arXiv subjects

Hahn Lheem

Publications and source records attributed to Hahn Lheem.

3 recordsLinked to original sources

arXiv2026

Intertwining Operators for Siegel Parabolics over Finite Fields

We consider degenerate principal series representations $\operatorname{Ind}_P^G\chi$ over finite fields, where $G$ is a classical subgroup of $\operatorname{GL}_{2n}$, and $P$ is the Siegel parabolic subgroup. For example, we show that this representation is always multiplicity-free and irreducible for generic characters $\chi$. We then discuss a particular intertwining operator $I$ on $\operatorname{Ind}_P^G\chi$ and its related combinatorics. Firstly, this operator $I$ produces families of diagonalizable antitriangular matrices with well-behaved eigenvalues. Secondly, applying $I$ to a special vector in $\operatorname{Ind}_P^G\chi$ leads us to various matrix Gauss sums, whose evaluations imply an explicit equidistribution result of the trace and determinant of symmetric and alternating invertible matrices.

math.RT↗

arXiv2019

Exponents of Jacobians of Graphs and Regular Matroids

Let $G$ be a finite undirected multigraph with no self-loops. The Jacobian $\operatorname{Jac}(G)$ is a finite abelian group associated with $G$ whose cardinality is equal to the number of spanning trees of $G$. There are only a finite number of biconnected graphs $G$ such that the exponent of $\operatorname{Jac}(G)$ equals $2$ or $3$. The definition of a Jacobian can also be extended to regular matroids as a generalization of graphs. We prove that there are finitely many connected regular matroids $M$ such that $\operatorname{Jac}(M)$ has exponent $2$ and characterize all such matroids.

math.CO↗

arXiv2018

On Characteristics of Hyperfields Obtained as Quotients of Finite Fields

Hyperstructures are a natural extension of regular algebraic structures in which one of the operations, known as the hyperoperation, is multivalued; a hyperfield is such an extension on a field. M. Krasner (1962) proved that the quotient $\mathbb{F}_p/G$, where $G$ is a subgroup of units in $\mathbb{F}_p$ is a hyperfield. The characteristic of a field may be explicitly determined from the order of the field, but there are no existing generalizations for determining the characteristic of a hyperfield of the form $\mathbb{F}_p/G$. We show that for odd primes $p$, there exists an explicit form for the characteristic of the hyperfield $\mathbb{F}_p/G$ and $|G|=1,2,3,4$. Finally, we prove a general form of the characteristic for hyperfields where $|G|$ is prime.

math.RA↗