scholarly journals Gelfand–Graev Characters of the Finite Unitary Groups

10.37236/235 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Nathaniel Thiem ◽  
C. Ryan Vinroot
Keyword(s):  
Finite Groups ◽  
Unitary Group ◽  
Symmetric Functions ◽  
Point Of View ◽  
Unitary Groups ◽  
Character Theory ◽  
Characteristic Map ◽  
Finite Unitary Groups ◽  
Combinatorial Point ◽  
Groups Of Lie Type

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.

The Character Theory of Finite Groups of Lie Type

2020 ◽  
Author(s):  
Meinolf Geck ◽  
Gunter Malle
Keyword(s):  
Finite Groups ◽  
Character Theory ◽  
Groups Of Lie Type

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 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 A sharp upper bound for the size of Lusztig series

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Bessenrodt ◽  
Alexandre Zalesski
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
Simple Formula ◽  
Character Theory ◽  
Classical Groups ◽  
Irreducible Characters ◽  
Maximal Size ◽  
Groups Of Lie Type

AbstractThe paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group 𝐺 of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group 𝐺 in terms of its Lie rank and defining characteristic. When 𝐺 is specified as G(q) and 𝑞 is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of 𝐺.

scholarly journals On the characteristic map of finite unitary groups

2007 ◽  
Vol 210 (2) ◽  
pp. 707-732 ◽  
Author(s):  
Nathaniel Thiem ◽  
C. Ryan Vinroot
Keyword(s):  
Unitary Groups ◽  
Characteristic Map ◽  
Finite Unitary Groups

Inequivalent Transitive Factorizations into Transpositions

2001 ◽  
Vol 53 (4) ◽  
pp. 758-779 ◽  
Author(s):  
I. P. Goulden ◽  
D. M. Jackson ◽  
F. G. Latour
Keyword(s):  
Symmetric Functions ◽  
Point Of View ◽  
Algebraic Setting ◽  
Two Factors ◽  
Combinatorial Point ◽  
Ramified Coverings

AbstractThe question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions.It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factorizations, where two factorizations are equivalent if they differ only by the interchange of adjacent factors that commute. We obtain an explicit and elegant result for the number of such factorizations of permutations with precisely two factors. The approach used is a combinatorial one that rests on two constructions.We believe that this approach, and the combinatorial primitives that have been developed for the “cut and join” analysis, will also assist with the general case.

Linear Algebraic Groups and Finite Groups of Lie Type

2009 ◽  
Author(s):  
Gunter Malle ◽  
Donna Testerman
Keyword(s):  
Finite Groups ◽  
Algebraic Groups ◽  
Linear Algebraic Groups ◽  
Groups Of Lie Type
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