I know that a minority of plants are fully dioecious, with adult plants producing either only male or only female flowers. When is this sexual differentiation determined? Is the seed of such a plant already male or female? Is it a simple genetic difference?
The short answer is that in dioecious plants the sex of an individual plant is determined by its chromosomes.
Several familiar crop plants are dioecious: asparagus, kiwi, hop, papaya, spinach and yam. The evidence indicates that dioecy has evolved independently in different plant families. I refer you to
Matsunuga & Kawano (2001) Sex determination by sex chromosomes in dioecius plants. Plant Biol 3: 481-488.
I have summarised below some of the introductory information in this review.
First some background for context: In mammals there is an XY system in which maleness is determined by an active Y chromosome i.e. the Y chromosome carries male sex determination gene(s). Males are said to be heterogametic. In Drosophila there is also an XY system, with heterogametic males, but in this case what determines sex is the ratio of X chromosomes to autosomes. This ratio exerts its effects through the Sxl gene. In this system the Y chromosome does not carry male-determining genes.
In flowering plants, in dioecious species, the usual situation is that males are heterogametic. However, there is variability in the underlying mechanism of sex determination. So for example in hop (Humulus lupulus) there is a pair of sex chromosomes that can be distinguished by their appearance. However there is no active male determination but rather the X to autosomal ratio determines sex (as in Drosophila). Conversely, in asparagus (Asparagus officinalis), sex is determined by a pair of chromosomes designated pair 5. The two chromosomes are morphologically indistinguishable but genetic evidence indicates that there is active sex determination in this species - one member of pair 5 acts like the mammalian Y chromosome.