Degrees, dates and the drift
Every sign is cut into three parts of ten degrees. This is the first in Virgo, from zero to ten degrees, and it opens the sign.
The Sun spends about ten days there, generally from around 23 August to around 2 September. Neither end is fixed. Because the tropical year overruns three hundred and sixty five days by a fraction, ingress times move through the calendar and can vary by up to a day and a half between years. That matters most at the opening boundary, which is also the boundary with Leo, so a birthday on 22 or 23 August is worth checking rather than assuming.
The Sun by one count, Mercury by the other
The Chaldean order assigns rulers in a fixed loop, Mars, Sun, Venus, Mercury, Moon, Saturn, Jupiter, starting at zero degrees of Aries and running straight through the thirty six decans. Counting forward, the first decan of Virgo comes out on the Sun.
The triplicity method distributes each sign's decans among the signs of the same element in order. The earth signs run Taurus, Virgo, Capricorn, so the first earth decan of Virgo goes to Virgo itself, and its ruler is Mercury.
The two answers here are simply different rather than opposed, but they do point at different things, and most writing on decans quotes one without naming the scheme behind it. If you have seen this stretch called solar somewhere and Mercurial somewhere else, that is why.
What the tradition takes from each
Virgo on Virgo is the triplicity reading, and it is taken to mean the sign shows without the tempering the later decans supply. No Capricorn structure, no Taurus patience, just the sign's own habits: sorting, correcting, close attention to how something is actually done rather than how it is described. The same disposition read unkindly becomes fussing over detail that nobody else has noticed and no one has asked about.
The Chaldean assignment of the Sun is the more interesting claim, because the Sun in Virgo has no traditional dignity, and older texts on this face are correspondingly restrained. What the Sun is read as adding is visibility and a stake in the outcome: work done in the open rather than quietly, and a degree of personal identification with the standard being applied. Where Mercury describes the method, the Sun describes the person insisting on it.
Other placements, and keeping the technique in proportion
Decans are not a solar technique. An Ascendant at four degrees of Virgo sits in this decan, so does a Moon at nine, and older practice paid close attention to the Ascendant's decan in particular. The rulers above apply to whatever point occupies the degrees.
On what decans are worth: they came into astrology from Egyptian star lists, groups of stars used to mark the hours of the night, and were fitted to the ten degree divisions of the zodiac later, at which point the planetary rulers were attached. That is a real and old lineage. It is also a refinement. The sign, the houses and the aspects do the structural work in a reading, and a decan adjusts the shading rather than setting the picture. Anyone presenting a decan as the key to a chart is claiming more for it than the tradition ever did.
The check above computes your true solar degree from your birth date and names both rulers, which is the only reliable way to settle a birth near either end of these ten degrees.
Questions people ask about this
What dates are the first decan of Virgo?
Roughly 23 August to 2 September, with up to a day and a half of movement between years. The earlier edge doubles as the Leo boundary, so births around 22 and 23 August should be checked by degree.
Is this decan ruled by the Sun or by Mercury?
The Chaldean order gives it to the Sun; the triplicity scheme keeps it with Virgo and so with Mercury. Both are in use, and an article quoting only one is usually not saying which method it followed.
Does the Sun ruling this decan mean anything special?
The tradition reads it as adding visibility and personal investment to a stretch of the zodiac otherwise concerned with method. It is not treated as a dignity, since the Sun holds no rulership or exaltation in Virgo.
What is a decan?
A ten degree slice of a sign, three to each sign and thirty six in all. The divisions come from Egyptian star lists used to tell the hours at night, and the planetary rulers were added by the later tradition.
Can I use decans for my Ascendant?
Yes. The Ascendant, Moon and any planet all fall in a decan, and traditional practice read the Ascendant's decan closely. The Sun is simply the placement most readers already know.
How much does the decan change a Virgo reading?
Less than the sign, the houses or the aspects do. Decan work is a genuine part of the tradition but a second-order refinement, and everything above is interpretation rather than something measured.