We classify all strongly real conjugacy classes of the finite unitary group U(n, F q ) when q is odd. In particular, we show that g ∈ U(n, F q ) is strongly real if and only if g is an element of some embedded orthogonal group O ± (n, F q ). Equivalently, g is strongly real in U(n, F q ) if and only if g is real and every elementary divisor of g of the form (t ± 1) 2m has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group Sp(2n, F q ), q odd, and a generating function for the number of strongly real classes in U(n, F q ), q odd, and we also give partial results on strongly real classes in U(n, F q ) when q is even.2010 Mathematics Subject Classification: 20G40, 20E45

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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.

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