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C. Ryan Vinroot

Publications and source records attributed to C. Ryan Vinroot.

At least 19 recordsLinked to original sources
arXiv2025

Counting factorizations of Singer cycles in linear and unitary groups

We count factorizations of Singer cycles as products of reflections in the families of special and general unitary and linear groups over a finite field. In the case of minimum-length factorizations, the resulting answer is a striking product formula resembling the count for minimum-length factorizations of Coxeter elements into reflections in complex reflections groups. Moreover, for minimum length, the answers for the unitary and linear groups exhibit the phenomenon of Ennola duality, where the number of factorizations in a unitary group over the field $\mathbb{F}_q$ is given by replacing `$q$' with `$-q$' in the corresponding answer for a linear group. We use the character theory of these groups to make this count, and in particular we employ the Deligne--Lusztig theory of characters for finite reductive groups.

math.CO↗
arXiv2024

Galois automorphisms and a unique Jordan decomposition in the case of connected centralizer

We show that the Jordan decomposition of characters of finite reductive groups can be chosen so that if the centralizer of the relevant semisimple element in the dual group is connected, then the map is Galois-equivariant. Further, in this situation, we show that there is a unique Jordan decomposition satisfying conditions analogous to those of Digne--Michel's unique Jordan decomposition in the connected center case.

math.GR↗
arXiv2022

Some Reality Properties of Finite Simple Orthogonal Groups

We prove several reality properties for finite simple orthogonal groups. For any prime power $q$ and $m\geq 1$, we show that all real conjugacy classes are strongly real in the simple groups $\mathrm{P}Ω^{\pm}(4m+2,q), m \geq 1$, except in the case $\mathrm{P}Ω^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$, and we construct weakly real classes in this exceptional case for any $m$. We also show that no irreducible complex character of $\mathrm{P}Ω^{\pm}(n,q)$ can have Frobenius-Schur indicator $-1$, except possibly in the case $\mathrm{P}Ω^{-}(4m+2,q)$ with $q \equiv 3(\mathrm{mod} \; 4)$.

math.GR↗
arXiv2019

A computational approach to the Frobenius-Schur indicators of finite exceptional groups

We prove that the finite exceptional groups $F_4(q)$, $E_7(q)_{\mathrm{ad}}$, and $E_8(q)$ have no irreducible complex characters with Frobenius-Schur indicator $-1$, and we list exactly which irreducible characters of these groups are not real-valued. We also give an exact list of complex irreducible characters of the Ree groups ${^2 F_4}(q^2)$ which are not real-valued, and we show the only character of this group which has Frobenius-Schur indicator $-1$ is the cuspidal unipotent character $χ_{21}$ found by M. Geck.

math.RT↗
arXiv2019

A note on dual modules and the transpose

It is a classical result in matrix algebra that any square matrix over a field can be conjugated to its transpose by a symmetric matrix. For $F$ a non-Archimedean local field, Tupan used this to give an elementary proof that transpose inverse takes each irreducible smooth representation of ${\rm GL}_n(F)$ to its dual. We re-prove the matrix result and related observations using module-theoretic arguments. In addition, we write down a generalization that applies to central simple algebras with an involution of the first kind. We use this generalization to extend Tupan's method of argument to ${\rm GL}_n(D)$ for $D$ a quaternion division algebra over $F$.

math.RA↗
arXiv2019

On the number of irreducible real-valued characters of a finite group

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)| is at most f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k = 5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters.

math.GR↗
arXiv2018

Totally orthogonal finite simple groups

We prove that if $G$ is a finite simple group, then all irreducible complex representations of $G$ by be realized over the real numbers if and only if every element of $G$ may be written as a product of two involutions in $G$. This follows from our result that if $q$ is a power of $2$, then all irreducible complex representations of the orthogonal groups $\mathrm{O}^{\pm}(2n, \mathbb{F}_q)$ may be realized over the real numbers. We also obtain generating functions for the sums of degrees of several sets of unipotent characters of finite orthogonal groups, and we obtain a twisted version of our main result for a broad family of finite classical groups.

math.RT↗
arXiv2018

Galois group action and Jordan decomposition of characters of finite reductive groups with connected center

Let $\mathbf{G}$ be a connected reductive group with connected center defined over $\mathbb{F}_q$, with Frobenius morphism F. Given an irreducible complex character $χ$ of $\mathbf{G}^F$ with its Jordan decomposition, and a Galois automorphism $σ\in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, we give the Jordan decomposition of the image ${^σχ}$ of $χ$ under the action of $σ$ on its character values.

math.RT↗
arXiv2018

On the number of real classes in the finite projective linear and unitary groups

We show that for any $n$ and $q$, the number of real conjugacy classes in $\mathrm{PGL}(n, \mathbb{F}_q)$ is equal to the number of real conjugacy classes of $\mathrm{GL}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SL}(n, \mathbb{F}_q)$, refining a result of Lehrer, and extending the result of Gill and Singh that this holds when $n$ is odd or $q$ is even. Further, we show that this quantity is equal to the number of real conjugacy classes in $\mathrm{PGU}(n, \mathbb{F}_q)$, and equal to the number of real conjugacy classes of $\mathrm{U}(n, \mathbb{F}_q)$ which are contained in $\mathrm{SU}(n, \mathbb{F}_q)$, refining results of Gow and Macdonald. We also give a generating function for this common quantity.

math.GR↗
arXiv2017

Real representations of finite symplectic groups over fields of characteristic two

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\mathrm{Sp}(2n, \mathbb{F}_q)$, or of $\mathrm{SO}(2n+1, \mathbb{F}_q)$, for any prime power $q$.

math.RT↗
arXiv2017

On involutions and indicators of finite orthogonal groups

We study the numbers of involutions and their relation to Frobenius-Schur indicators in the groups $\mathrm{SO}^{\pm}(n,q)$ and $Ω^{\pm}(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius-Schur indicators are 1), and we are able to show this holds for all finite simple groups $G$ other than the groups $\mathrm{Sp}(2n,q)$ with $q$ even or $Ω^{\pm}(4m,q)$ with $q$ even. We prove computationally that for small $n$ and $m$ this statement indeed holds for these groups by equating their character degree sums to the number of involutions. We also prove a result on a certain twisted indicator for the groups $\mathrm{SO}^{\pm}(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating function and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\mathrm{SO}^{\pm}(n,q)$ and $Ω^{\pm}(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

math.GR↗
arXiv2016

A factorization result for classical and similitude groups

For most classical and similitude groups, we show that each element can be written as a product of two transformations that a) preserve or almost preserve the underlying form and b) whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras and Waldspurger on the existence of automorphisms of $p$-adic classical groups that take each irreducible smooth representations to its dual.

math.RT↗
arXiv2015

Real Classes of Finite Special Unitary Groups

We classify all real and strongly real classes of the finite special unitary group $SU_n(q)$. Unless $q \equiv 3 (mod 4)$ and $n |4$, the classification of real classes is similar to that of the finite special linear group $SL_n(q)$. We relate strong reality in $SU_n(q)$ to strong reality in the finite special orthogonal groups $SO^\pm_n(q)$, and we classify real and strongly real classes in the last case for the group $SO^\pm_n(q)$, when q is odd and $n \equiv 2 (mod 4)$.

math.GR↗
arXiv2015

Kostka multiplicity one for multipartitions

If $[λ(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $μ$ is a partition of $n$, we study the number $K_{[λ(j)]μ}$ of sequences of semistandard Young tableaux of shape $[λ(j)]$ and total weight $μ$. We show that the numbers $K_{[λ(j)] μ}$ occur naturally as the multiplicities in certain permutation representations of wreath products. The main result is a set of conditions on $[λ(j)]$ and $μ$ which are equivalent to $K_{[λ(j)] μ} = 1$, generalizing a theorem of Berenshte\uın and Zelevinski\uı. We also show that the questions of whether $K_{[λ(j)] μ} > 0$ or $K_{[λ(j)] μ} = 1$ can be answered in polynomial time, expanding on a result of Narayanan. Finally, we give an application to multiplicities in the degenerate Gel'fand-Graev representations of the finite general linear group, and we show that the problem of determining whether a given irreducible representation of the finite general linear group appears with nonzero multiplicity in a given degenerate Gel'fand-Graev representation, with their partition parameters as input, is $NP$-complete.

math.CO↗
arXiv2013

Generating functions for real character degree sums of finite general linear and unitary groups

We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof that every real-valued character has Frobenius-Schur indicator 1, and we obtain some q-series identities. For the finite unitary group, we expand the generating function in terms of values of Hall-Littlewood functions, and we obtain combinatorial expressions for the character degree sums of real-valued characters with Frobenius-Schur indicator 1 or -1.

math.GR↗