I am not going to give an 'authoritative' answer but I will tell you my opinion.
The commandment of tzit-tzit , the instruction to attach fringes, only applies to a special kind of garment... Tzit-Tzits must only be attached to a garment which has four corners.
The High Priest has special garments which may not include a four-corner garment.
Here is a list of the requirements for tzitzit:
http://www.chabad.org/library/article_cdo/aid/536771/jewish/The-Fringes-and-Garment.htmThe Garment
* Only a garment which covers most of the body requires tzitzit. A scarf, for example, is therefore exempt.
* Any garment which has four or more corners is obligated in tzitzit. If the garment has more than four corners, tzitzit fringes are only attached to the four corners furthest from each other.
* Only a garment which has two corners in the front of its wearer and two in the back must have tzitzit.
* A rounded edge is not considered as a corner.
* A garment only requires tzitzit if and when it is actually worn.
* Only a four-cornered garment which is of a woven material is obligated in tzitzit. A four-cornered plastic or leather garment does not require tzitzit.
* The noted 13th century halachic authority Rabbi Meir of Rothenberg contended that one does not satisfy the biblical obligation of tzitzit unless the garment and fringes are of the same material. Since all manufactured tzitzit fringes are made of wool, according to Rabbi Meir, one would not fulfill any mitzvah when wearing a cotton or silk tallit or tzitzit garment. Though Rabbi Meir's is a minority ruling and not binding, ideally one should strive to recite the blessing on a woolen tallit or tzitzit.
* The tallit or tzitzit garment must be at least 19x19 inches. A child's tzitzit can be smaller.
Also the very reason for the commandment of tzitzits involves controlling base human feelings by commanding:
http://www.chabad.org/library/article_cdo/aid/758547/jewish/Do-I-need-to-let-my-tzitzit-hang-out.htmWe read in the Torah, "This shall be fringes for you, and when you see it, you will remember all the commandments of the Lord to perform them. And you will not turn after your heart and after your eyes, after which you tend to stray …"1
Since the only time the Kohen Gadol wore the special garments was during the service in the Holy of Holies there should be no issue of turning after your heart and your eyes, so possibly there is no real need for the Kohen Gadols garments to have tzitzits..
This is my opinion...