arXiv subjects

Pál Hegedűs

Publications and source records attributed to Pál Hegedűs.

4 recordsLinked to original sources

arXiv2025

Descent set distribution for permutations with cycles of only odd or only even lengths

It is known that the number of permutations in the symmetric group $S_{2n}$ with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent sets: the number of permutations in $S_{2n}$ with a prescribed descent set and all cycles of odd lengths is equal to the number of permutations with the complementary descent set and all cycles of even lengths. There is also a variant for $S_{2n+1}$. The proof uses generating functions for character values and applies a new identity on higher Lie characters.

math.CO↗

arXiv2018

Finite groups with large Noether number are almost cyclic

Noether, Fleischmann and Fogarty proved that if the characteristic of the underlying field does not divide the order $|G|$ of a finite group $G$, then the polynomial invariants of $G$ are generated by polynomials of degrees at most $|G|$. Let $β(G)$ denote the largest indispensable degree in such generating sets. Cziszter and Domokos recently described finite groups $G$ with $|G|/β(G)$ at most $2$. We prove an asymptotic extension of their result. Namely, $|G|/β(G)$ is bounded for a finite group $G$ if and only if $G$ has a characteristic cyclic subgroup of bounded index. In the course of the proof we obtain the following surprising result. If $S$ is a finite simple group of Lie type or a sporadic group then we have $β(S) \leq {|S|}^{39/40}$. We ask a number of questions motivated by our results.

math.GR↗

arXiv2013

Transposable character tables, dual groups

One way of expressing the self-duality $A\cong \Hom(A,\mathbb{C})$ of Abelian groups is that their character tables are self-transpose (in a suitable ordering). Noncommutative groups fail to satisfy this property. In this paper we extend the duality to some noncommutative groups considering when the character table of a finite group is close to being the transpose of the character table for some other group. We find that groups dual to each other have dual normal subgroup lattices. We show that our concept of duality cannot work for non-nilpotent groups and we describe $p$-group examples.

math.GR↗

arXiv2012

Determination of Conjugacy Class Sizes from Products of Characters

Robinson showed that the character degrees are determined by knowing, for all $n$, the number of ways that the identity can be expressed as a product of $n$ commutators. Earlier, Strunkov showed that the existence of characters of $p$-defect 0 can be determined by counting solutions to certain equations involving commutators and conjugates. In this paper, we prove analogs to Robinson's and Strunkov's theorems by switching conjugacy classes and characters. We show that counting the multiplicity of the trivial character in certain products of characters determines the conjugacy class sizes and existence of conjugacy classes with $p$-defect 0.

math.GR↗