arXiv subjects

A. K. M. Selim Reza

Publications and source records attributed to A. K. M. Selim Reza.

4 recordsLinked to original sources

arXiv2026

Vector-Valued Invariants Associated with All Irreducible Representations for a Finite Group

We investigate the complex reflection group $\mathfrak{G}$ associated with the octahedral group, identified as the ninth entry in the Shephard-Todd classification. We determine all irreducible representations of $\mathfrak{G}$ and compute the character table. Moreover, for each representation, we compute the module of vector-valued invariants and relate it to the fundamental invariants of the octahedral group. Additionally, we derive explicit dimension formulas for the corresponding rings of invariants.

math.RT↗

arXiv2025

Computation of Vector-Valued Invariants for a Finite Complex Reflection Group

We consider the complex reflection group $ \mathcal{G} $, identified as No. 8 in the Shephard-Todd classification. In this paper, we present computations of the vector-valued invariants associated with various representations of $ \mathcal{G} $. Additionally, we investigate the structure of the corresponding invariant rings.

math.RA↗

arXiv2025

On the Structure of Multilinear Invariants of a Finite Unitary Reflection Group

We study the space of multilinear invariants $ V_f $ of degree $ f $ for a specified finite unitary reflection group. A subspace $ W_f $ of typical invariants is also introduced. We note that the dimension of $ W_f $ is given by Catalan number. We explore both spaces for $ f \leq 5 $, noting that their dimensions differ based on the value of $ f $. We explicitly determine the bases for both spaces, and then we establish the relationship between the vectors of the two bases.

math.CO↗

arXiv2024

Weight Enumerators of codes over $\mathbb{F}_2$ and over $\mathbb{Z}_4$

Weight enumerators are important tools for deciphering the algebraic structure of the related code spaces and for understanding group actions on these spaces. Our study focuses on symmetrized weight enumerators of pairs of Type II codes over the finite field $\mathbb{F}_{2}$ and the ring $\mathbb{Z}_{4}$. These pairs have been examined as invariants for a specified group. In particular, we concentrate on the scenarios where the space of the invariant ring is of degree 8 and 16. Our findings show that in certain situations, the ring produced by the symmetrized weight enumerators precisely matches with the invariant ring of the designated group. This coincidence points to a profound relationship between the invariant ring's structure and the algebraic characteristics of the weight enumerators.

math.CO↗