Tafsir of Ad-Dhariyat 51:13

That is, they are burned. The root of al-fitnah (trial/testing) is the melting of precious metals to reveal their impurities; then, it was used to signify burning, tormenting, and the like.

"The Day" (yawm) is in the accusative case as an adverbial of time for an omitted element, which is indicated by the discourse occurring as a response to a question, and it is genitively linked to the nominal sentence that follows it. That is: "The Day of Judgment shall occur on the day they are, etc., exposed to the Fire."

Al-Zajjaj said: It is an adverbial for an omitted element that serves as the predicate for an implied subject; that is, "It [the event] is occurring or existing on the day they are, etc." It is also permissible that "the Day" itself is the predicate of an omitted subject, and the fatha is a vowel of indeclinability due to its addition to a non-fixed entity—the nominal sentence—for sentences are, in their origin, similar to such [nouns], though there is a detailed dispute regarding this between the Basrans and Kufans in Sharh al-Tashil. That is, "It is the day they are..."

As for the pronoun, it was said: It refers to the time of occurrence, thus this discourse stands in the place of an answer, similar to [the Quranic example]: "They will say, 'To Allah'" in response to "Who is the Lord of the heavens and the earth?" Because the implied question is: "At what time will it occur?" and its original answer is "On such and such a day." When you say, "The time of its occurrence is on such and such a day," it stands in its place.

It is also permissible that the pronoun refers to "the Day," and the discourse is an answer in terms of meaning; thus, the estimation is: The Day of Recompense is the day of the tormenting of the disbelievers. Its being in the nominative position as a predicate for an omitted subject is supported by the recitation of Ibn Abi 'Ablah and al-Za'farani, " yawmu hum" (the Day—in nominative case).

Some grammarians claimed that "the Day" is a substitute (badal) for "the Day of Judgment," and its fatha on the reading of the majority is due to indeclinability. "The Day" and what is within its scope from the sentence is the speech of the questioners, which they said in mockery. It was narrated according to the meaning; had it been narrated according to the literal wording, it would have been said: "The day we are exposed to the Fire," which is extremely far-fetched, as is not hidden. And His saying, the Exalted...