scholarly journals On the characteristic map of finite unitary groups

2007 ◽  
Vol 210 (2) ◽  
pp. 707-732 ◽  
Author(s):  
Nathaniel Thiem ◽  
C. Ryan Vinroot
10.37236/235 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Nathaniel Thiem ◽  
C. Ryan Vinroot

Gelfand–Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions, we study degenerate Gelfand–Graev characters of the finite unitary group from a combinatorial point of view. In particular, we give the values of Gelfand–Graev characters at arbitrary elements, recover the decomposition multiplicities of degenerate Gelfand–Graev characters in terms of tableau combinatorics, and conclude with some multiplicity consequences.


10.37236/2574 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Zachary Gates ◽  
Brian Goldman ◽  
C. Ryan Vinroot

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.


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

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

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Zachary Gates ◽  
Anupam Singh ◽  
C. Ryan Vinroot

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


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