The given cross is between a heterozygous tall plant (Tt) and a homozygous dwarf plant (tt). The genetic cross is represented as:
\[
\text{Tt} \times \text{tt}
\]
The possible offspring from this cross can be determined by a Punnett square:
| | T | t |
|-------|----|----|
| t | Tt | tt |
| t | Tt | tt |
From the Punnett square:
- 50\% of the progeny will be Tt (tall), and
- 50\% of the progeny will be tt (dwarf).
Thus, the percentage of progeny having the dwarf character (tt) is 50%.