vol.56 No.1

Ricci curvature for locally finite graphs, as proposed by Lin, Lu and Yau, provides a useful isomorphism invariant. A Matching Condition was introduced as a key tool for computation of this Ricci curvature. The scope of the Matching Condition is quite broad, but it does not cover all cases. Thus the current paper introduces extended versions of the Matching Condition, and applies them to the computation of the Ricci curvature of a class of circulants determined by certain number-theoretic data. The classical Matching Condition is also applied to determine the Ricci curvature for other families of circulants, along with Cayley graphs of abelian groups that are generated by the complements of (unions of) subgroups.

LINEAR AND NON-LINEAR LOOP-TRANSVERSAL CODES IN ERROR-CORRECTION AND GRAPH DOMINATION

Dagli, Mehmet,Im, Bokhee,Smith, Jonathan D.H. Korean Mathematical Society 2020 대한수학회보 Vol.57 No.2

Loop transversal codes take an alternative approach to the theory of error-correcting codes, placing emphasis on the set of errors that are to be corrected. Hitherto, the loop transversal code method has been restricted to linear codes. The goal of the current paper is to extend the conceptual framework of loop transversal codes to admit nonlinear codes. We present a natural example of this nonlinearity among perfect single-error correcting codes that exhibit efficient domination in a circulant graph, and contrast it with linear codes in a similar context.