On the decomposition numbers of the finite unitary groups in non-defining characteristic

1991 ◽  
Vol 207 (1) ◽  
pp. 83-89 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Unitary Groups ◽  
Decomposition Numbers ◽  
Finite Unitary Groups

scholarly journals On the Number of Partition Weights with Kostka Multiplicity One

10.37236/2574 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Zachary Gates ◽  
Brian Goldman ◽  
C. Ryan Vinroot
Keyword(s):  
Positive Integer ◽  
Ordered Set ◽  
Unitary Groups ◽  
Character Theory ◽  
Partition Functions ◽  
Young Tableaux ◽  
Multiplicity One ◽  
Restricted Partition ◽  
Finite Unitary Groups ◽  
Kostka Number

Given a positive integer $n$, and partitions $\lambda$ and $\mu$ of $n$, let $K_{\lambda \mu}$ denote the Kostka number, which is the number of semistandard Young tableaux of shape $\lambda$ and weight $\mu$.  Let $J(\lambda)$ denote the number of $\mu$ such that $K_{\lambda \mu} = 1$.  By applying a result of Berenshtein and Zelevinskii, we obtain a formula for $J(\lambda)$ in terms of restricted partition functions, which is recursive in the number of distinct part sizes of $\lambda$.  We use this to classify all partitions $\lambda$ such that $J(\lambda) = 1$ and all $\lambda$ such that $J(\lambda) = 2$.  We then consider signed tableaux, where a semistandard signed tableau of shape $\lambda$ has entries from the ordered set $\{0 < \bar{1} < 1 < \bar{2} < 2 < \cdots \}$, and such that $i$ and $\bar{i}$ contribute equally to the weight.  For a weight $(w_0, \mu)$ with $\mu$ a partition, the signed Kostka number $K^{\pm}_{\lambda,(w_0, \mu)}$ is defined as the number of semistandard signed tableaux of shape $\lambda$ and weight $(w_0, \mu)$, and $J^{\pm}(\lambda)$ is then defined to be the number of weights $(w_0, \mu)$ such that $K^{\pm}_{\lambda, (w_0, \mu)} = 1$.  Using different methods than in the unsigned case, we find that the only nonzero value which $J^{\pm}(\lambda)$ can take is $1$, and we find all sequences of partitions with this property.  We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.

scholarly journals The (2,3)-generation of the finite unitary groups

2020 ◽  
Vol 549 ◽  
pp. 319-345
Author(s):  
M.A. Pellegrini ◽  
M.C. Tamburini Bellani
Keyword(s):  
Unitary Groups ◽  
Finite Unitary Groups

scholarly journals The characters of the finite unitary groups

1977 ◽  
Vol 49 (1) ◽  
pp. 167-171 ◽  
Author(s):  
George Lusztig ◽  
Bhama Srinivasan
Keyword(s):  
Unitary Groups ◽  
Finite Unitary Groups

Categorical actions on unipotent representations of finite unitary groups

2019 ◽  
Vol 129 (1) ◽  
pp. 129-197
Author(s):  
O. Dudas ◽  
M. Varagnolo ◽  
E. Vasserot
Keyword(s):  
Unitary Groups ◽  
Finite Unitary Groups ◽  
Unipotent Representations

scholarly journals Strongly real classes in finite unitary groups of odd characteristic

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Zachary Gates ◽  
Anupam Singh ◽  
C. Ryan Vinroot
Keyword(s):  
Unitary Group ◽  
Conjugacy Classes ◽  
Unitary Groups ◽  
Finite Unitary Groups

Abstract.We classify all strongly real conjugacy classes of the finite unitary group U(

scholarly journals Involution centralisers in finite unitary groups of odd characteristic

2020 ◽  
Vol 545 ◽  
pp. 245-299
Author(s):  
S.P. Glasby ◽  
Cheryl E. Praeger ◽  
Colva M. Roney-Dougal
Keyword(s):  
Unitary Groups ◽  
Finite Unitary Groups
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