Allen Herman

Permanent URI for this collection

https://hdl.handle.net/10294/9088

Professor
Office: CW 307.13
E-mail: allen.herman@uregina.ca
Phone: 306-585-4487
Website: http://uregina.ca/~hermana/

Current classes
MATH 301 Introduction to Mathematical Logic, MATH 323 Modern Algebra I

Research interests
Algebras and representation theory

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Browsing Allen Herman by Issue Date

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Now showing 1 - 4 of 4
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    Orders of Torsion Units of integral reality-based algebras with rational multiplicities
    (2018) Herman, Allen; Singh, Gurmail
    A reality-based algebra (RBA) is a finite-dimensional associative algebra with involution over C whose distinguished basis B contains 1 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 B, 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 1 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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    Corrigendum to “The recognition problem for table algebras and reality-based algebras” [J. Algebra 479 (2017) 173–191]
    (2019) Herman, Allen; Muzychuk, Mikhail; Xu, Bangteng
    This note reports and corrects an error in the above article.
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    A survey of semisimple algebras in algebraic combinatorics
    (Springer Nature, 2021-10-21) Herman, Allen
    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 $\Delta$ that sums to the identity matrix. The special case when $\Delta$ 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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    The Involutive Double Coset Property for String C-goups of Affine Type
    (University of Calgary, 2024) Allen Herman; Roqayia Shalabi
    In this article we complete the classification of infinite affine Coxeter group types with the property that every double coset relative to the first parabolic subgroup is represented by an involution. This involutive double coset property was established earlier for the Coxeter groups of type $\tilde{C}_2$ and $\tilde{G}_2$, we complete the classification by showing it also holds for type $\tilde{F}_4$ and the types $\tilde{C}_n$ for all $n$. As this property is inherited by all string $C$-groups of these types, it follows that the corresponding abstract regular polytopes will have polyhedral realization cones.