arXiv subjects

Qiong Guo

Publications and source records attributed to Qiong Guo.

7 recordsLinked to original sources

arXiv2026

Coadjoint orbits of Row Closed Subgroups of $U_n(q)$

Let $n$ be a natural number, let $q$ be a prime power, and let $U_n(q)$ denote the group of unitriangular $n\times n$ matrices over the finite field $\mathbb{F}_q$ with $q$ elements. Row closed and column closed subgroups $U$ of $U_n(q)$ are special pattern subgroups obtained by deleting entire rows or, respectively, entire columns (apart from the diagonal entries). The supercharacters of the Andr\'{e}-Yan supercharacter theory of $U$ are afforded by the orbit modules arising from a monomial action of $U$ on the character group of the Lie algebra of $U$. We classify the corresponding orbits for row closed subgroups $U$, that is, we determine all orbits and identify which of the associated orbit modules are isomorphic and which afford orthogonal supercharacters. The classification for column closed subgroups follows from the mirror map, which reflects matrices across the antidiagonal.

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arXiv2019

Orbit method for $p$-Sylow subgroups of finite classical groups

For the $p$-Sylow subgroups $U$ of the finite classical groups of untwisted Lie type, $p$ an odd prime, we construct a monomial $\mathbb C U$-module $M$ which is isomorphic to the regular representation of $\mathbb C G$ by a modification of Kirillov's orbit method called monomial linearisation. We classify a certain subclass of orbits of the $U$-action on the monomial basis of $M$ consisting of so called staircase orbits and show, that every orbit module in $M$ is isomorphic to a staircase one. Finally we decompose the André-Neto supercharacters of $U$ into a sum of $U$-characters afforded by staircase orbit modules contained in $M$.

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arXiv2018

On coadjoint orbits for $p$-Sylow subgroups of finite classical groups

Kirillov's orbit theory provides a powerful tool for the investigation of irreducible unitary representations of many classes of Lie groups. In a previous paper we used a modification hereof, called monomial linearisation, to construct a monomial basis of the regular representation of $p$-Sylow subgroups $U$ of the finite classical groups of untwisted type. In this sequel to this article we determine the stabilizers of special orbit generators and show, that for the groups of Lie type ${\mathfrak{B}}_n$ and ${\mathfrak{D}}_n$ a subclass of the orbit modules decompose the $U$-modules affording the André-Neto supercharacters into a direct sum of submodules. Moreover these special orbit modules are either isomorphic or have no irreducible constituent in common, and each irreducible $U$ module is up to isomorphism constituent of precisely one of these.

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arXiv2017

On monomial linearisation and supercharacters of pattern subgroups

Column closed pattern subgroups $U$ of the finite upper unitriangular groups $U_n(q)$ are defined as sets of matrices in $U_n(q)$ having zeros in a prescribed set of columns besides the diagonal ones. We explain Jedlitschky's construction of monomial linearisation and apply this to $C U$ yielding a generalisation of Yan's coadjoint cluster representations. Then we give a complete classification of the resulting supercharacters, by describing the resulting orbits and determining the Hom-spaces between orbit modules.

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arXiv2016

$U_n(q)$ acting on flags and supercharacters

Let $U=U_n(q)$ be the group of lower unitriangular $n \times n$-matrices with entries in the field $\mathbb F_q$ with $q$ elements for some prime power $q$ and $n \in \mathbb N$. We investigate the restriction to $U$ of the permutation action of $GL_n(q)$ on flags in the natural $GL_n(q)$-module $\mathbb F_q^n$. Applying our results to the special case of flags of length two we obtain a complete decomposition of the permutation representation of $GL_n(q)$ on the cosets of maximal parabolic subgroups into irreducible $\mathbb C U$-modules.

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arXiv2013

Irreducible constituents of minimal degree in supercharacters of the finite unitriangular groups

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of André-Yan supercharacters and classify the supercharacters having constituents whose degree assume this lower bound. Moreover we show that the number of distinct irreducible characters of $U$ meeting this condition is a polynomial in $(q-1)$ with nonnegative integral coefficients and exhibit monomial sources for those.

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arXiv2013

On the U-module Structure of the Unipotent Specht Modules of Finite General Linear Groups

Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $λ$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by $S^λ$. It is conjectured that for $c\in \Z_{\geqslant 0}$ the number of irreducible constituents of dimension $q^c$ of the restriction $\RRes^{G}_U(S^λ)$ of $S^λ$ to $U$ is a polynomial in $q$ with integer coefficients depending only on $c$ and $λ$, not on $q$. In the special case of the partition $λ=(1^n)$ this implies a longstanding (still open) conjecture of Higman \cite{higman}, stating that the number of conjugacy classes of $U$ should be a polynomial in $q$ with integer coefficients depending only on $n$ not on $q$. In this paper we prove the conjecture in the case that $λ=(n-m,m)$ $(0\leqslant m \leqslant n/2)$ is a 2-part partition. As a consequence, we obtain a new representation theoretic construction of the standard basis of $S^λ$ (over fields of characteristic coprime to $q$) defined by M. Brandt, R. Dipper, G. James and S. Lyle in \cite{brandt2}, \cite{dj1} and an explanation of the rank polynomials appearing there.

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