Central Torsion Units of Integral Reality-Based Algebras With a Positive Degree Map

Herman, Allen; Singh, Gurmail

doi:10.24330/ieja.296156

Cited by 4 publications

(6 citation statements)

References 5 publications

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“…Gurmail Singh and the authour have established the analogue of one of the most basic results about units of integral group rings: Theorem 8. [34] Let B be the standard basis of a SITA. Then the central torsion units of ZB are trivial: (i.e.…”

Section: The Algebrasmentioning

confidence: 99%

A survey of semisimple algebras in algebraic combinatorics

Herman

2021

Indian J Pure Appl Math

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This is a survey of semisimple algebras of current interest in algebraic combinatorics, with a focus on questions which we feel will be new and interesting to experts in group algebras, integral representation theory, and computational algebra. The algebras arise primarily in two families: coherent algebras and subconstituent (aka. Terwilliger) algebras. Coherent algebras are subalgebras of full matrix algebras having a basis of 01-matrices satisfying the conditions that it be transpose-closed, sum to the all 1's matrix, and contain a subset ∆ that sums to the identity matrix. The special case when ∆ is a singleton is the important case of an adjacency algebra of a finite association scheme. A Terwilliger algebra is a semisimple extension of a coherent algebra by a set of diagonal 01-matrices determined canonically from its basis elements and a choice of row. We will survey the current state of knowledge of the complex, real, rational, modular, and integral representation theory of these semisimple algebras, indicate their connections with other areas of mathematics, and present several open questions.

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Section: The Algebrasmentioning

confidence: 99%

A survey of semisimple algebras in algebraic combinatorics

Herman

2021

Indian J Pure Appl Math

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“…In particular, it has nice applications to string theory, statistical mechanics, and condensed matter physics, see [4] and [7]. Modular data give rise to fusion rings, C-algebras and C * -algebras, see [2], [3], [5] and [6]. These rings and algebras are interesting topics of study in their own right.…”

Section: Introductionmentioning

confidence: 99%

Diagonal torsion matrices associated with modular data

Singh

2021

ADM

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Modular data are commonly studied in mathematics and physics. A modular datum defines a finite-dimensional representation of the modular group SL2(Z). Cuntz (2007) defined isomorphic integral modular data. Here we discuss isomorphic integral and non-integral modular data as well as non-isomorphic but closely related modular data. In this paper, we give some insights into diagonal torsion matrices associated to modular data.

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“…integral group rings to torsion units of ZB where B is a standard basis of an integral RBA with standard character; i.e. in the situation where the multiplicity of every irreducible character of CB in the standard feasible trace of CB is a positive integer (see [12], [13], [16], and [11]). After reviewing background terminology and properties for RBAs in Section 2, in Section 3 we adapt the approach of [13] to the situation where the multiplicities are rational.…”

Section: Introductionmentioning

confidence: 99%

Orders of torsion units of integral reality-based algebras with rational multiplicities

Singh

Herman

2018

J. Algebra Appl.

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A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over [Formula: see text] whose distinguished basis [Formula: see text] contains [Formula: see text] and is closed under pseudo-inverse. An integral RBA is one whose structure constants in its distinguished basis are integers. If the algebra has a one-dimensional representation taking positive values on [Formula: see text], then we say that the RBA has a positive degree map. These RBAs have a standard feasible trace, and the multiplicities of the irreducible characters in the standard feasible trace are the multiplicities of the RBA. In this paper, we show that for integral RBAs with positive degree map whose multiplicities are rational, any finite subgroup of torsion units whose elements are all of degree [Formula: see text] and have algebraic integer coefficients must have order dividing a certain positive integer determined by the degree map and the multiplicities. The paper concludes with a thorough investigation of the properties of RBAs that force multiplicities to be rational.

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Central Torsion Units of Integral Reality-Based Algebras With a Positive Degree Map

Abstract: Abstract. A reality-based algebra (RBA) is a finite-dimensional associative algebra that has a distinguished basis B containing 1 A , where 1 A is the iden-

Cited by 4 publications

References 5 publications

A survey of semisimple algebras in algebraic combinatorics

A survey of semisimple algebras in algebraic combinatorics

Diagonal torsion matrices associated with modular data

Orders of torsion units of integral reality-based algebras with rational multiplicities

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