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Allen Herman

Publications and source records attributed to Allen Herman.

27 records · Page 2Linked to original sources
arXiv2017

Noncommutative reality-based algebras of rank 6

We classify the RBA-bases of $6$-dimensional noncommutative semisimple algebras for which the algebra has a positive degree map. We show that these RBAs are parametrized by seven real numbers, the first four of which are positive and the remaining three arbitrary. Our classification gives formulas for their standard bases and structure constants. Using these we give a list of all noncommutative integral table algebras of rank 6 with order up to 150. Four in the list are primitive, but we show these cannot be realized as adjacency algebras of association schemes. In the last section of the paper we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.

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arXiv2016

Representations of McLain groups

Basic modules of McLain groups $M=M(Λ,\leq, R)$ are defined and investigated. These are (possibly infinite dimensional) analogues of André's supercharacters of $U_n(q)$. The ring $R$ need not be finite or commutative and the field underlying our representations is essentially arbitrary: we deal with all characteristics, prime or zero, on an equal basis. The set $Λ$, totally ordered by $\leq$, is allowed to be infinite. We show that distinct basic modules are disjoint, determine the dimension of the endomorphism algebra of a basic module, find when a basic module is irreducible, and exhibit a full decomposition of a basic module as direct sum of irreducible submodules, including their multiplicities. Several examples of this decomposition are presented, and a criterion for a basic module to be multiplicity-free is given. In general, not every irreducible module of a McLain group is a constituent of a basic module.

math.RT↗
arXiv2016

The status of the Zassenhaus conjecture for small groups

We identify all small groups of order up to 288 in the GAP Library for which the Zassenhaus conjecture on rational conjugacy of units of finite order in the integral group ring cannot be established by an existing method. The groups must first survive all theoretical sieves and all known restrictions on partial augmentations (the HeLP$^+$ method). Then two new computational methods for verifying the Zassenhaus conjecture are applied to the unresolved cases, which we call the quotient method and the partially central unit construction method. To the cases that remain we attempt an assortment of special arguments available for units of certain orders and the lattice method. In the end, the Zassenhaus conjecture is verified for all groups of order less than 144 and we give a list of all remaining cases among groups of orders 144 to 287.

math.RA↗
arXiv2016

The recognition problem for table algebras and reality-based algebras

Given a finite-dimensional noncommutative semisimple algebra $A$ with involution, we show that $A$ always has an RBA-basis. We look for an RBA-basis that has integral or rational structure constants, and ask if the RBA admits a positive degree map. For RBAs that have a positive degree map, we try to find an RBA-basis with nonnegative structure constants to determine if there is a generalized table algebra structure. We settle these questions for the algebras $\mathbb{C} \oplus M_n(\mathbb{C})$, $n \ge 2$.

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arXiv2016

Zeta functions for tensor products of locally coprime integral adjacency algebras of association schemes

The zeta function of an integral lattice $Λ$ is the generating function $ζ_Λ(s) = \sum\limits_{n=0}^{\infty} a_n n^{-s}$, whose coefficients count the number of left ideals of $Λ$ of index $n$. We derive a formula for the zeta function of $Λ_1 \otimes Λ_2$, where $Λ_1$ and $Λ_2$ are $\mathbb{Z}$-orders contained in finite-dimensional semisimple $\mathbb{Q}$-algebras that satisfy a "locally coprime" condition. We apply the formula obtained above to $\mathbb{Z}S \otimes \mathbb{Z}T$ and obtain the zeta function of the adjacency algebra of the direct product of two finite association schemes $(X,S)$ and $(Y,T)$ in several cases where the $\mathbb{Z}$-orders $\mathbb{Z}S$ and $\mathbb{Z}T$ are locally coprime and their zeta functions are known.

math.RA↗
arXiv2014

Schur indices in GAP: wedderga 4.6+

We describe a algorithms and their implementations that calculate local and global Schur indices of ordinary irreducible characters of finite groups, cyclotomic algebras over abelian number fields, and rational quaternion algebras. These functions are available with the latest release of the GAP package wedderga, versions 4.6 and higher.

math.RA↗
arXiv2013

The Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring

We find all irreducible constituents of the Weil representation of a unitary group $U_m(A)$ of rank $m$ associated to a ramified quadratic extension $A$ of a finite, commutative, local and principal ring $R$ of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when $R$ is a field.

math.RT↗
arXiv2007

The Schur group of an abelian number field

We characterize the maximum $r$-local index of a Schur algebra over an abelian number field $K$ in terms of global information determined by the field $K$, for $r$ an arbitrary rational prime. This completes and unifies previous results of Janusz and Pendergrass.

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