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Ayush Udeep

Publications and source records attributed to Ayush Udeep.

11 recordsLinked to original sources
arXiv2026

On the representation dimension of finite $p$-groups

For a finite group $G$, the representation dimension $\delta(G)$ is the least dimension of a faithful complex representation of $G$. We prove that $\delta(H\times K) = \delta(H) + \delta(K)$ whenever $H$ and $K$ are $p$-groups for a fixed prime $p$, and show by example that this additivity can fail more generally for nilpotent groups. We also determine $\delta(G)$ for several important classes of non-abelian $p$-groups, {\it viz.} VZ groups, Camina groups, and metacyclic groups; and compute $\delta(G)$ for all groups of order $p^6$ ($p\geq 5$), organized by their isoclinism family.

math.GR↗
arXiv2026

Codegrees of Irreducible Characters of VZ and Camina $p$-Groups

The character codegree of an irreducible character of a finite group $G$ is given by the index of its kernel in $G$ upon the character degree. We compute the codegrees of irreducible characters of VZ and Camina $p$-groups, and also obtain the character codegrees set of $p$-groups of order $\leq p^5$, where $p$ is an odd prime.

math.GR↗
arXiv2025

Representations of Skew Left Braces of order $pq$

In this paper, we study the irreducible representations of skew braces of order \( pq \), which is equivalent to studying the representation theory of groups of order \( p^2q^2 \) arising from skew left braces, where \( p > q \) are primes. To achieve this, we classify all semidirect product groups \( \Lambda_A \) associated with skew left braces $A$ of order \( pq \), up to isomorphism.

math.GR↗
arXiv2024

Exceptional groups of order $p^6$ for primes $p\geq 5$

The minimal faithful permutation degree $\mu(G)$ of a finite group $G$ is the least integer $n$ such that $G$ is isomorphic to a subgroup of the symmetric group $S_n$. If $G$ has a normal subgroup $N$ such that $\mu(G/N) > \mu(G)$, then $G$ is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically 0. We identify $(11p+107)/2$ such groups and conjecture that there are no others.

math.GR↗
arXiv2024

Representations of skew braces

In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace $(A, \cdot, \circ)$ induces an action $\lambda^{\op}: (A, \circ) \to \Aut (A, \cdot)$, which gives rise to the group $\Lambda_{A^{\op}} = (A, \cdot) \rtimes_{\lambda^{\op}} (A, \circ)$. We prove that if $A$ and $B$ are isoclinic skew left braces, then $\Lambda_{A^{\op}}$ and $\Lambda_{B^{\op}}$ are also isoclinic under some mild restrictions on the centers of the respective groups. Our key observation is that there is a one-to-one correspondence between the set of equivalence classes of irreducible representations of $(A, \cdot, \circ)$ and that of the group $\Lambda_{A^{\op}}$. We obtain a decomposition of the induced representation of the additive group $(A, \cdot)$ and of the multiplicative group $(A, \circ)$ corresponding to the regular representation of the group $\Lambda_{A^{\op}}$. As examples, we compute the dimensions of the irreducible representations for several skew left braces with prime power orders.

math.GR↗
arXiv2024

Various Representation Dimensions associated with a Finite Group

To a finite group $G$, one can associate several notions of dimensions (or degrees). In this survey, we attempt to bring together some of the notions of dimensions or degrees defined using representations of the group in General Linear Groups and permutation groups. These are embedding degree, minimal faithful irreducible character degree, minimal faithful permutation representation degree, minimal faithful quasi-permutation representation degree and essential dimension. We briefly present the progress in understanding these notions and the related problems.

math.RT↗
arXiv2023

On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of $p$-groups

For a finite group $G$, we denote by $c(G)$, the minimal degree of faithful representation of $G$ by quasi-permutation matrices over the complex field $\mathbb{C}$. For an irreducible character $\chi$ of $G$, the codegree of $\chi$ is defined as $\cod(\chi) = |G/ \ker(\chi)|/ \chi(1)$. In this article, we establish equality between $c(G)$ and a $\mathbb{Q}_{\geq 0}$-sum of codegrees of some irreducible characters of a non-abelian $p$-group $G$ of odd order. We also study the relation between $c(G)$ and irreducible character codegrees for various classes of non-abelian $p$-groups, such as, $p$-groups with cyclic center, maximal class $p$-groups, GVZ $p$-groups, and others.

math.GR↗
arXiv2023

Minimal Faithful Quasi-Permutation Representation Degree of p-Groups with Cyclic Center

For a finite group G, we denote by $\mu(G)$, and c(G), the minimal degree of faithful permutation representation of G, and the minimal degree of faithful representation of G by quasi-permutation matrices over the complex field C, respectively. In this article, we study $\mu(G)$, and c(G) for various classes of finite non-abelian p-groups with cyclic center. We prove a result for normally monomial p-groups with cyclic center which generalizes a result of Behravesh for finite p-groups of nilpotency class 2 with cyclic center [5, Theorem 4.12]. We also compute minimal degrees for some classes of metabelian p-groups.

math.GR↗