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Yom Tov clothing:
One is to wear clean and fresh clothing for Rosh Hashanah. Some Poskim however write that one should not wear the same level of elegant clothing as he wears on Yom Tov, such as silk clothing. Rather he is to wear clean white colored clothing. [Practically the custom today is to wear ones regular Yom Tov clothing.]
Why we wear elegant clothing on a day of judgment:
The reason we wear clean and elegant clothing on Rosh Hashanah is to show that we trust in Hashem that he will perform a miracle on our behalf. In the words of the Midrash: Rebbi Simon stated: It says in the verse “Who is like your nation?” Rebbe Chanina and Rebbe Yehoshua say: What nation is like the Jewish nation which knows the way G-d acts. The worldly custom is that a man which has a court case [regarding a capital offence] wears black clothing, grows out his beard, and does not cut his nails being that he does not know what the verdict will be. The Jewish people however are not like this. They wear clean white clothing, make themselves look representable, cut their nails, eat, drink, and are merry. This is because they know Hashem will perform a miracle for them.
 Tur 581 from Midrash; Michaber 581/4; Admur 529/2 regarding that one is obligated in Kavod on Yom Tov just like Shabbos; 529/5 that this includes Rosh Hashanah; Admur 262/3 that one is obligated on Shabbos to wear elegant clothing; 529/7 that one is to wear more elegant clothing on Yom Tov than on Shabbos.
 Rashal; Bach 581; Taz 581/5; M”A 597; Elya Raba 581/36; Kaf Hachaim 581/79; M”E 581/55
 As one is to have the fear of judgment upon him. [ibid]
 Mateh Moshe; Tur 581 from Midrash; Bach brought in Kaf Hachaim ibid
The reason: This is to resemble the dead and arouse on in repentance. [Mateh Moshe] Others write it is to emulate the verse of being white without sin. [Bach]
 Shaareiy Teshuvah 581/6; Luach Kolel Chabad; See 529/7 that one is to wear more elegant clothing on Yom Tov than on Shabbos and in 529/5 Admur includes Rosh Hashanah within this matter.
 Mateh Moshe brought in Kaf Hachaim ibid; Mateh Efraim 581/55; 584/3 that in some communities the entire community wears a Kittel while in his community only the Chazan wears a Kittel. See Otzer Minhagei Chabad 83; In Admur 610/9 the custom of wearing a Kittel is only brought regarding Yom Kippur.
 Sefer Haminhagim p. 118 [English]; Otzer Minhagei Chabad 8
The reason: The reason we do not wear a Kittel on Rosh Hashanah is because Admur in 610/9 writes the reason for wearing it is to emulate angels, and this matter is not relevant on Rosh Hashanah. [Hamelech Bemisibo 1/115; Otzer ibid]
Does the Chazan wear a Kittel? No one in the congregation wears a Kittel; neither the congregation, Chazan or Baal Tokeia. [Otzer ibid] It was once witnessed that the Chazan in the times of the Rebbe Rashab wore a Kittel on Rosh Hashanah and the Rebbe Rashab did not mention anything. The Rebbe however explains that it is not the custom of the Rabbeim to interfere with the matters of the Shul. [Hamelech Bemisibo 1/115]
Clothing of Tzaddikim: The Rebbe would wear on Rosh Hashanah a certain garment that belonged to the Tzemach Tzedek. [See Otzer Minhagei Chabad 86]
 Tur 581
 Brought in Tur ibid