arXiv subjects

Saber Ahmed

Publications and source records attributed to Saber Ahmed.

5 recordsLinked to original sources

arXiv2026

The $E_6$ Restricted Hyperplane Arrangement and its $E_7$ Shadow: Weyl Transport on a Minuscule Bruhat Poset

We study the restricted fan cut inside the dual fundamental Weyl chamber by the weights of a $27$-dimensional minuscule representation of $E_6$; the two such representations are dual and give the same arrangement. Only $11$ of the $27$ weights have kernels meeting its interior, and we prove that they determine the entire fan. It has exactly $14$ chambers and $18$ extreme rays, every chamber is a six-dimensional simplicial cone, and we determine all facets, rays, and incidence relations. The chamber count was previously obtained by Diaconescu and Entin; the simplicial structure, extreme rays, and incidence data are new. The geometry of the $27$ lines on a cubic surface then explains and organizes the resulting chamber architecture. We also enumerate all faces, compute both characteristic polynomials---the arrangement is not supersolvable---together with lattice indices and projective chamber volumes, and describe the oriented matroid. Our main result is representation-theoretic. A distinguished $14$-element visible subposet of the minuscule $\mathbf{56}$ of $E_7$, defined entirely inside $E_7$, has Hasse diagram equal to the chamber adjacency graph of the $E_6$ arrangement. Three canonical $7+7$ splittings of it, of types $A_7$, $D_7$, and $E_7$, are the visible traces of Levi-center $\mathfrak{u}(1)$-charge decompositions of the $\mathbf{56}$ and reproduce the three level-$8$ decompositions on the $E_6$ side. More strongly, the simple-root labels on its covers, transported by minimal-length coset representatives, recover chamber by chamber all six facets and, globally, the $11$ active weight hyperplanes and the boundary walls of the dual Weyl chamber. Thus the $E_7$ shadow records not merely the chamber graph but, once matched with the independent $E_6$ classification, the full local wall architecture of $I(E_6,\mathbf{27})$.

math.RT↗

arXiv2025

Convexity of Neural Codes with Four Maximal Codewords

Place cells are neurons that act as biological position sensors, associated with and firing in response to regions of an environment to situate an organism in space. These associations are recorded in (combinatorial) neural codes, motivating the following mathematical question: Which neural codes are generated by a collection of convex open sets in Euclidean space? Giusti and Itskov showed that a necessary condition for convexity is the absence of ``local obstructions." This necessary condition is, in fact, sufficient for certain families of codes. One such family consists of all codes with up to three maximal codewords. In this article, we investigate codes with four maximal codewords, showing that for many such codes, convexity is characterized by the absence of local obstructions, whereas for other such codes, convexity is characterized by the absence of local obstructions and a second type of obstruction, a ``wheel". Key to our analysis is a case-by-case investigation based on the nerve complex of the set of maximal codewords of a neural code. Up to symmetry, there are 20 possible nerves; and our results fully characterize convexity in 15 of the 20 cases.

math.CO↗

arXiv2025

Identifiability of directed-cycle and catenary linear compartment models

A parameter of a mathematical model is structurally identifiable if it can be determined from noiseless experimental data. Here, we examine the identifiability properties of two important classes of linear compartmental models: directed-cycle models and catenary models (models for which the underlying graph is a directed cycle or a bidirected path, respectively). Our main result is a complete characterization of the directed-cycle models for which every parameter is (generically locally) identifiable. Additionally, for catenary models, we give a formula for their input-output equations. Such equations are used to analyze identifiability, so we expect our formula to support future analyses into the identifiability of catenary models. Our proofs rely on prior results on input-output equations, and we also use techniques from linear algebra and graph theory.

math.CO↗

arXiv2023

Quantized enveloping superalgebra of type $P$

We introduce a new quantized enveloping superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ attached to the Lie superalgebra ${\mathfrak{p}}_n$ of type $P$. The superalgebra $\mathfrak{U}_q{\mathfrak{p}}_n$ is a quantization of a Lie bisuperalgebra structure on ${\mathfrak{p}}_n$ and we study some of its basic properties. We also introduce the periplectic $q$-Brauer algebra and prove that it is the centralizer of the $\mathfrak{U}_q {\mathfrak{p}}_n$-module structure on ${\mathbb C}(n|n)^{\otimes l}$. We end by proposing a definition for a new periplectic $q$-Schur superalgebra.

math.QA↗

arXiv2022

Highest Weight Modules Over The Quantum Periplectic Superalgebra of Type $P$

In this paper, we begin the study of highest weight representations of the quantized enveloping superalgebra ${\mathfrak U}_q {\mathfrak p}_n$ of type $P$. We introduce a Drinfeld-Jimbo representation and establish a triangular-decomposition of ${\mathfrak U}_q {\mathfrak p}_n$. We explain how to relate modules over ${\mathfrak U}_q {\mathfrak p}_n$ to modules over ${\mathfrak p}_n$, the Lie superalgebra of type $P$, and we prove that the category of tensor modules over ${\mathfrak U}_q {\mathfrak p}_n$ is not semisimple.

math.RT↗