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Farid Aliniaeifard

Publications and source records attributed to Farid Aliniaeifard.

24 records · Page 2Linked to original sources
arXiv2018

Pattern groups and a poset based Hopf monoid

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

math.CO↗
arXiv2018

The structure of normal lattice supercharacter theories

The character theory of finite groups has numerous basic questions that are often already quite involved: enumerating of irreducible characters, their character formulas, point-wise product decompositions, and restriction/induction between groups. A supercharacter theory is a framework for simplifying the character theory of a finite group, while ideally not losing all important information. This paper studies one such theory that straddles the gap between retaining valuable group information while reducing the above fundamental questions to more combinatorial lattice constructions.

math.RT↗
arXiv2016

Supercharacters for Normal Supercharacter Theory

In order to find a tractable theory to substitute for the wild character theory of the group of $n\times n$ unipotent upper-triangular matrices over a finite field $\mathbb{F}_q$, André and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set $S$ of normal subgroups of $G$. We call such supercharacter theory the normal supercharacter theory generated by $S$. It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. The normal supercharacter theory provides many substitutions for wild character theories. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of $G$ are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, We justify that normal supercharacter theories cannot be obtained by preceding supercharacter theory constructions.

math.RA↗
arXiv2015

Normal Supercharacter Theory

There are two main constructions of supercharacter theories for a group $ G $. The first, defined by Diaconis and Isaacs, comes from the action of a group $A$ via automorphisms on our given group $G$. The second, defined by Hendrickson, is combining a supercharacter theories of a normal subgroup $N$ of $G$ with a supercharacter theory of $G/N$. In this paper we construct a supercharacter theory from an arbitrary set of normal subgroups of $G$. We show that when consider the set of all normal subgroups of G, the corresponding supercharacter theory is related to a partition of $G$ given by certain values on the central idempotents. Also, we show the supercharacter theories that we construct can not be obtained via automorphisms or a single normal subgroup.

math.RA↗
arXiv2015

Noetherian Rings Whose Annihilating-Ideal Graphs Have finite Genus

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ such that two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We characterize commutative Noetherian rings $R$ whose annihilating-ideal graphs have finite genus $γ(\Bbb{AG}(R))$. It is shown that if $R$ is a Noetherian ring such that $0<γ(\Bbb{AG}(R))<\infty$, then $R$ has only finitely many ideals.

math.RA↗
arXiv2011

Rings Whose Annihilating-Ideal Graphs Have Positive Genus

Let $R$ be a commutative ring and ${\Bbb{A}}(R)$ be the set of ideals with non-zero annihilators. The annihilating-ideal graph of $R$ is defined as the graph ${\Bbb{AG}}(R)$ with the vertex set ${\Bbb{A}}(R)^*={\Bbb{A}}\setminus\{(0)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. We investigate commutative rings $R$ whose annihilating-ideal graphs have positive genus $γ(\Bbb{AG}(R))$. It is shown that if $R$ is an Artinian ring such that $γ(\Bbb{AG}(R))<\infty$, then $R$ has finitely many ideals or $(R,\mathfrak{m})$ is a Gorenstein ring with maximal ideal $\mathfrak{m}$ and ${\rm v.dim}_{R/{\mathfrak{m}}}{\mathfrak{m}}/{\mathfrak{m}}^{2}=2$. Also, for any two integers $g\geq 0$ and $q>0$, there are finitely many isomorphism classes of Artinian rings $R$ satisfying the conditions: (i) $γ(\Bbb{AG}(R)) < g$ and (ii) $|R/{\mathfrak{m}}| \leq q$ for every maximal ideal ${\mathfrak{m}}$ of $R$. Also, it is shown that if $R$ is a non-domain Noetherian local ring such that $γ(\Bbb{AG}(R))<\infty$, then either $R$ is a Gorenstein ring or $R$ is an Artinian ring with finitely many ideals.

math.AC↗